Resolution of a high-order parabolic equation in conical time-dependent domains of R3R3

Abstract

New results on the existence, uniqueness and maximal regularity of a solution are given for a two-space dimensional high-order parabolic equation set in conical time-dependent domains. The study is performed in the framework of anisotropic weighted Sobolev spaces. Our method is based on the technique of decomposition of domains.

Keywords

High-order parabolic equations; Conical domains; Anisotropic weighted Sobolev spaces

2010 Mathematics Subject Classification

35K05; 35K55

1. Introduction

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}

be an open set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R^3}
defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Q=\lbrace (t,x_1,x_2)\in R^3:(x_1,x_2)\in \Omega _t,0<t<T\rbrace

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T}

is a finite positive number and for a fixed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t}
in the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ]0,T[}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega _t}

is a bounded domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R^2}
defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Omega _t=\lbrace (x_1,x_2)\in R^2:0\leq \sqrt{x_1^2+x_2^2}<\varphi (t)\rbrace \mbox{.}

Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi }

is a continuous real-valued function defined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,T]}

, Lipschitz continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,T]}

and such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \varphi (t)>0

for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\in \quad ]0,T]} . We assume that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \varphi (0)=0\mbox{,}

(1.1)

In Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q} , consider the boundary value problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} \partial _tu+(-1)^m\sum_{j=1}^2\partial _{x_j}^{2m}u=f\in L_{\omega }^2(Q)\mbox{,}\\ \partial _{x_j}^ku\vert _{\partial Q\setminus \varsupsetneq \Gamma _T}=0\mbox{, }\quad k=0,\ldots ,m-1;j=1,2\mbox{,} \end{array}

(1.3)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m\in N^{{_\ast}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial Q}

is the boundary of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Gamma _T}
is the part of the boundary of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=T}

. Here, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(Q)}

is the space of square-integrable functions on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}
with the measure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega dtdx_1dx_2}

, where the weight Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega }

is a real-valued function defined on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [0,T]}

, differentiable on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ]0,T]} , such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \forall t\in [0,T]:\omega (t)>0\mbox{,}

(1.4)

The difficulty related to this kind of problems comes from the fact that the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}

considered here is nonstandard since it shrinks at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=0\quad (\varphi (0)=0)}

, which prevents the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}

to be transformed into a regular domain without the appearance of some degenerate terms in the parabolic equation, see for example Sadallah  [15].

In this work, we will prove that Problem (1.3) has a solution with optimal regularity, that is a solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u}

belonging to the anisotropic weighted Sobolev space

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle H_{0,\omega }^{1,2m}(Q):=\lbrace u\in H_{\omega }^{1,2m}(Q):\partial _{x_j}^ku\vert _{\partial Q\setminus \varsupsetneq \Gamma _T}=0\mbox{, }k=0\\\displaystyle \ldots ,m-1;j=1,2\rbrace \mbox{,}\end{array}

with

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where

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The space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{\omega }^{1,2m}(Q)}

is equipped with the natural norm, that is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert u\Vert _{H_{\omega }^{1,2m}(Q)}=(\Vert \partial _tu\Vert _{L_\omega ^2(Q)}^2+\\\displaystyle +\sum _{\vert \alpha \vert \leq 2m}\underset{L_{\omega }^2(Q)}{\overset{2}{\Vert \partial ^{\alpha }u\Vert }})^{1/2}\mbox{.}\end{array}

Remark 1.1.

The boundary conditions of Problem (1.3) are equivalent to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{\nu }^ku\vert _{\partial Q\setminus \varsupsetneq \Gamma _T}=0\mbox{,}\quad k=0,\ldots ,m-1\mbox{,}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial _{\nu }}

stands for the normal derivative. This equivalence can be proved, for instance, by induction. So Problem (1.3) is also equivalent to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} \partial _tu+(-1)^m\sum_{j=1}^2\partial _{x_j}^{2m}u=f\in L_{\omega }^2(Q)\mbox{,}\\ \partial _{\nu }^ku\vert _{\partial Q\setminus \varsupsetneq \Gamma _T}=0\mbox{,}\quad k=0,\ldots ,m-1\mbox{.} \end{array}

(1.6)

Observe that the number of the boundary conditions in (1.3) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2m} , but they are not independent, while in (1.6), there are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m}

independent boundary conditions.

Our main result is

Theorem 1.1.

Let us assume that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi } satisfies condition (1.1) and the weight function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega } verifies assumptions and . Then, the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2m} -th order parabolic operator

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): L=\partial _t+(-1)^m\sum_{j=1}^2\partial _{x_j}^{2m}

is an isomorphism from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{0,\omega }^{1,2m}(Q)} into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(Q)} if one of the following conditions is satisfied

(1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi } is an increasing function in a neighborhood of 0,

(2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi } verifies the condition (1.2).

The case Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m=1}

corresponding to a second-order parabolic equation is studied in  [16] and  [9] both in bi-dimensional and multidimensional cases. We can find in Sadallah [15] a study of such kind of problems in the case of one space variable. Further references on the analysis of higher-order parabolic problems in non-cylindrical domains are: Baderko [1] and [2], Cherepova  [4] and [5], Labbas and Sadallah  [10], Galaktionov [6], Mikhailov  [13] and [14] and Kheloufi  [8].

The organization of this paper is as follows. In Section 2, first we prove a uniqueness result for Problem (1.3), then we derive some technical lemmas which will allow us to prove an energy type estimate (in a sense to be defined later). In Section 3, we divide the proof of Theorem 1.1 into four steps:

Case of a truncated domain,
An energy type estimate in small in time case,
Passage to the limit,
Case of a large in time conical type domain.

2. Preliminaries

Proposition 2.1.

Under the assumptions and on the weight function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega } , Problem (1.3) is uniquely solvable.

Proof.

Let us consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u\in H_{0,\omega }^{1,2m}(Q)}

a solution of Problem (1.3) with a null right-hand side term. So,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _tu+(-1)^m\sum_{j=1}^2\partial _{x_j}^{2m}u=0\quad \mbox{in }Q\mbox{.}

In addition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u}

fulfils the boundary conditions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{x_j}^ku\vert _{\partial Q\setminus \varsupsetneq \Gamma _T}=0\mbox{,}\quad k=0,1,\ldots ,m-1;\quad j=1,2\mbox{.}

Using Green’s formula, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \int _Q(\partial _tu+(-1)^m\sum _{j=1}^2\partial _{x_j}^{2m}u)u\quad \omega (t)dt\quad dx_1dx_2=\\\displaystyle =\int _{\partial Q}[\frac{1}{2}\vert u\vert ^2\nu _t+\sum _{j=1}^2\sum _{k=0}^{m-1}(\partial _{x_j}^{2m-k-1}u.\partial _{x_j}^ku)(-\\\displaystyle -1)^{k+m}\nu _{x_j}]\omega (t)d\sigma +\int _Q(\vert \partial _{x_1}^mu\vert ^2+\\\displaystyle +\vert \partial _{x_2}^mu\vert ^2)dtdx_1dx_2-\int _Q\frac{1}{2}\vert u\vert ^2\omega ^{'}(t)dtdx_1dx_2\mbox{,}\end{array}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu _t} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu _{x_1}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu _{x_2}}

are the components of the unit outward normal vector at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial Q}

. Taking into account the boundary conditions, all the boundary integrals vanish except Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \int _{\partial Q}\vert u\vert ^2\omega (t)\nu _t\quad d\sigma } . We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \int _{\partial Q}\vert u\vert ^2\omega (t)\nu _td\sigma =\int _{\Gamma _T}\vert u\vert ^2\omega (T)dx_1dx_2\mbox{.}

Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \int _Q(\partial _tu+(-1)^m\sum _{j=1}^2\partial _{x_j}^{2m}u)u\quad \omega (t)dt\quad dx_1dx_2=\\\displaystyle =\int _{\Gamma _T}\frac{1}{2}\vert u\vert ^2\omega (T)dx_1dx_2-\int _Q\frac{1}{2}\vert u\vert ^2\omega ^{'}(t)dtdx_1dx_2+\\\displaystyle +\int _Q(\vert \partial _{x_1}^mu\vert ^2+\vert \partial _{x_2}^mu\vert ^2)dtdx_1dx_2\mbox{ .}\end{array}

Consequently

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \int _Q(\partial _tu+(-1)^m\sum_{j=1}^2\partial _{x_j}^{2m}u)u\quad \omega (t)dt\quad dx_1dx_2=0

yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \int _Q(\vert \partial _{x_1}^mu\vert ^2+\vert \partial _{x_2}^mu\vert ^2)dt\mbox{ }dx_1dx_2=0\mbox{,}

because

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \int _{\Gamma _T}\frac{1}{2}\vert u\vert ^2\omega (t)dx_1dx_2-\int _Q\frac{1}{2}\vert u\vert ^2\omega ^{{'}}(t)dt\mbox{ }dx_1dx_2\geq 0

thanks to the conditions and . This implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert \partial _{x_1}^mu\vert ^2+\vert \partial _{x_2}^mu\vert ^2=0}

and consequently Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial _{x_1}^{2m}u=\partial _{x_2}^{2m}u=0}

. Then, the hypothesis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial _tu+(-1)^m\sum _{j=1}^2\partial _{x_j}^{2m}u=0}

gives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial _tu=0}

. Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u}

is constant. The boundary conditions imply that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u=0}
in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}

. This proves the uniqueness of the solution of Problem (1.3). □

Remark 2.1.

In the sequel, we will be interested only in the question of the existence of the solution of Problem (1.3).

The following result is well known (see, for example, [12])

Lemma 2.1.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(0,1)} be the unit disk of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R^2} . Then, the operator

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle A:H^{2m}(B(0,1))\cap H_0^m(B(0,1))\longrightarrow L^2(B(0\\\displaystyle 1))\mbox{,}v\mapsto Av=(-1)^m\sum _{j=1}^2\partial _{x_j}^{2m}v\end{array}

is an isomorphism. Moreover, there exists a constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C>0} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert v\Vert _{H^2m(B(0,1))}\leq C\Vert Av\Vert _{L^2(B(0,1))}\quad \forall v\in H^{2m}(B(0\\\displaystyle 1))\cap H_0^m(B(0,1))\end{array}

In the above lemma, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H^{2m}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_0^m}
are the usual Sobolev spaces defined, for instance, in Lions–Magenes  [12]. In Section  3, we will need the following result.

Lemma 2.2.

For a fixed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\in \quad ]0,T[} , there exists a constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C>0} such that for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u\in H^{2m}(\Omega _t)} , we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert \partial _{x_j}^lu\Vert _{L^2(\Omega _t)}^2\leq C\varphi ^{2(2m-l)}(t)\Vert Au\Vert _{L^2(\Omega _t)}^2\quad l=0,1\\\displaystyle \ldots ,2m-1;j=1,2\end{array}

Proof.

It is a direct consequence of Lemma 2.1. Indeed, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\in \quad ]0,T[}

and define the following change of variables

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): B(0,1)\rightarrow \Omega _t\mbox{,}\quad (x_1,x_2)\longmapsto (\varphi (t)x_1,\varphi (t)x_2)=(x_1^{{'}},x_2^{{'}})\mbox{.}

Set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v(x_1,x_2)=u(x_1^{{'}},x_2^{{'}})} , then if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v\in H^{2m}(B(0,1))} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u}

belongs to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H^{2m}(\Omega _t)}

. For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle j=1,2} , we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert \partial _{x_j}^lv\Vert _{L^2(B(0,1))}^2=\int _{B(0,1)}(\partial _{x_j}^lv)^2(x_1\\\displaystyle x_2)dx_1dx_2=\int _{\Omega _t}(\partial _{x_j^'}^lu)^2(x_1^{'}\\\displaystyle x_2^{'})\varphi ^{2l}(t)\frac{1}{\varphi ^2(t)}dx_1^{'}dx_2^{'}=\varphi ^{2l-2}(t)\int _{\Omega _t}(\partial _{x_j^'}^lu)^2(x_1^{'}\\\displaystyle x_2^{'})dx_1^{'}dx_2^{'}=\varphi ^{2l-2}(t)\Vert \partial _{x_j^'}^lu\Vert _{L^2(\Omega _t)}^2\end{array}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle l\in \lbrace 0,1,\ldots ,2m-1\rbrace } . On the other hand, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert Av\Vert _{L^2(B(0,1))}^2=\int _{B(0,1)}[(-1)^m\sum _{j=1}^2\partial _{x_j}^{2m}v(x_1\\\displaystyle x_2)]^2dx_1dx_2=\int _{\Omega _t}(\sum _{j=1}^2\varphi ^{2m}(t)\partial _{x_j^'}^{2m}u)^2(x_1^{'}\\\displaystyle x_2^{'})\frac{1}{\varphi ^2(t)}dx_1^{'}dx_2^{'}=\varphi ^{4m-2}(t)\int _{\Omega _t}(\sum _{j=1}^2\partial _{x_j^'}^{2m}u)^2(x_1^{'}\\\displaystyle x_2^{'})dx_1^{'}dx_2^{'}=\varphi ^{4m-2}(t)\Vert Au\Vert _{L^2(\Omega _t)}^2\mbox{.}\end{array}

Using the inequality

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{ccc} \Vert \partial _{x_j}^lv\Vert _{L^2(B(0,1))}^2 & \leq & C\Vert Av\Vert _{L^2(B(0,1))}^2 \end{array}

of Lemma 2.1, we obtain the desired inequality

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{ccc} \Vert \partial _{x_j^{{'}}}^lu\Vert _{L^2(\Omega _t)}^2 & \leq & C\varphi ^{2(2m-l)}(t)\Vert Au\Vert _{L^2(\Omega _t)}^2\mbox{.} \end{array}

□

Remark 2.2.

In Lemma 2.2 we can replace Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert .\Vert _{L^2}}

by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Vert .\Vert _{L_{\omega }^2}}

.

3. Proof of Theorem 1.1

3.1. Case of a truncated domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_n}

In this subsection, we replace Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}

by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_n,n\in N^{{_\ast}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{n}<T}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Q_n=\lbrace (t,x_1,x_2)\in Q:\frac{1}{n}<t<T\rbrace \mbox{.}

Theorem 3.1.

For each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\in N^{{_\ast}}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{n}<T} , the problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} \partial _tu_n+(-1)^m\sum_{j=1}^2\partial _{x_j}^{2m}u_n=f_n\in L_{\omega }^2(Q_n)\mbox{ }\\ \partial _{x_j}^ku_n\vert _{\partial Q_n\setminus \varsupsetneq \Gamma _T}=0\quad k=0,1,\ldots ,m-1;\quad j=1,2\mbox{,} \end{array}

(3.1)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_n=f\vert _{Q_n}} admits a unique solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_n\in H_{\omega }^{1,2m}(Q_n)} .

Proof of Theorem 3.1.

The change of variables

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{ccc} (t,x_1,x_2) & \mapsto & (t,y_1,y_2)=(t,\frac{x_1}{\varphi (t)},\frac{x_2}{\varphi (t)}) \end{array}

transforms Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_n}

into the cylinder Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_n=]\frac{1}{n},T[\times B(0,1)}

, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(0,1)}

is the unit disk of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R^2}

. Putting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_n(t,x_1,x_2)=v_n(t,y_1,y_2)}

 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_n(t,x_1,x_2)=g_n(t,y_1,y_2)}

, then Problem (3.1) is transformed, in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_n}

into the following variable-coefficient parabolic problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} \partial _tv_n+\frac{(-1)^m}{\varphi ^{2m}(t)}\sum_{j=1}^2\partial _{y_j}^{2m}v_n+\frac{\varphi ^{{'}}(t)}{\varphi (t)}\sum_{j=1}^2y_j\partial _{y_j}v_n=g_n\\ \partial _{y_j}^kv_n\vert _{\partial P_n\setminus \varsupsetneq \Sigma _T}=0\mbox{,}\quad k=0,1,\ldots ,m-1;\mbox{ }j=1,2 \end{array}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Sigma _T}

is the part of the boundary of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_n}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=T}

. The above change of variables conserves the spaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{\omega }^{1,2m}}
because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{(-1)^m}{\varphi ^{2m}(t)}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{\varphi ^{{'}}(t)}{\varphi (t)}}
are bounded functions when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\in \quad ]\frac{1}{n},T[}

. In other words

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): f_n\in L_{\omega }^2(Q_n)\Longleftrightarrow g_n\in L_{\omega }^2(P_n)\mbox{,}\quad u_n\in H_{\omega }^{1,2m}(Q_n)\Longleftrightarrow v_n\in H_{\omega }^{1,2m}(P_n)\mbox{.}

Proposition 3.1.

For each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\in N^{{_\ast}}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{n}<T} , the following operator is compact

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\varphi ^{{'}}(t)}{\varphi (t)}\sum_{j=1}^2y_j\partial _{y_j}:H_{0,\omega }^{1,2m}(P_n)\longrightarrow L_{\omega }^2(P_n)\mbox{.}

Proof.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_n}

has the “horn property” of Besov (see  [3]). So, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle j=1,2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{y_j}:H_{0,\omega }^{1,2m}(P_n)\longrightarrow H_{\omega }^{1-\frac{1}{2m},2m-1}(P_n)\mbox{,}\quad v\longmapsto \partial _{y_j}v\mbox{,}

is continuous. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_n}

is bounded, the canonical injection is compact from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{\omega }^{1-\frac{1}{2m},2m-1}(P_n)}
into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(P_n)}
(see for instance [3]), where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle H_{\omega }^{1-\frac{1}{2m},2m-1}(P_n)=L^2(\frac{1}{n},T;H^{2m-1}(B(0,1)))\cap H^{1-\frac{1}{2m}}(\frac{1}{n}\\\displaystyle T;L^2(B(0,1)))\mbox{.}\end{array}

For the complete definitions of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H^{r,s}}

Hilbertian Sobolev spaces, see for instance  [12]. Consider the composition

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{y_j}:H_{0,\omega }^{1,2m}(P_n)\rightarrow H_{\omega }^{1-\frac{1}{2m},2m-1}(P_n)\rightarrow L_{\omega }^2(P_n)\mbox{,}\quad v\mapsto \partial _{y_j}v\mapsto \partial _{y_j}v\mbox{,}

then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial _{y_j}}

is a compact operator from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{0,\omega }^{1,2m}(P_n)}
into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(P_n)}

. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{\varphi ^{{'}}(t)}{\varphi (t)}}

is a bounded function for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{n}<t<T}

, the operators Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{\varphi ^{{'}}(t)y_j}{\varphi (t)}\partial _{y_j}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle j=1,2}

are also compact from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{0,\omega }^{1,2m}(P_n)}
into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(P_n)}

. Consequently,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\varphi ^{{'}}(t)}{\varphi (t)}\sum_{j=1}^Ny_j\partial _{y_j}

is compact from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{0,\omega }^{1,2m}(P_n)}

into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(P_n)}

. □

So, thanks to Proposition 3.1, to complete the proof of Theorem 3.1, it is sufficient to show that the operator

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _t+\frac{(-1)^m}{\varphi ^{2m}(t)}\sum_{j=1}^N\partial _{y_j}^{2m}

is an isomorphism from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{0,\omega }^{1,2m}(P_n)}

into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(P_n)}

.

Lemma 3.1.

For each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\in N^{{_\ast}}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{n}<T} , the operator

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _t+\frac{(-1)^m}{\varphi ^{2m}(t)}\sum_{j=1}^2\partial _{y_j}^{2m}

is an isomorphism from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{0,\omega }^{1,2m}(P_n)} into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(P_n)} .

Proof.

Thanks to Remark 1.1, the problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} \partial _tv_n+\frac{(-1)^m}{\varphi ^{2m}(t)}\sum_{j=1}^2\partial _{y_j}^{2m}v_n=g_n\\ \partial _{y_j}^kv_n\vert _{\partial P_n\setminus \varsupsetneq \Sigma _T}=0\mbox{,}\quad k=0,1,\ldots ,m-1;\mbox{ }j=1,2\mbox{,} \end{array}

is equivalent to the following problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} \partial _tv_n+\frac{(-1)^m}{\varphi ^{2m}(t)}\sum_{j=1}^2\partial _{y_j}^{2m}v_n=g_n\\ \partial _{\nu }^kv_n\vert _{\partial P_n\setminus \varsupsetneq \Sigma _T}=0\mbox{,}\quad k=0,\ldots ,m-1\mbox{.} \end{array}

Since the coefficient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{\varphi ^{2m}(t)}}

is bounded in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \overline{P_n}}

, the optimal regularity is given by Ladyzhenskaya, Solonnikov and Ural’tseva [11]. □

We shall need the following result in order to justify the calculus of this section.

Lemma 3.2.

For each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\in N^{{_\ast}}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{n}<T} , the space

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace v\in H^{2m}(P_n):\partial _{x_j}^kv\vert _{\partial _pP_n}=0\mbox{ }k=0,1,\ldots ,m-1;\quad j=1,2\rbrace

is dense in the space

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace v\in H^{1,2m}(P_n):\partial _{x_j}^kv\vert _{\partial _pP_n}=0\mbox{ }k=0,1,\ldots ,m-1;\quad j=1,2\rbrace

Here, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial _pP_n} is the parabolic boundary of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_n} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H^{2m}} stands for the usual Sobolev space defined, for instance, in Lions–Magenes [12].

The proof of the above lemma may be found in [12].

Remark 3.1.

In Lemma 3.2, we can replace Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_n}

by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_n}
with the help of the change of variables defined above.

3.2. Case of a “small” conical domain

Now, we return to the conical domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}

and we suppose that the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi }
satisfies conditions  and .

For each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\in N^{{_\ast}}}

such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{1}{n}<T}

, we denote Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_n=f\vert _{Q_n}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_n\in H_{\omega }^{1,2m}(Q_n)}
the solution of Problem (1.3) in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_n}

. Such a solution exists by Theorem 3.1.

Proposition 3.2.

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} small enough, there exists a constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K_1} independent of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Vert u_n\Vert _{H_{\omega }^{1,2m}(Q_n)}\leq K_1\Vert f_n\Vert _{L_{\omega }^2(Q_n)}\leq K_1\Vert f\Vert _{L_{\omega }^2(Q)}

Remark 3.2.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon >0}

be a real number which we will choose small enough. The hypothesis (1.2) implies the existence of a real number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T>0}
small enough such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \forall t\in (0,T)\mbox{,}\quad \vert \varphi ^{{'}}(t)\varphi ^m(t)\vert \leq \epsilon \mbox{.}

(3.2)

In order to prove Proposition 3.2, we need the following result which is a consequence of Lemma 2.2 and Grisvard–Looss [7, Theorem 2.2].

Lemma 3.3.

There exists a constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C>0} independent of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \sum_{\vert \alpha \vert =2m}\underset{L_{\omega }^2(Q_n)}{\overset{2}{\Vert \partial ^{\alpha }u_n\Vert }}\leq C\Vert Au_n\Vert _{L_{\omega }^2(Q_n)}^2

Proof of Proposition 3.2.

Let us denote the inner product in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(Q_n)}

by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left\langle .,.\right\rangle }

, then we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert f_n\Vert _{L_\omega ^2(Q_n)}^2=\left\langle \partial _tu_n+Au_n\right.\\\left.\displaystyle \partial _tu_n+Au_n\right\rangle =\Vert \partial _tu_n\Vert _{L_\omega ^2(Q_n)}^2+\Vert Au_n\Vert _{L_\omega ^2(Q_n)}^2+2\left\langle \partial _tu_n\right.\\\left.\displaystyle Au_n\right\rangle \mbox{.}\end{array}

Estimation ofFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2\left\langle \partial _tu_n,Au_n\right\rangle }

We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{c}\partial _tu_n.Au_n=\sum _{j=1}^2[\sum _{k=0}^{m-1}\partial _{x_j}(\partial _{x_j}^k\partial _tu_n.\partial _{x_j}^{2m-k-1}u_n)(-\\\displaystyle -1)^{k+m}+\frac{1}{2}\partial _t(\partial _{x_j}^mu_n)^2]\mbox{.} \end{array}

Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle 2\left\langle \partial _tu_n\right.\\\left.\displaystyle Au_n\right\rangle =2\int _{Q_n}\partial _tu_n.Au_n.\omega (t)dtdx_1dx_2=\\\displaystyle =2\int _{Q_n}\sum _{j=1}^2\sum _{k=0}^{m-1}\partial _{x_j}(\partial _{x_j}^k\partial _tu_n.\partial _{x_j}^{2m-k-1}u_n)(-\\\displaystyle -1)^{k+m}.\omega (t)dtdx_1dx_2+\\\displaystyle +\int _{Q_n}\partial _t\sum _{j=1}^2(\partial _{x_j}^mu_n)^2.\omega (t)dtdx_1dx_2=\\\displaystyle =2\int _{\partial Q_n}\sum _{j=1}^2\sum _{k=0}^{m-1}(\partial _{x_j}^k\partial _tu_n.\partial _{x_j}^{2m-k-1}u_n)(-\\\displaystyle -1)^{k+m}\nu _{x_j}.\omega (t)d\sigma +\int _{\partial Q_n}\sum _{j=1}^2(\partial _{x_j}^mu_n)^2\nu _t.\omega (t)d\sigma -\\\displaystyle -\int _{Q_n}\sum _{j=1}^2(\partial _{x_j}^mu_n)^2.\omega ^{'}(t)dtdx_1dx_2\end{array}

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu _t,\nu _{x_1},\nu _{x_2}}

are the components of the unit outward normal vector at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial Q_n}

. We shall rewrite the boundary integral making use of the boundary conditions. On the part of the boundary of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_n}

 where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=\frac{1}{n}}

, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial _{x_j}^ku_n=0,k=0,\ldots ,m-1;j=1,2}

and consequently the corresponding boundary integral vanishes. On the part of the boundary where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=T}

, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu _{x_j}=0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle j=1,2}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu _t=1}

. Accordingly, the corresponding boundary integral

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \int _{\Gamma _T}\sum_{j=1}^2(\partial _{x_j}^mu_n)^2(T,x_1,x_2).\omega (T)dx_1dx_2

is nonnegative. On the part Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Gamma _1}

of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial Q_n}
defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Gamma _1=\lbrace (t,x_1,x_2):\sqrt{x_1^2+x_2^2}=\varphi (t)\rbrace \mbox{,}

we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \nu _t=\frac{-\varphi ^{{'}}(t)}{\sqrt{1+(\varphi ^{{'}})^2(t)}}\mbox{,}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{x_j}^ku_n(t,\varphi (t)cos\theta ,\varphi (t)sin\theta )=0\mbox{,}\quad k=0,\ldots ,m-1\mbox{;}j=1,2\mbox{.}

Let us denote

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle I=2\int _{\Gamma _1}\sum _{j=1}^2\sum _{k=0}^{m-1}(\partial _{x_j}^k\partial _tu_n.\partial _{x_j}^{2m-k-1}u_n)(-\\\displaystyle -1)^{k+m}\nu _{x_j}.\omega (t)d\sigma \mbox{.}\end{array}

We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle I=2\int _{\Gamma _1}\sum _{j=1}^2(\partial _tu_n.\partial _{x_j}^{2m-1}u_n)(-1)^m\nu _{x_j}.\omega (t)d\sigma +\\\displaystyle +2\int _{\Gamma _1}\sum _{j=1}^2\sum _{k=1}^{m-2}(\partial _{x_j}^k\partial _tu_n.\partial _{x_j}^{2m-k-1}u_n)(-\\\displaystyle -1)^{k+m}\nu _{x_j}.\omega (t)d\sigma -\\\displaystyle -2\int _{\Gamma _1}\sum _{j=1}^2(\partial _{x_j}^{m-1}\partial _tu_n.\partial _{x_j}^mu_n)\nu _{x_j}.\omega (t)d\sigma =I_0+\\\displaystyle +I_1+I_{m-1}\mbox{.}\end{array}

(a) Estimation ofFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_0=2\int _{\Gamma _1}\sum _{j=1}^2(\partial _tu_n.\partial _{x_j}^{2m-1}u_n)(-1)^m\nu _{x_j}.\omega (t)d\sigma }

We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u_n(t,\varphi (t)cos\theta ,\varphi (t)sin\theta )=0\mbox{.}

Differentiating with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t} , we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _tu_n=-\varphi ^{{'}}(t)(cos\theta .\partial _{x_1}u_n+sin\theta .\partial _{x_2}u_n)=0\mbox{.}

So, the boundary integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_0}

vanishes.

(b) Estimation ofFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \begin{array}{l}\displaystyle I_1=2\int _{\Gamma _1}\sum _{j=1}^2\sum _{k=1}^{m-2}(\partial _{x_j}^k\partial _tu_n.\partial _{x_j}^{2m-k-1}u_n)(-\\\displaystyle -1)^{k+m}\nu _{x_j}.\omega (t)d\sigma \end{array}}

We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{x_j}^ku_n(t,\varphi (t)cos\theta ,\varphi (t)sin\theta )=0\mbox{,}\quad k=1,\ldots ,m-2;j=1,2\mbox{.}

Differentiating with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t} , we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \partial _t\partial _{x_1}^ku_n=-\varphi ^{'}(t)[cos\theta .\partial _{x_1}^{k+1}u_n+\\\displaystyle +sin\theta .\partial _{x_2}\partial _{x_1}^ku_n]\mbox{,}k=1,\ldots ,m-2\end{array}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \partial _t\partial _{x_2}^ku_n=-\varphi ^{'}(t)[cos\theta .\partial _{x_1}\partial _{x_2}^ku_n+\\\displaystyle +sin\theta .\partial _{x_2}^{k+1}u_n]\mbox{,}k=1,\ldots ,m-2\mbox{.}\end{array}

The Dirichlet boundary conditions on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Gamma _1}

lead to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _t\partial _{x_1}^ku_n=-\varphi ^{{'}}(t)sin\theta .\partial _{x_2}\partial _{x_1}^ku_n\mbox{,}\quad k=1,\ldots ,m-2

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _t\partial _{x_2}^ku_n=-\varphi ^{{'}}(t)cos\theta .\partial _{x_1}\partial _{x_2}^ku_n\mbox{,}\quad k=1,\ldots ,m-2\mbox{.}

Now, differentiating the formula

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{x_j}^ku_n(t,\varphi (t)cos\theta ,\varphi (t)sin\theta )=0\mbox{, }\quad k=1,\ldots ,m-2;j=1,2

with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta } , we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): sin\theta .\partial _{x_1}^{k+1}u_n=cos\theta .\partial _{x_2}\partial _{x_1}^ku_n\mbox{,}\quad k=1,\ldots ,m-2

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): cos\theta .\partial _{x_2}^{k+1}u_n=sin\theta .\partial _{x_1}\partial _{x_2}^ku_n\mbox{,}\quad k=1,\ldots ,m-2\mbox{.}

The Dirichlet boundary conditions on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Gamma _1}

lead to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{x_1}\partial _{x_2}^ku_n=\partial _{x_2}\partial _{x_1}^ku_n=0\mbox{,}\quad k=1,\ldots ,m-2

and consequently

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _t\partial _{x_1}^ku_n=\partial _t\partial _{x_2}^ku_n=0\mbox{,}\quad k=1,\ldots ,m-2\mbox{.}

So, the boundary integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_1}

vanishes.

(c) Estimation ofFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle I_{m-1}=-2\int _{\Gamma _1}\sum _{j=1}^2(\partial _{x_j}^{m-1}\partial _tu_n.\partial _{x_j}^mu_n)\nu _{x_j}.\omega (t)d\sigma }

We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{x_j}^{m-1}u_n(t,\varphi (t)cos\theta ,\varphi (t)sin\theta )=0\mbox{,}\quad j=1,2\mbox{.}

Differentiating with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t} , we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \partial _t\partial _{x_1}^{m-1}u_n=-\varphi ^{'}(t)[(cos\theta .\partial _{x_1}^mu_n+\\\displaystyle +sin\theta .\partial _{x_2}\partial _{x_1}^{m-1}u_n)]\end{array}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \partial _t\partial _{x_2}^{m-1}u_n=-\varphi ^{'}(t)[cos\theta .\partial _{x_1}\partial _{x_2}^{m-1}u_n+\\\displaystyle +sin\theta .\partial _{x_2}^mu_n]\mbox{.}\end{array}

Differentiating with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \theta } , we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): cos\theta .\partial _{x_2}\partial _{x_1}^{m-1}u_n=sin\theta .\partial _{x_1}^mu_n

Taking into account these relationships we deduce

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle I_{m-1}=2\int _0^{2\pi }\int _{\frac{1}{n}}^T[cos^2\theta .\partial _{x_1}^mu_n+\\\displaystyle +sin\theta cos\theta .\partial _{x_1}\partial _{x_2}^{m-1}u_n]\partial _{x_1}^mu_n\varphi ^{'}(t)\varphi (t).\omega (t)dtd\theta +\\\displaystyle +2\int _0^{2\pi }\int _{\frac{1}{n}}^T[cos\theta sin\theta .\partial _{x_2}\partial _{x_1}^{m-1}u_n+\\\displaystyle +sin^2\theta .\partial _{x_2}^mu_n]\partial _{x_2}^mu_n\varphi ^{'}(t)\varphi (t).\omega (t)dtd\theta =\\\displaystyle =2\int _0^{2\pi }\int _{\frac{1}{n}}^T[(\partial _{x_1}^mu_n)^2+\\\displaystyle +(\partial _{x_2}^mu_n)^2]\varphi ^{'}(t)\varphi (t).\omega (t)dtd\theta \mbox{.}\end{array}

Finally

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle 2\left\langle \partial _tu_n\right.\\\left.\displaystyle Au_n\right\rangle =\int _0^{2\pi }\int _{\frac{1}{n}}^T(\sum _{j=1}^2(\partial _{x_j}^mu_n)^2)\varphi ^{'}(t)\varphi (t).\omega (t)dtd\theta +\\\displaystyle +\int _{\Gamma _T}(\sum _{j=1}^2(\partial _{x_j}^mu_n)^2)(T,x_1\\\displaystyle x_2).\omega (T)dx_1dx_2-\\\displaystyle -\int _{Q_n}(\sum _{j=1}^2(\partial _{x_j}^mu_n)^2).\omega ^{'}(t)dtdx_1dx_2\mbox{.}\end{array}

(3.3)

Remark 3.3.

Observe that the integrals

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \int _{\Gamma _T}(\sum_{j=1}^2(\partial _{x_j}^mu_n)^2)(T,x_1,x_2).\omega (T)dx_1dx_2

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\int _{Q_n}(\sum_{j=1}^2(\partial _{x_j}^mu_n)^2).\omega ^{{'}}(t)dtdx_1dx_2\mbox{,}

which appear in the last formula are nonnegative thanks to the assumptions and on the weight function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \omega } . This is a good sign for our estimate because we can deduce immediately

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert f_n\Vert _{L_\omega ^2(Q_n)}^2\geq \Vert \partial _tu_n\Vert _{L_\omega ^2(Q_n)}^2+\Vert Au_n\Vert _{L_\omega ^2(Q_n)}^2+\\\displaystyle +\int _0^{2\pi }\int _{\frac{1}{n}}^T(\sum _{j=1}^2(\partial _{x_j}^mu_n)^2)\varphi ^{'}(t)\varphi (t).\omega (t)dtd\theta \mbox{.}\end{array}

So, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi }

is an increasing function in the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (\frac{1}{n},T)}

, then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \int _0^{2\pi }\int _{\frac{1}{n}}^T(\sum_{j=1}^2(\partial _{x_j}^mu_n)^2)\varphi ^{{'}}(t)\varphi (t).\omega (t)dtd\theta \geq 0\mbox{.}

Consequently,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Vert f_n\Vert _{L_{\omega }^2(Q_n)}^2\geq \Vert \partial _tu_n\Vert _{L_{\omega }^2(Q_n)}^2+\Vert Au_n\Vert _{L_{\omega }^2(Q_n)}^2\mbox{.}

(3.4)

But, thanks to Lemma 2.2 and since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \varphi }

is bounded in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (0,T)}

, there exists a constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C^{{'}}>0}

such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert \partial _{x_j}^lu_n\Vert _{L_\omega ^2(Q_n)}^2\leq C^{'}\Vert Au_n\Vert _{L_\omega ^2(Q_n)}^2\mbox{,}\quad l=0,1,\ldots \\\displaystyle 2m-1;j=1,2\mbox{.}\end{array}

Taking into account Lemma 3.3 and estimate (3.4), this proves the desired estimate of Proposition 3.2.

So, it remains to establish the estimate of Proposition 3.2 under the hypothesis (1.2). For this purpose, we need the following lemma

Lemma 3.4.

One has

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle 2\left\langle \partial _tu_n\right.\\\left.\displaystyle Au_n\right\rangle =2\int _{Q_n}\frac{\varphi ^{'}}{\varphi }(\sum _{j=1}^2x_j\partial _{x_j}^mu_n)Au_n.\omega (t)dtdx_1dx_2+\\\displaystyle +\int _{\Gamma _T}\sum _{j=1}^2(\partial _{x_j}^mu_n)^2(T,x_1,x_2).\omega (T)dx_1dx_2\mbox{.}\end{array}

Proof.

This result can be obtained by following step by step the proof of [9, Lemma 3.4]. □

Now, we continue the proof of Proposition 3.2. We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \vert \int _{Q_n}\frac{\varphi ^{{'}}}{\varphi }(\sum_{j=1}^2x_j\partial _{x_j}^mu_n)Au_n.\omega (t)dtdx_1dx_2\vert \leq \Vert Au_n\Vert _{L_{\omega }^2(Q_n)}\sum_{j=1}^2\Vert \frac{\varphi ^{{'}}}{\varphi }x_j\partial _{x_j}^mu_n\Vert _{L_{\omega }^2(Q_n)}\mbox{,}

but Lemma 2.2 yields for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle j=1,2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert \frac{\varphi ^{'}}{\varphi }x_j\partial _{x_j}^mu_n\Vert _{L_\omega ^2(Q_n)}^2=\\\displaystyle =\int _{\frac{1}{n}}^T\varphi ^{'2}(t)\int _{\Omega _t}(\frac{x_j}{\varphi (t)})^2(\partial _{x_j}^mu_n)^2.\omega (t)dtdx_1dx_2\leq \int _{\frac{1}{n}}^T\varphi ^{'2}(t)\int _{\Omega _t}(\partial _{x_j}^mu_n)^2.\omega (t)dtdx_1dx_2\leq C^2\int _{\frac{1}{n}}^T(\varphi ^m(t)\varphi ^{'}(t))^2\int _{\Omega _t}(Au_n)^2.\omega (t)dtdx_1dx_2\leq C^2\epsilon ^2\Vert Au_n\Vert _{L_\omega ^2(Q_n)}^2\mbox{,}\end{array}

since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (\varphi ^m(t)\varphi ^{{'}}(t))\leq \epsilon }

thanks to the condition (3.2). Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \vert \int _{Q_n}\frac{\varphi ^{{'}}}{\varphi }(\sum_{j=1}^2x_j\partial _{x_j}^mu_n)Au_n.\omega (t)dtdx_1dx_2\vert \leq 2C\epsilon \Vert Au_n\Vert _{L_{\omega }^2(Q_n)}^2\mbox{.}

Therefore, Lemma 3.4 shows that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \vert 2\left\langle \partial _tu_n\right.\\\left.\displaystyle Au_n\right\rangle \vert \geq -2\vert \int _{Q_n}\frac{\varphi ^{'}}{\varphi }(\sum _{j=1}^2x_j\partial _{x_j}^mu_n)Au_n.\omega (t)dtdx_1dx_2\vert +\\\displaystyle +\int _{\Gamma _T}\sum _{j=1}^2(\partial _{x_j}^mu_n)^2(T,x_1\\\displaystyle x_2).\omega (T)dx_1dx_2.\geq -4C\epsilon \Vert Au_n\Vert _{L_\omega ^2(Q_n)}^2\mbox{.}\end{array}

Hence

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert f_n\Vert _{L_\omega ^2(Q_n)}^2=\Vert \partial _tu_n\Vert _{L_\omega ^2(Q_n)}^2+\Vert Au_n\Vert _{L_\omega ^2(Q_n)}^2+\\\displaystyle +2\left\langle \partial _tu_n\right.\\\left.\displaystyle Au_n\right\rangle \geq \Vert \partial _tu_n\Vert _{L_\omega ^2(Q_n)}^2+(1-4C\epsilon )\Vert Au_n\Vert _{L_\omega ^2(Q_n)}^2\mbox{.}\end{array}

Then, it is sufficient to choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon }

such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1-4C\epsilon >0}
to get a constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K_0>0}
independent of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n}
such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Vert f_n\Vert _{L_{\omega }^2(Q_n)}\geq K_0\Vert u_n\Vert _{H_{\omega }^{1,2m}(Q_n)}\mbox{,}

and since

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Vert f_n\Vert _{L_{\omega }^2(Q_n)}\leq \Vert f\Vert _{L_{\omega }^2(Q_n)}\mbox{,}

there exists a constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K_1>0} , independent of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n}

satisfying

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Vert u_n\Vert _{H_{\omega }^{1,2m}(Q_n)}\leq K_1\Vert f_n\Vert _{L_{\omega }^2(Q_n)}\leq K_1\Vert f\Vert _{L_{\omega }^2(Q)}\mbox{.}

This completes the proof of Proposition 3.2.

3.3. Passage to the limit

Choose a sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (Q_n)_{n\in N^{{_\ast}}}}

of the domains defined above (see Section  3.1), such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_n\subseteq Q}

. Then, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_n\rightarrow Q} , as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\rightarrow \infty } . Consider the solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_n\in H_{\omega }^{1,2m}(Q_n)}

of the Cauchy–Dirichlet problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} \partial _tu_n+(-1)^m\sum_{j=1}^2\partial _{x_j}^{2m}u_n=f_n\in L_{\omega }^2(Q_n)\mbox{,}\\ \partial _{x_j}^ku_n\vert _{\partial Q_n\setminus \varsupsetneq \Gamma _T}=0\mbox{,}\quad k=0,1,\ldots ,m-1;\mbox{ }j=1,2\mbox{,} \end{array}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_n=f\vert _{Q_n}} . Such a solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_n}

exists by Theorem 3.1. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tilde{u_n}}
be the 0-extension of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_n}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}

. In virtue of Proposition 3.2 for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T}

small enough, we know that there exists a constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C}
such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{l}\displaystyle \Vert \tilde{u_n}\Vert _{L_\omega ^2(Q)}+\Vert \tilde{\partial _tu_n}\Vert _{L_\omega ^2(Q)}+\\\displaystyle +\sum _{1\leq \vert \alpha \vert \leq 2m}\Vert \tilde{\partial ^{\alpha }u_n}\Vert _{\underset{\omega }{\overset{2}{L}}(Q)}\leq C\Vert f\Vert _{\underset{\omega }{\overset{2}{L}}(Q)}\mbox{.}\end{array}

This means that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tilde{u_n}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tilde{\partial _tu_n},\tilde{\partial ^{\alpha }u_n}}

for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1\leq \vert \alpha \vert \leq 2m}
are bounded functions in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(Q)}

. So, for a suitable increasing sequence of integers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_k} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k=1,2,\ldots } , there exist functions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u\mbox{, }v\mbox{ and }v_{\alpha }\quad 1\leq \vert \alpha \vert \leq 2m

in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(Q)}

such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{array}{ccccc} \tilde{u_{n_k}} & \rightharpoonup & u & \mbox{weakly in }L_{\omega }^2(Q)\mbox{,} & k\rightarrow \infty \\ \tilde{\partial _tu_{n_k}} & \rightharpoonup & v & \mbox{weakly in }L_{\omega }^2(Q)\mbox{,} & k\rightarrow \infty \\ \tilde{\partial ^{\alpha }u_{n_k}} & \rightharpoonup & v_{\alpha }\quad & \mbox{weakly in }L_{\omega }^2(Q)\mbox{,} & k\rightarrow \infty \mbox{,} \end{array}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 1\leq \vert \alpha \vert \leq 2m} . Clearly,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): v=\partial _tu\mbox{,}\quad v_{\alpha }=\partial ^{\alpha }u\mbox{,}\quad 1\leq \vert \alpha \vert \leq 2m

in the sense of distributions in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}

and so in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(Q)}

. So, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u\in H_{\omega }^{1,2m}(Q)}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _tu+(-1)^m\sum_{j=1}^2\partial _{x_j}^{2m}u=f\quad \mbox{in }Q\mbox{.}

On the other hand, the solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u}

satisfies the boundary conditions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{x_j}^ku_n\vert _{\partial Q_n\setminus \varsupsetneq \Gamma _T}=0\mbox{,}\quad k=0,1,\ldots ,m-1;\quad j=1,2\mbox{,}

since

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \forall n\in N^{{_\ast}}\mbox{,}\quad u\vert _{Q_n}=u_n\mbox{.}

This proves the existence of a solution to Problem (1.3). This ends the proof of Theorem 1.1 in the case of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T}

small enough.

3.4. The general case

Assume that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}

satisfies (1.1). In the case where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T}
is not small enough, we set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q=D_1\cup D_2\cup \Gamma _{T_1}}
where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): D_1=\lbrace (t,x_1,x_2)\in Q:0<t<T_1\rbrace

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T_1}

small enough. In the sequel, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f}
stands for an arbitrary fixed element of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_{\omega }^2(Q)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_i=f\vert _{D_i}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i=1} , 2. We know that (see Section 3.3) there exists a unique solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w_1\in H_{\omega }^{1,2m}(D_1)}

of the problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} \partial _tw_1+(-1)^m\sum_{j=1}^2\partial _{x_j}^{2m}w_1=f_1\in L_{\omega }^2(D_1)\mbox{,}\\ \partial _{x_j}^kw_1\vert _{\partial D_1\setminus \varsupsetneq \Gamma _{T_1}}=0\mbox{,}\quad k=0,\ldots ,m-1;\mbox{ }j=1,2\mbox{.} \end{array}

(3.5)

Hereafter, we denote the trace Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w_1\vert _{\Gamma _{T_1}}}

by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \psi }
which is in the Sobolev space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle H_{\omega }^m(\Gamma _{T_1})}
because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w_1\in H_{\omega }^{1,2m}(D_1)}
(see  [12]). Now, consider the following problem in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D_2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} \partial _tw_2+(-1)^m\sum_{j=1}^2\partial _{x_j}^{2m}w_2=f_2\quad \in L_{\omega }^2(D_2)\mbox{,}\\ w_2\vert _{\Gamma _{T_1}}=\psi \mbox{, }\\ \partial _{x_j}^kw_2\vert _{\partial D_2\setminus \varsupsetneq (\Gamma _{T_1}\cup \Gamma _T)}=0\mbox{, }\quad k=0,\ldots ,m-1;\quad j=1,2\mbox{.} \end{array}

(3.6)

We use the following result, which is a consequence of [12, Theorem 4.3, Vol. 2], to solve Problem (3.6).

Proposition 3.3.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R} be the cylinder Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ]0,T[\quad \times \quad B(0,1)} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B(0,1)} is the unit disk of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R^2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f\in L_{\omega }^2(R)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u_0\in H_{\omega }^m(\gamma _0)} . Then, the problem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \lbrace \begin{array}{c} \partial _tu+(-1)^m\sum_{j=1}^2\partial _{x_j}^{2m}u=f\quad \mbox{ }R\mbox{,}\\ u\vert _{\gamma _0}=u_0\\ \partial _{x_j}^ku\vert _{\gamma _1}=0\mbox{ }\quad k=0,\ldots ,m-1;\quad j=1,2\mbox{,} \end{array}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma _0=\lbrace 0\rbrace \times B(0,1)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma _1=]0,T[\quad \times \quad \partial B(0,1)} , admits a (unique) solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u\in H_{\omega }^{1,2m}(R)} if and only if the following compatibility conditions are fulfilled

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{x_j}^ku_0\vert _{\partial \gamma _0}=0\mbox{ }\quad k=0,\ldots ,m-1;\quad j=1,2\mbox{.}

Thanks to the transformation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): (t,x_1,x_2)\longmapsto (t,y_1,y_2)=(t,\varphi (t)x_1,\varphi (t)x_2)\mbox{,}

we deduce the following result:

Proposition 3.4.

Problem (3.6) admits a (unique) solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w_2\in H_{\omega }^{1,2m}(D_2)} if and only if the following compatibility conditions are fulfilled

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{x_j}^k\psi \vert _{\partial \Gamma _{T_1}}=0\mbox{ }\quad k=0,\ldots ,m-1;\quad j=1,2\mbox{.}

Remark 3.4.

We can observe that the boundary conditions of Problems (3.5) and (3.6) yield

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): w_1\vert _{\Gamma _{T_1}}=w_2\vert _{\Gamma _{T_1}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \partial _{x_j}^kw_i\vert _{\Gamma _{T_1}}\in H_{\omega }^{1-\frac{1}{2m}}(\Gamma _{T_1});\quad i,j=1,2} . Then the compatibility conditions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{x_j}^k\psi \vert _{\partial \Gamma _{T_1}}=0\mbox{, }\quad k=0,\ldots ,m-1;\quad j=1,2

are satisfied since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w_1\vert _{\Gamma _{T_1}}=\psi } .

Now, define the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u}

in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q}
by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): u:=\lbrace \begin{array}{c} w_1\quad \mbox{in }D_1\\ w_2\quad \mbox{in }D_2 \end{array}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w_2}
are the solutions of Problem (3.5) and Problem (3.6) respectively. Observe that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w_1\vert _{\Gamma _{T_1}}=w_2\vert _{\Gamma _{T_1}}}

, see Remark 3.4, so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \partial _{x_j}^kw_1\vert _{\Gamma _{T_1}}=\partial _{x_j}^kw_2\vert _{\Gamma _{T_1}}\mbox{,}\quad k=0,\ldots ,m-1;\quad j=1,2\mbox{.}

This implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u\in H_{\omega }^{1,2m}(Q)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle u}
is the (unique) solution of Problem (1.3) for an arbitrary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T}

. This ends the proof of Theorem 1.1.

Acknowledgment

We are thankful to the referee for the valuable remarks which led to an improvement of the original manuscript.

References

[1] E.A. Baderko; The solvability of boundary value problems for higher order parabolic equations in domains with curvilinear lateral boundaries; Differ. Uravn., 10 (12) (1976), pp. 1781–1792
[2] E.A. Baderko, On the solution of boundary value problems for linear parabolic equations of arbitrary order in noncylindrical domains by the method of boundary integral equations (Ph.D. thesis), Moscow, 1992.
[3] V. Besov; Continuation of functions from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle L_p^l} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle W_p^l}
Proc. Steklov Inst. Math., 89 (1967), pp. 5–17
[4] M.F. Cherepova; On the solvability of boundary value problems for a higher order parabolic equation with growing coefficients; Dokl. Math., 74 (3) (2006), pp. 819–820
[5] M.F. Cherepova; Regularity of solutions of boundary value problems for a second order parabolic equation in weighted Hölder spaces; Differ. Equ., 1 (49) (2013), pp. 79–87
[6] V.A. Galaktionov; On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach; Nonlinear Differential Equations Appl., 5 (16) (2009), pp. 597–655
[7] P. Grisvard, G. Looss; Problèmes aux limites unilatéraux dans des domaines non réguliers; J. Equ. Dériv. Part. (1976), pp. 1–26
[8] A. Kheloufi; On a fourth order parabolic equation in a nonregular domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R^3}
Mediterr. J. Math. (2015) http://dx.doi.org/10.1007/s00009-014-0429-7
[9] A. Kheloufi, B.K. Sadallah; Study of the heat equation in a symmetric conical type domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R^{N+1}}
Math. Methods Appl. Sci., 37 (2014), pp. 1807–1818
[10] R. Labbas, B.K. Sadallah; Smoothness of the solution of a fourth order parabolic equation in a polygonal domain; Int. J. Appl. Math., 1 (1999), pp. 75–90
[11] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’tseva; Linear and Quasilinear Equations of Parabolic Type; A.M.S, Providence, Rhode Island (1968)
[12] J.L. Lions, E. Magenes; Problèmes aux limites non homogènes et applications, 1, 2; Dunod, Paris (1968)
[13] V.P. Mikhailov; The Dirichlet problem for a parabolic equation; I, Mat. Sb., 61 (103) (1963), pp. 40–64
[14] V.P. Mikhailov; The Dirichlet problem for a parabolic equation; II, Mat. Sb., 62 (104) (1963), pp. 140–159
[15] B.K. Sadallah; Etude d’un problème Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 2m} -parabolique dans des domaines plan non rectangulaires; Boll. Unione Mat. Ital., 2B (5) (1983), pp. 51–112
[16] B.K. Sadallah; Study of a parabolic problem in a conical domain; Math. J. Okayama Univ., 56 (2014), pp. 157–169

Abstract

Keywords

2010 Mathematics Subject Classification

1. Introduction

Remark 1.1.

Theorem 1.1.

2. Preliminaries

Proposition 2.1.

Proof.

Remark 2.1.

Lemma 2.1.

Lemma 2.2.

Proof.

Remark 2.2.

3. Proof of Theorem 1.1

3.1. Case of a truncated domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle Q_n}

Theorem 3.1.

Proof of Theorem 3.1.

Proposition 3.1.

Proof.

Lemma 3.1.

Proof.

Lemma 3.2.

Remark 3.1.

3.2. Case of a “small” conical domain

Proposition 3.2.

Remark 3.2.

Lemma 3.3.

Proof of Proposition 3.2.

Remark 3.3.

Lemma 3.4.

Proof.

3.3. Passage to the limit

3.4. The general case

Proposition 3.3.

Proposition 3.4.

Remark 3.4.

Acknowledgment

References

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