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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 ${\textstyle Q}$ be an open set of ${\textstyle R^{3}}$ defined by

${\displaystyle Q=\lbrace (t,x_{1},x_{2})\in R^{3}:(x_{1},x_{2})\in \Omega _{t},0<t<T\rbrace }$

where ${\textstyle T}$ is a finite positive number and for a fixed ${\textstyle t}$ in the interval ${\textstyle ]0,T[}$, ${\textstyle \Omega _{t}}$ is a bounded domain of ${\textstyle R^{2}}$ defined by

${\displaystyle \Omega _{t}=\lbrace (x_{1},x_{2})\in R^{2}:0\leq {\sqrt {x_{1}^{2}+x_{2}^{2}}}<\varphi (t)\rbrace {\mbox{.}}}$

Here ${\textstyle \varphi }$ is a continuous real-valued function defined on ${\textstyle [0,T]}$, Lipschitz continuous on ${\textstyle [0,T]}$ and such that

${\displaystyle \varphi (t)>0}$

for every ${\textstyle t\in \quad ]0,T]}$. We assume that

${\displaystyle \varphi (0)=0{\mbox{,}}}$

(1.1)

In ${\textstyle Q}$, consider the boundary value problem

${\displaystyle \lbrace {\begin{array}{c}\partial _{t}u+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}u=f\in L_{\omega }^{2}(Q){\mbox{,}}\\\partial _{x_{j}}^{k}u\vert _{\partial Q\setminus \varsupsetneq \Gamma _{T}}=0{\mbox{,  }}\quad k=0,\ldots ,m-1;j=1,2{\mbox{,}}\end{array}}}$

(1.3)

where ${\textstyle m\in N^{_{\ast }}}$, ${\textstyle \partial Q}$ is the boundary of ${\textstyle Q}$ and ${\textstyle \Gamma _{T}}$ is the part of the boundary of ${\textstyle Q}$ where ${\textstyle t=T}$. Here, ${\textstyle L_{\omega }^{2}(Q)}$ is the space of square-integrable functions on ${\textstyle Q}$ with the measure ${\textstyle \omega dtdx_{1}dx_{2}}$, where the weight ${\textstyle \omega }$ is a real-valued function defined on ${\textstyle [0,T]}$, differentiable on ${\textstyle ]0,T]}$, such that

${\displaystyle \forall t\in [0,T]:\omega (t)>0{\mbox{,}}}$

(1.4)

The difficulty related to this kind of problems comes from the fact that the domain ${\textstyle Q}$ considered here is nonstandard since it shrinks at ${\textstyle t=0\quad (\varphi (0)=0)}$, which prevents the domain ${\textstyle Q}$ 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 ${\textstyle u}$ belonging to the anisotropic weighted Sobolev space

${\displaystyle {\begin{array}{l}\displaystyle H_{0,\omega }^{1,2m}(Q):=\lbrace u\in H_{\omega }^{1,2m}(Q):\partial _{x_{j}}^{k}u\vert _{\partial Q\setminus \varsupsetneq \Gamma _{T}}=0{\mbox{,  }}k=0\\\displaystyle \ldots ,m-1;j=1,2\rbrace {\mbox{,}}\end{array}}}$

with

${\displaystyle H_{\omega }^{1,2m}(Q)=\lbrace u:\partial _{t}u,\partial ^{\alpha }u\in L_{\omega }^{2}(Q),\vert \alpha \vert \leq 2m\rbrace }$

where

${\displaystyle \alpha =(i_{1},i_{2})\in N^{2}{\mbox{,}}\quad \vert \alpha \vert =i_{1}+i_{2}{\mbox{,}}\quad \partial ^{\alpha }u=\partial _{x_{1}}^{i_{1}}\partial _{x_{2}}^{i_{2}}u{\mbox{.}}}$

The space ${\textstyle H_{\omega }^{1,2m}(Q)}$ is equipped with the natural norm, that is

${\displaystyle \Vert u\Vert _{H_{\omega }^{1,2m}(Q)}=(\Vert \partial _{t}u\Vert _{L_{\omega }^{2}(Q)}^{2}+\sum _{\vert \alpha \vert \leq 2m}{\underset {L_{\omega }^{2}(Q)}{\overset {2}{\Vert \partial ^{\alpha }u\Vert }}})^{1/2}{\mbox{.}}}$

Remark 1.1.

The boundary conditions of Problem (1.3) are equivalent to

${\displaystyle \partial _{\nu }^{k}u\vert _{\partial Q\setminus \varsupsetneq \Gamma _{T}}=0{\mbox{,}}\quad k=0,\ldots ,m-1{\mbox{,}}}$

where ${\textstyle \partial _{\nu }}$ stands for the normal derivative. This equivalence can be proved, for instance, by induction. So Problem (1.3) is also equivalent to

${\displaystyle \lbrace {\begin{array}{c}\partial _{t}u+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}u=f\in L_{\omega }^{2}(Q){\mbox{,}}\\\partial _{\nu }^{k}u\vert _{\partial Q\setminus \varsupsetneq \Gamma _{T}}=0{\mbox{,}}\quad k=0,\ldots ,m-1{\mbox{.}}\end{array}}}$

(1.6)

Observe that the number of the boundary conditions in (1.3) is ${\textstyle 2m}$, but they are not independent, while in (1.6), there are ${\textstyle m}$ independent boundary conditions.

Our main result is

Theorem 1.1.

Let us assume that  ${\textstyle \varphi }$satisfies condition   (1.1)   and the weight function  ${\textstyle \omega }$verifies assumptions    and . Then, the  ${\textstyle 2m}$-th order parabolic operator

${\displaystyle L=\partial _{t}+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}}$

is an isomorphism from  ${\textstyle H_{0,\omega }^{1,2m}(Q)}$into  ${\textstyle L_{\omega }^{2}(Q)}$if one of the following conditions is satisfied

(1)  ${\textstyle \varphi }$is an increasing function in a neighborhood of 0,

(2)  ${\textstyle \varphi }$verifies the condition   (1.2).

The case ${\textstyle m=1}$ 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  ${\textstyle \omega }$, Problem   (1.3)   is uniquely solvable.

Proof.

Let us consider ${\textstyle u\in H_{0,\omega }^{1,2m}(Q)}$ a solution of Problem (1.3) with a null right-hand side term. So,

${\displaystyle \partial _{t}u+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}u=0\quad {\mbox{in  }}Q{\mbox{.}}}$

In addition ${\textstyle u}$ fulfils the boundary conditions

${\displaystyle \partial _{x_{j}}^{k}u\vert _{\partial Q\setminus \varsupsetneq \Gamma _{T}}=0{\mbox{,}}\quad k=0,1,\ldots ,m-1;\quad j=1,2{\mbox{.}}}$

Using Green’s formula, we have

${\displaystyle {\begin{array}{l}\displaystyle \int _{Q}(\partial _{t}u+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}u)u\quad \omega (t)dt\quad dx_{1}dx_{2}=\\\displaystyle =\int _{\partial Q}[{\frac {1}{2}}\vert u\vert ^{2}\nu _{t}+\sum _{j=1}^{2}\sum _{k=0}^{m-1}(\partial _{x_{j}}^{2m-k-1}u.\partial _{x_{j}}^{k}u)(-\\\displaystyle -1)^{k+m}\nu _{x_{j}}]\omega (t)d\sigma +\int _{Q}(\vert \partial _{x_{1}}^{m}u\vert ^{2}+\\\displaystyle +\vert \partial _{x_{2}}^{m}u\vert ^{2})dtdx_{1}dx_{2}-\int _{Q}{\frac {1}{2}}\vert u\vert ^{2}\omega ^{'}(t)dtdx_{1}dx_{2}{\mbox{,}}\end{array}}}$

where ${\textstyle \nu _{t}}$, ${\textstyle \nu _{x_{1}}}$, ${\textstyle \nu _{x_{2}}}$ are the components of the unit outward normal vector at ${\textstyle \partial Q}$. Taking into account the boundary conditions, all the boundary integrals vanish except ${\textstyle \int _{\partial Q}\vert u\vert ^{2}\omega (t)\nu _{t}\quad d\sigma }$. We have

${\displaystyle \int _{\partial Q}\vert u\vert ^{2}\omega (t)\nu _{t}d\sigma =\int _{\Gamma _{T}}\vert u\vert ^{2}\omega (T)dx_{1}dx_{2}{\mbox{.}}}$

Then

${\displaystyle {\begin{array}{l}\displaystyle \int _{Q}(\partial _{t}u+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}u)u\quad \omega (t)dt\quad dx_{1}dx_{2}=\\\displaystyle =\int _{\Gamma _{T}}{\frac {1}{2}}\vert u\vert ^{2}\omega (T)dx_{1}dx_{2}-\int _{Q}{\frac {1}{2}}\vert u\vert ^{2}\omega ^{'}(t)dtdx_{1}dx_{2}+\\\displaystyle +\int _{Q}(\vert \partial _{x_{1}}^{m}u\vert ^{2}+\vert \partial _{x_{2}}^{m}u\vert ^{2})dtdx_{1}dx_{2}{\mbox{  .}}\end{array}}}$

Consequently

${\displaystyle \int _{Q}(\partial _{t}u+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}u)u\quad \omega (t)dt\quad dx_{1}dx_{2}=0}$

yields

${\displaystyle \int _{Q}(\vert \partial _{x_{1}}^{m}u\vert ^{2}+\vert \partial _{x_{2}}^{m}u\vert ^{2})dt{\mbox{   }}dx_{1}dx_{2}=0{\mbox{,}}}$

because

${\displaystyle \int _{\Gamma _{T}}{\frac {1}{2}}\vert u\vert ^{2}\omega (t)dx_{1}dx_{2}-\int _{Q}{\frac {1}{2}}\vert u\vert ^{2}\omega ^{'}(t)dt{\mbox{   }}dx_{1}dx_{2}\geq 0}$

thanks to the conditions  and . This implies that ${\textstyle \vert \partial _{x_{1}}^{m}u\vert ^{2}+\vert \partial _{x_{2}}^{m}u\vert ^{2}=0}$ and consequently ${\textstyle \partial _{x_{1}}^{2m}u=\partial _{x_{2}}^{2m}u=0}$. Then, the hypothesis ${\textstyle \partial _{t}u+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}u=0}$ gives ${\textstyle \partial _{t}u=0}$. Thus, ${\textstyle u}$ is constant. The boundary conditions imply that ${\textstyle u=0}$ in ${\textstyle Q}$. 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  ${\textstyle B(0,1)}$be the unit disk of  ${\textstyle R^{2}}$. Then, the operator

${\displaystyle {\begin{array}{l}\displaystyle A:H^{2m}(B(0,1))\cap H_{0}^{m}(B(0,1))\longrightarrow L^{2}(B(0\\\displaystyle 1)){\mbox{,}}v\mapsto Av=(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}v\end{array}}}$

is an isomorphism. Moreover, there exists a constant  ${\textstyle C>0}$such that

${\displaystyle {\begin{array}{l}\displaystyle \Vert v\Vert _{H^{2}m(B(0,1))}\leq C\Vert Av\Vert _{L^{2}(B(0,1))}{\mbox{,}}\quad \forall v\in H^{2m}(B(0\\\displaystyle 1))\cap H_{0}^{m}(B(0,1)){\mbox{.}}\end{array}}}$

In the above lemma, ${\textstyle H^{2m}}$ and ${\textstyle H_{0}^{m}}$ 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  ${\textstyle t\in \quad ]0,T[}$, there exists a constant  ${\textstyle C>0}$such that for each  ${\textstyle u\in H^{2m}(\Omega _{t})}$, we have

${\displaystyle {\begin{array}{l}\displaystyle \Vert \partial _{x_{j}}^{l}u\Vert _{L^{2}(\Omega _{t})}^{2}\leq C\varphi ^{2(2m-l)}(t)\Vert Au\Vert _{L^{2}(\Omega _{t})}^{2}{\mbox{,}}\quad l=0,1\\\displaystyle \ldots ,2m-1;j=1,2{\mbox{.}}\end{array}}}$

Proof.

It is a direct consequence of Lemma 2.1. Indeed, let ${\textstyle t\in \quad ]0,T[}$ and define the following change of variables

${\displaystyle B(0,1)\rightarrow \Omega _{t}{\mbox{,}}\quad (x_{1},x_{2})\longmapsto (\varphi (t)x_{1},\varphi (t)x_{2})=(x_{1}^{'},x_{2}^{'}){\mbox{.}}}$

Set ${\textstyle v(x_{1},x_{2})=u(x_{1}^{'},x_{2}^{'})}$, then if ${\textstyle v\in H^{2m}(B(0,1))}$, ${\textstyle u}$ belongs to ${\textstyle H^{2m}(\Omega _{t})}$. For ${\textstyle j=1,2}$, we have

${\displaystyle {\begin{array}{l}\displaystyle \Vert \partial _{x_{j}}^{l}v\Vert _{L^{2}(B(0,1))}^{2}=\int _{B(0,1)}(\partial _{x_{j}}^{l}v)^{2}(x_{1}\\\displaystyle x_{2})dx_{1}dx_{2}=\int _{\Omega _{t}}(\partial _{x_{j}^{'}}^{l}u)^{2}(x_{1}^{'}\\\displaystyle x_{2}^{'})\varphi ^{2l}(t){\frac {1}{\varphi ^{2}(t)}}dx_{1}^{'}dx_{2}^{'}=\varphi ^{2l-2}(t)\int _{\Omega _{t}}(\partial _{x_{j}^{'}}^{l}u)^{2}(x_{1}^{'}\\\displaystyle x_{2}^{'})dx_{1}^{'}dx_{2}^{'}=\varphi ^{2l-2}(t)\Vert \partial _{x_{j}^{'}}^{l}u\Vert _{L^{2}(\Omega _{t})}^{2}\end{array}}}$

where ${\textstyle l\in \lbrace 0,1,\ldots ,2m-1\rbrace }$. On the other hand, we have

${\displaystyle {\begin{array}{l}\displaystyle \Vert Av\Vert _{L^{2}(B(0,1))}^{2}=\int _{B(0,1)}[(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}v(x_{1}\\\displaystyle x_{2})]^{2}dx_{1}dx_{2}=\int _{\Omega _{t}}(\sum _{j=1}^{2}\varphi ^{2m}(t)\partial _{x_{j}^{'}}^{2m}u)^{2}(x_{1}^{'}\\\displaystyle x_{2}^{'}){\frac {1}{\varphi ^{2}(t)}}dx_{1}^{'}dx_{2}^{'}=\varphi ^{4m-2}(t)\int _{\Omega _{t}}(\sum _{j=1}^{2}\partial _{x_{j}^{'}}^{2m}u)^{2}(x_{1}^{'}\\\displaystyle x_{2}^{'})dx_{1}^{'}dx_{2}^{'}=\varphi ^{4m-2}(t)\Vert Au\Vert _{L^{2}(\Omega _{t})}^{2}{\mbox{.}}\end{array}}}$

Using the inequality

${\displaystyle {\begin{array}{ccc}\Vert \partial _{x_{j}}^{l}v\Vert _{L^{2}(B(0,1))}^{2}&\leq &C\Vert Av\Vert _{L^{2}(B(0,1))}^{2}\end{array}}}$

of Lemma 2.1, we obtain the desired inequality

${\displaystyle {\begin{array}{ccc}\Vert \partial _{x_{j}^{'}}^{l}u\Vert _{L^{2}(\Omega _{t})}^{2}&\leq &C\varphi ^{2(2m-l)}(t)\Vert Au\Vert _{L^{2}(\Omega _{t})}^{2}{\mbox{.}}\end{array}}}$

 □

Remark 2.2.

In Lemma 2.2 we can replace ${\textstyle \Vert .\Vert _{L^{2}}}$ by ${\textstyle \Vert .\Vert _{L_{\omega }^{2}}}$.

3. Proof of Theorem 1.1

3.1. Case of a truncated domain ${\textstyle Q_{n}}$3.1. Case of a truncated domain Q n {\textstyle Q {n}}

In this subsection, we replace ${\textstyle Q}$ by ${\textstyle Q_{n},n\in N^{_{\ast }}}$ and ${\textstyle {\frac {1}{n}}<T}$:

${\displaystyle Q_{n}=\lbrace (t,x_{1},x_{2})\in Q:{\frac {1}{n}}<t<T\rbrace {\mbox{.}}}$

Theorem 3.1.

For each  ${\textstyle n\in N^{_{\ast }}}$such that  ${\textstyle {\frac {1}{n}}<T}$, the problem

${\displaystyle \lbrace {\begin{array}{c}\partial _{t}u_{n}+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}u_{n}=f_{n}\in L_{\omega }^{2}(Q_{n}){\mbox{,   }}\\\partial _{x_{j}}^{k}u_{n}\vert _{\partial Q_{n}\setminus \varsupsetneq \Gamma _{T}}=0{\mbox{,}}\quad k=0,1,\ldots ,m-1;\quad j=1,2{\mbox{,}}\end{array}}}$

(3.1)

where  ${\textstyle f_{n}=f\vert _{Q_{n}}}$admits a unique solution  ${\textstyle u_{n}\in H_{\omega }^{1,2m}(Q_{n})}$.

Proof of Theorem 3.1.

The change of variables

${\displaystyle {\begin{array}{ccc}(t,x_{1},x_{2})&\mapsto &(t,y_{1},y_{2})=(t,{\frac {x_{1}}{\varphi (t)}},{\frac {x_{2}}{\varphi (t)}})\end{array}}}$

transforms ${\textstyle Q_{n}}$ into the cylinder ${\textstyle P_{n}=]{\frac {1}{n}},T[\times B(0,1)}$, where ${\textstyle B(0,1)}$ is the unit disk of ${\textstyle R^{2}}$. Putting ${\textstyle u_{n}(t,x_{1},x_{2})=v_{n}(t,y_{1},y_{2})}$  and ${\textstyle f_{n}(t,x_{1},x_{2})=g_{n}(t,y_{1},y_{2})}$, then Problem (3.1) is transformed, in ${\textstyle P_{n}}$ into the following variable-coefficient parabolic problem

${\displaystyle \lbrace {\begin{array}{c}\partial _{t}v_{n}+{\frac {(-1)^{m}}{\varphi ^{2m}(t)}}\sum _{j=1}^{2}\partial _{y_{j}}^{2m}v_{n}+{\frac {\varphi ^{'}(t)}{\varphi (t)}}\sum _{j=1}^{2}y_{j}\partial _{y_{j}}v_{n}=g_{n}\\\partial _{y_{j}}^{k}v_{n}\vert _{\partial P_{n}\setminus \varsupsetneq \Sigma _{T}}=0{\mbox{,}}\quad k=0,1,\ldots ,m-1;{\mbox{   }}j=1,2\end{array}}}$

where ${\textstyle \Sigma _{T}}$ is the part of the boundary of ${\textstyle P_{n}}$ where ${\textstyle t=T}$. The above change of variables conserves the spaces ${\textstyle L_{\omega }^{2}}$ and ${\textstyle H_{\omega }^{1,2m}}$ because ${\textstyle {\frac {(-1)^{m}}{\varphi ^{2m}(t)}}}$ and ${\textstyle {\frac {\varphi ^{'}(t)}{\varphi (t)}}}$ are bounded functions when ${\textstyle t\in \quad ]{\frac {1}{n}},T[}$. In other words

${\displaystyle f_{n}\in L_{\omega }^{2}(Q_{n})\Longleftrightarrow g_{n}\in L_{\omega }^{2}(P_{n}){\mbox{,}}\quad u_{n}\in H_{\omega }^{1,2m}(Q_{n})\Longleftrightarrow v_{n}\in H_{\omega }^{1,2m}(P_{n}){\mbox{.}}}$

Proposition 3.1.

For each  ${\textstyle n\in N^{_{\ast }}}$such that  ${\textstyle {\frac {1}{n}}<T}$, the following operator is compact

${\displaystyle {\frac {\varphi ^{'}(t)}{\varphi (t)}}\sum _{j=1}^{2}y_{j}\partial _{y_{j}}:H_{0,\omega }^{1,2m}(P_{n})\longrightarrow L_{\omega }^{2}(P_{n}){\mbox{.}}}$

Proof.

${\textstyle P_{n}}$ has the “horn property” of Besov (see  [3]). So, for ${\textstyle j=1,2}$

${\displaystyle \partial _{y_{j}}:H_{0,\omega }^{1,2m}(P_{n})\longrightarrow H_{\omega }^{1-{\frac {1}{2m}},2m-1}(P_{n}){\mbox{,}}\quad v\longmapsto \partial _{y_{j}}v{\mbox{,}}}$

is continuous. Since ${\textstyle P_{n}}$ is bounded, the canonical injection is compact from ${\textstyle H_{\omega }^{1-{\frac {1}{2m}},2m-1}(P_{n})}$ into ${\textstyle L_{\omega }^{2}(P_{n})}$ (see for instance [3]), where

${\displaystyle {\begin{array}{l}\displaystyle H_{\omega }^{1-{\frac {1}{2m}},2m-1}(P_{n})=L^{2}({\frac {1}{n}},T;H^{2m-1}(B(0,1)))\cap H^{1-{\frac {1}{2m}}}({\frac {1}{n}}\\\displaystyle T;L^{2}(B(0,1))){\mbox{.}}\end{array}}}$

For the complete definitions of the ${\textstyle H^{r,s}}$ Hilbertian Sobolev spaces, see for instance  [12]. Consider the composition

${\displaystyle \partial _{y_{j}}:H_{0,\omega }^{1,2m}(P_{n})\rightarrow H_{\omega }^{1-{\frac {1}{2m}},2m-1}(P_{n})\rightarrow L_{\omega }^{2}(P_{n}){\mbox{,}}\quad v\mapsto \partial _{y_{j}}v\mapsto \partial _{y_{j}}v{\mbox{,}}}$

then ${\textstyle \partial _{y_{j}}}$ is a compact operator from ${\textstyle H_{0,\omega }^{1,2m}(P_{n})}$ into ${\textstyle L_{\omega }^{2}(P_{n})}$. Since ${\textstyle {\frac {\varphi ^{'}(t)}{\varphi (t)}}}$ is a bounded function for ${\textstyle {\frac {1}{n}}<t<T}$, the operators ${\textstyle {\frac {\varphi ^{'}(t)y_{j}}{\varphi (t)}}\partial _{y_{j}}}$, ${\textstyle j=1,2}$ are also compact from ${\textstyle H_{0,\omega }^{1,2m}(P_{n})}$ into ${\textstyle L_{\omega }^{2}(P_{n})}$. Consequently,

${\displaystyle {\frac {\varphi ^{'}(t)}{\varphi (t)}}\sum _{j=1}^{N}y_{j}\partial _{y_{j}}}$

is compact from ${\textstyle H_{0,\omega }^{1,2m}(P_{n})}$ into ${\textstyle L_{\omega }^{2}(P_{n})}$. □

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

${\displaystyle \partial _{t}+{\frac {(-1)^{m}}{\varphi ^{2m}(t)}}\sum _{j=1}^{N}\partial _{y_{j}}^{2m}}$

is an isomorphism from ${\textstyle H_{0,\omega }^{1,2m}(P_{n})}$ into ${\textstyle L_{\omega }^{2}(P_{n})}$.

Lemma 3.1.

For each  ${\textstyle n\in N^{_{\ast }}}$such that  ${\textstyle {\frac {1}{n}}<T}$, the operator

${\displaystyle \partial _{t}+{\frac {(-1)^{m}}{\varphi ^{2m}(t)}}\sum _{j=1}^{2}\partial _{y_{j}}^{2m}}$

is an isomorphism from  ${\textstyle H_{0,\omega }^{1,2m}(P_{n})}$into  ${\textstyle L_{\omega }^{2}(P_{n})}$.

Proof.

Thanks to Remark 1.1, the problem

${\displaystyle \lbrace {\begin{array}{c}\partial _{t}v_{n}+{\frac {(-1)^{m}}{\varphi ^{2m}(t)}}\sum _{j=1}^{2}\partial _{y_{j}}^{2m}v_{n}=g_{n}\\\partial _{y_{j}}^{k}v_{n}\vert _{\partial P_{n}\setminus \varsupsetneq \Sigma _{T}}=0{\mbox{,}}\quad k=0,1,\ldots ,m-1;{\mbox{   }}j=1,2{\mbox{,}}\end{array}}}$

is equivalent to the following problem

${\displaystyle \lbrace {\begin{array}{c}\partial _{t}v_{n}+{\frac {(-1)^{m}}{\varphi ^{2m}(t)}}\sum _{j=1}^{2}\partial _{y_{j}}^{2m}v_{n}=g_{n}\\\partial _{\nu }^{k}v_{n}\vert _{\partial P_{n}\setminus \varsupsetneq \Sigma _{T}}=0{\mbox{,}}\quad k=0,\ldots ,m-1{\mbox{.}}\end{array}}}$

Since the coefficient ${\textstyle {\frac {1}{\varphi ^{2m}(t)}}}$ is bounded in ${\textstyle {\overline {P_{n}}}}$, 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  ${\textstyle n\in N^{_{\ast }}}$such that  ${\textstyle {\frac {1}{n}}<T}$, the space

${\displaystyle \lbrace v\in H^{2m}(P_{n}):\partial _{x_{j}}^{k}v\vert _{\partial _{p}P_{n}}=0{\mbox{,   }}k=0,1,\ldots ,m-1;\quad j=1,2\rbrace }$

is dense in the space

${\displaystyle \lbrace v\in H^{1,2m}(P_{n}):\partial _{x_{j}}^{k}v\vert _{\partial _{p}P_{n}}=0{\mbox{,   }}k=0,1,\ldots ,m-1;\quad j=1,2\rbrace {\mbox{.}}}$

Here,  ${\textstyle \partial _{p}P_{n}}$is the parabolic boundary of  ${\textstyle P_{n}}$and  ${\textstyle H^{2m}}$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 ${\textstyle P_{n}}$ by ${\textstyle Q_{n}}$ 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 ${\textstyle Q}$ and we suppose that the function ${\textstyle \varphi }$ satisfies conditions  and .

For each ${\textstyle n\in N^{_{\ast }}}$ such that ${\textstyle {\frac {1}{n}}<T}$, we denote ${\textstyle f_{n}=f\vert _{Q_{n}}}$ and ${\textstyle u_{n}\in H_{\omega }^{1,2m}(Q_{n})}$ the solution of Problem (1.3) in ${\textstyle Q_{n}}$. Such a solution exists by Theorem 3.1.

Proposition 3.2.

For  ${\textstyle T}$small enough, there exists a constant  ${\textstyle K_{1}}$independent of  ${\textstyle n}$such that

${\displaystyle \Vert u_{n}\Vert _{H_{\omega }^{1,2m}(Q_{n})}\leq K_{1}\Vert f_{n}\Vert _{L_{\omega }^{2}(Q_{n})}\leq K_{1}\Vert f\Vert _{L_{\omega }^{2}(Q)}{\mbox{.}}}$

Remark 3.2.

Let ${\textstyle \epsilon >0}$ be a real number which we will choose small enough. The hypothesis (1.2) implies the existence of a real number ${\textstyle T>0}$ small enough such that

${\displaystyle \forall t\in (0,T){\mbox{,}}\quad \vert \varphi ^{'}(t)\varphi ^{m}(t)\vert \leq \epsilon {\mbox{.}}}$

(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  ${\textstyle C>0}$independent of  ${\textstyle n}$such that

${\displaystyle \sum _{\vert \alpha \vert =2m}{\underset {L_{\omega }^{2}(Q_{n})}{\overset {2}{\Vert \partial ^{\alpha }u_{n}\Vert }}}\leq C\Vert Au_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}{\mbox{.}}}$

Proof of Proposition 3.2.

Let us denote the inner product in ${\textstyle L_{\omega }^{2}(Q_{n})}$ by ${\textstyle \left\langle .,.\right\rangle }$, then we have

${\displaystyle {\begin{array}{l}\displaystyle \Vert f_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}=\left\langle \partial _{t}u_{n}+Au_{n}\right.\\\left.\displaystyle \partial _{t}u_{n}+Au_{n}\right\rangle =\Vert \partial _{t}u_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}+\Vert Au_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}+2\left\langle \partial _{t}u_{n}\right.\\\left.\displaystyle Au_{n}\right\rangle {\mbox{.}}\end{array}}}$

Estimation of${\textstyle 2\left\langle \partial _{t}u_{n},Au_{n}\right\rangle }$: We have

${\displaystyle {\begin{array}{c}\partial _{t}u_{n}.Au_{n}=\sum _{j=1}^{2}[\sum _{k=0}^{m-1}\partial _{x_{j}}(\partial _{x_{j}}^{k}\partial _{t}u_{n}.\partial _{x_{j}}^{2m-k-1}u_{n})(-\\\displaystyle -1)^{k+m}+{\frac {1}{2}}\partial _{t}(\partial _{x_{j}}^{m}u_{n})^{2}]{\mbox{.}}\end{array}}}$

Then

${\displaystyle {\begin{array}{l}\displaystyle 2\left\langle \partial _{t}u_{n}\right.\\\left.\displaystyle Au_{n}\right\rangle =2\int _{Q_{n}}\partial _{t}u_{n}.Au_{n}.\omega (t)dtdx_{1}dx_{2}=\\\displaystyle =2\int _{Q_{n}}\sum _{j=1}^{2}\sum _{k=0}^{m-1}\partial _{x_{j}}(\partial _{x_{j}}^{k}\partial _{t}u_{n}.\partial _{x_{j}}^{2m-k-1}u_{n})(-\\\displaystyle -1)^{k+m}.\omega (t)dtdx_{1}dx_{2}+\\\displaystyle +\int _{Q_{n}}\partial _{t}\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2}.\omega (t)dtdx_{1}dx_{2}=\\\displaystyle =2\int _{\partial Q_{n}}\sum _{j=1}^{2}\sum _{k=0}^{m-1}(\partial _{x_{j}}^{k}\partial _{t}u_{n}.\partial _{x_{j}}^{2m-k-1}u_{n})(-\\\displaystyle -1)^{k+m}\nu _{x_{j}}.\omega (t)d\sigma +\int _{\partial Q_{n}}\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2}\nu _{t}.\omega (t)d\sigma -\\\displaystyle -\int _{Q_{n}}\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2}.\omega ^{'}(t)dtdx_{1}dx_{2}\end{array}}}$

with ${\textstyle \nu _{t},\nu _{x_{1}},\nu _{x_{2}}}$ are the components of the unit outward normal vector at ${\textstyle \partial Q_{n}}$. We shall rewrite the boundary integral making use of the boundary conditions. On the part of the boundary of ${\textstyle Q_{n}}$  where ${\textstyle t={\frac {1}{n}}}$, we have ${\textstyle \partial _{x_{j}}^{k}u_{n}=0,k=0,\ldots ,m-1;j=1,2}$ and consequently the corresponding boundary integral vanishes. On the part of the boundary where ${\textstyle t=T}$, we have ${\textstyle \nu _{x_{j}}=0}$, ${\textstyle j=1,2}$ and ${\textstyle \nu _{t}=1}$. Accordingly, the corresponding boundary integral

${\displaystyle \int _{\Gamma _{T}}\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2}(T,x_{1},x_{2}).\omega (T)dx_{1}dx_{2}}$

is nonnegative. On the part ${\textstyle \Gamma _{1}}$ of ${\textstyle \partial Q_{n}}$ defined by

${\displaystyle \Gamma _{1}=\lbrace (t,x_{1},x_{2}):{\sqrt {x_{1}^{2}+x_{2}^{2}}}=\varphi (t)\rbrace {\mbox{,}}}$

we have

${\displaystyle \nu _{t}={\frac {-\varphi ^{'}(t)}{\sqrt {1+(\varphi ^{'})^{2}(t)}}}{\mbox{,}}}$

and

${\displaystyle \partial _{x_{j}}^{k}u_{n}(t,\varphi (t)cos\theta ,\varphi (t)sin\theta )=0{\mbox{,}}\quad k=0,\ldots ,m-1{\mbox{;}}j=1,2{\mbox{.}}}$

Let us denote

${\displaystyle {\begin{array}{l}\displaystyle I=2\int _{\Gamma _{1}}\sum _{j=1}^{2}\sum _{k=0}^{m-1}(\partial _{x_{j}}^{k}\partial _{t}u_{n}.\partial _{x_{j}}^{2m-k-1}u_{n})(-\\\displaystyle -1)^{k+m}\nu _{x_{j}}.\omega (t)d\sigma {\mbox{.}}\end{array}}}$

We have

${\displaystyle {\begin{array}{l}\displaystyle I=2\int _{\Gamma _{1}}\sum _{j=1}^{2}(\partial _{t}u_{n}.\partial _{x_{j}}^{2m-1}u_{n})(-1)^{m}\nu _{x_{j}}.\omega (t)d\sigma +\\\displaystyle +2\int _{\Gamma _{1}}\sum _{j=1}^{2}\sum _{k=1}^{m-2}(\partial _{x_{j}}^{k}\partial _{t}u_{n}.\partial _{x_{j}}^{2m-k-1}u_{n})(-\\\displaystyle -1)^{k+m}\nu _{x_{j}}.\omega (t)d\sigma -\\\displaystyle -2\int _{\Gamma _{1}}\sum _{j=1}^{2}(\partial _{x_{j}}^{m-1}\partial _{t}u_{n}.\partial _{x_{j}}^{m}u_{n})\nu _{x_{j}}.\omega (t)d\sigma =I_{0}+\\\displaystyle +I_{1}+I_{m-1}{\mbox{.}}\end{array}}}$

(a) Estimation of${\textstyle I_{0}=2\int _{\Gamma _{1}}\sum _{j=1}^{2}(\partial _{t}u_{n}.\partial _{x_{j}}^{2m-1}u_{n})(-1)^{m}\nu _{x_{j}}.\omega (t)d\sigma }$:

We have

${\displaystyle u_{n}(t,\varphi (t)cos\theta ,\varphi (t)sin\theta )=0{\mbox{.}}}$

Differentiating with respect to ${\textstyle t}$, we obtain

${\displaystyle \partial _{t}u_{n}=-\varphi ^{'}(t)(cos\theta .\partial _{x_{1}}u_{n}+sin\theta .\partial _{x_{2}}u_{n})=0{\mbox{.}}}$

So, the boundary integral ${\textstyle I_{0}}$ vanishes.

(b) Estimation of${\textstyle {\begin{array}{l}\displaystyle I_{1}=2\int _{\Gamma _{1}}\sum _{j=1}^{2}\sum _{k=1}^{m-2}(\partial _{x_{j}}^{k}\partial _{t}u_{n}.\partial _{x_{j}}^{2m-k-1}u_{n})(-\\\displaystyle -1)^{k+m}\nu _{x_{j}}.\omega (t)d\sigma \end{array}}}$:

We have

${\displaystyle \partial _{x_{j}}^{k}u_{n}(t,\varphi (t)cos\theta ,\varphi (t)sin\theta )=0{\mbox{,}}\quad k=1,\ldots ,m-2;j=1,2{\mbox{.}}}$

Differentiating with respect to ${\textstyle t}$, we obtain

${\displaystyle {\begin{array}{l}\displaystyle \partial _{t}\partial _{x_{1}}^{k}u_{n}=-\varphi ^{'}(t)[cos\theta .\partial _{x_{1}}^{k+1}u_{n}+\\\displaystyle +sin\theta .\partial _{x_{2}}\partial _{x_{1}}^{k}u_{n}]{\mbox{,}}k=1,\ldots ,m-2\end{array}}}$

and

${\displaystyle {\begin{array}{l}\displaystyle \partial _{t}\partial _{x_{2}}^{k}u_{n}=-\varphi ^{'}(t)[cos\theta .\partial _{x_{1}}\partial _{x_{2}}^{k}u_{n}+\\\displaystyle +sin\theta .\partial _{x_{2}}^{k+1}u_{n}]{\mbox{,}}k=1,\ldots ,m-2{\mbox{.}}\end{array}}}$

The Dirichlet boundary conditions on ${\textstyle \Gamma _{1}}$ lead to

${\displaystyle \partial _{t}\partial _{x_{1}}^{k}u_{n}=-\varphi ^{'}(t)sin\theta .\partial _{x_{2}}\partial _{x_{1}}^{k}u_{n}{\mbox{,}}\quad k=1,\ldots ,m-2}$

and

${\displaystyle \partial _{t}\partial _{x_{2}}^{k}u_{n}=-\varphi ^{'}(t)cos\theta .\partial _{x_{1}}\partial _{x_{2}}^{k}u_{n}{\mbox{,}}\quad k=1,\ldots ,m-2{\mbox{.}}}$

Now, differentiating the formula

${\displaystyle \partial _{x_{j}}^{k}u_{n}(t,\varphi (t)cos\theta ,\varphi (t)sin\theta )=0{\mbox{,  }}\quad k=1,\ldots ,m-2;j=1,2}$

with respect to ${\textstyle \theta }$, we obtain

${\displaystyle sin\theta .\partial _{x_{1}}^{k+1}u_{n}=cos\theta .\partial _{x_{2}}\partial _{x_{1}}^{k}u_{n}{\mbox{,}}\quad k=1,\ldots ,m-2}$

and

${\displaystyle cos\theta .\partial _{x_{2}}^{k+1}u_{n}=sin\theta .\partial _{x_{1}}\partial _{x_{2}}^{k}u_{n}{\mbox{,}}\quad k=1,\ldots ,m-2{\mbox{.}}}$

The Dirichlet boundary conditions on ${\textstyle \Gamma _{1}}$ lead to

${\displaystyle \partial _{x_{1}}\partial _{x_{2}}^{k}u_{n}=\partial _{x_{2}}\partial _{x_{1}}^{k}u_{n}=0{\mbox{,}}\quad k=1,\ldots ,m-2}$

and consequently

${\displaystyle \partial _{t}\partial _{x_{1}}^{k}u_{n}=\partial _{t}\partial _{x_{2}}^{k}u_{n}=0{\mbox{,}}\quad k=1,\ldots ,m-2{\mbox{.}}}$

So, the boundary integral ${\textstyle I_{1}}$ vanishes.

(c) Estimation of${\textstyle I_{m-1}=-2\int _{\Gamma _{1}}\sum _{j=1}^{2}(\partial _{x_{j}}^{m-1}\partial _{t}u_{n}.\partial _{x_{j}}^{m}u_{n})\nu _{x_{j}}.\omega (t)d\sigma }$

We have

${\displaystyle \partial _{x_{j}}^{m-1}u_{n}(t,\varphi (t)cos\theta ,\varphi (t)sin\theta )=0{\mbox{,}}\quad j=1,2{\mbox{.}}}$

Differentiating with respect to ${\textstyle t}$, we obtain

${\displaystyle {\begin{array}{l}\displaystyle \partial _{t}\partial _{x_{1}}^{m-1}u_{n}=-\varphi ^{'}(t)[(cos\theta .\partial _{x_{1}}^{m}u_{n}+\\\displaystyle +sin\theta .\partial _{x_{2}}\partial _{x_{1}}^{m-1}u_{n})]\end{array}}}$

and

${\displaystyle {\begin{array}{l}\displaystyle \partial _{t}\partial _{x_{2}}^{m-1}u_{n}=-\varphi ^{'}(t)[cos\theta .\partial _{x_{1}}\partial _{x_{2}}^{m-1}u_{n}+\\\displaystyle +sin\theta .\partial _{x_{2}}^{m}u_{n}]{\mbox{.}}\end{array}}}$

Differentiating with respect to ${\textstyle \theta }$, we obtain

${\displaystyle cos\theta .\partial _{x_{2}}\partial _{x_{1}}^{m-1}u_{n}=sin\theta .\partial _{x_{1}}^{m}u_{n}}$

Taking into account these relationships we deduce

${\displaystyle {\begin{array}{l}\displaystyle I_{m-1}=2\int _{0}^{2\pi }\int _{\frac {1}{n}}^{T}[cos^{2}\theta .\partial _{x_{1}}^{m}u_{n}+\\\displaystyle +sin\theta cos\theta .\partial _{x_{1}}\partial _{x_{2}}^{m-1}u_{n}]\partial _{x_{1}}^{m}u_{n}\varphi ^{'}(t)\varphi (t).\omega (t)dtd\theta +\\\displaystyle +2\int _{0}^{2\pi }\int _{\frac {1}{n}}^{T}[cos\theta sin\theta .\partial _{x_{2}}\partial _{x_{1}}^{m-1}u_{n}+\\\displaystyle +sin^{2}\theta .\partial _{x_{2}}^{m}u_{n}]\partial _{x_{2}}^{m}u_{n}\varphi ^{'}(t)\varphi (t).\omega (t)dtd\theta =\\\displaystyle =2\int _{0}^{2\pi }\int _{\frac {1}{n}}^{T}[(\partial _{x_{1}}^{m}u_{n})^{2}+\\\displaystyle +(\partial _{x_{2}}^{m}u_{n})^{2}]\varphi ^{'}(t)\varphi (t).\omega (t)dtd\theta {\mbox{.}}\end{array}}}$

Finally

${\displaystyle {\begin{array}{l}\displaystyle 2\left\langle \partial _{t}u_{n}\right.\\\left.\displaystyle Au_{n}\right\rangle =\int _{0}^{2\pi }\int _{\frac {1}{n}}^{T}(\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2})\varphi ^{'}(t)\varphi (t).\omega (t)dtd\theta +\\\displaystyle +\int _{\Gamma _{T}}(\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2})(T,x_{1}\\\displaystyle x_{2}).\omega (T)dx_{1}dx_{2}-\\\displaystyle -\int _{Q_{n}}(\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2}).\omega ^{'}(t)dtdx_{1}dx_{2}{\mbox{.}}\end{array}}}$

(3.3)

Remark 3.3.

Observe that the integrals

${\displaystyle \int _{\Gamma _{T}}(\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2})(T,x_{1},x_{2}).\omega (T)dx_{1}dx_{2}}$

and

${\displaystyle -\int _{Q_{n}}(\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2}).\omega ^{'}(t)dtdx_{1}dx_{2}{\mbox{,}}}$

which appear in the last formula are nonnegative thanks to the assumptions  and  on the weight function ${\textstyle \omega }$. This is a good sign for our estimate because we can deduce immediately

${\displaystyle {\begin{array}{l}\displaystyle \Vert f_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}\geq \Vert \partial _{t}u_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}+\Vert Au_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}+\\\displaystyle +\int _{0}^{2\pi }\int _{\frac {1}{n}}^{T}(\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2})\varphi ^{'}(t)\varphi (t).\omega (t)dtd\theta {\mbox{.}}\end{array}}}$

So, if ${\textstyle \varphi }$ is an increasing function in the interval ${\textstyle ({\frac {1}{n}},T)}$, then

${\displaystyle \int _{0}^{2\pi }\int _{\frac {1}{n}}^{T}(\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2})\varphi ^{'}(t)\varphi (t).\omega (t)dtd\theta \geq 0{\mbox{.}}}$

Consequently,

${\displaystyle \Vert f_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}\geq \Vert \partial _{t}u_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}+\Vert Au_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}{\mbox{.}}}$

(3.4)

But, thanks to Lemma 2.2 and since ${\textstyle \varphi }$ is bounded in ${\textstyle (0,T)}$, there exists a constant ${\textstyle C^{'}>0}$ such that

${\displaystyle {\begin{array}{l}\displaystyle \Vert \partial _{x_{j}}^{l}u_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}\leq C^{'}\Vert Au_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}{\mbox{,}}\quad l=0,1,\ldots \\\displaystyle 2m-1;j=1,2{\mbox{.}}\end{array}}}$

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

${\displaystyle {\begin{array}{l}\displaystyle 2\left\langle \partial _{t}u_{n}\right.\\\left.\displaystyle Au_{n}\right\rangle =2\int _{Q_{n}}{\frac {\varphi ^{'}}{\varphi }}(\sum _{j=1}^{2}x_{j}\partial _{x_{j}}^{m}u_{n})Au_{n}.\omega (t)dtdx_{1}dx_{2}+\\\displaystyle +\int _{\Gamma _{T}}\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2}(T,x_{1},x_{2}).\omega (T)dx_{1}dx_{2}{\mbox{.}}\end{array}}}$

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

${\displaystyle \vert \int _{Q_{n}}{\frac {\varphi ^{'}}{\varphi }}(\sum _{j=1}^{2}x_{j}\partial _{x_{j}}^{m}u_{n})Au_{n}.\omega (t)dtdx_{1}dx_{2}\vert \leq \Vert Au_{n}\Vert _{L_{\omega }^{2}(Q_{n})}\sum _{j=1}^{2}\Vert {\frac {\varphi ^{'}}{\varphi }}x_{j}\partial _{x_{j}}^{m}u_{n}\Vert _{L_{\omega }^{2}(Q_{n})}{\mbox{,}}}$

but Lemma 2.2 yields for ${\textstyle j=1,2}$

${\displaystyle {\begin{array}{l}\displaystyle \Vert {\frac {\varphi ^{'}}{\varphi }}x_{j}\partial _{x_{j}}^{m}u_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}=\\\displaystyle =\int _{\frac {1}{n}}^{T}\varphi ^{'2}(t)\int _{\Omega _{t}}({\frac {x_{j}}{\varphi (t)}})^{2}(\partial _{x_{j}}^{m}u_{n})^{2}.\omega (t)dtdx_{1}dx_{2}\leq \int _{\frac {1}{n}}^{T}\varphi ^{'2}(t)\int _{\Omega _{t}}(\partial _{x_{j}}^{m}u_{n})^{2}.\omega (t)dtdx_{1}dx_{2}\leq C^{2}\int _{\frac {1}{n}}^{T}(\varphi ^{m}(t)\varphi ^{'}(t))^{2}\int _{\Omega _{t}}(Au_{n})^{2}.\omega (t)dtdx_{1}dx_{2}\leq C^{2}\epsilon ^{2}\Vert Au_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}{\mbox{,}}\end{array}}}$

since ${\textstyle (\varphi ^{m}(t)\varphi ^{'}(t))\leq \epsilon }$ thanks to the condition (3.2). Then

${\displaystyle \vert \int _{Q_{n}}{\frac {\varphi ^{'}}{\varphi }}(\sum _{j=1}^{2}x_{j}\partial _{x_{j}}^{m}u_{n})Au_{n}.\omega (t)dtdx_{1}dx_{2}\vert \leq 2C\epsilon \Vert Au_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}{\mbox{.}}}$

Therefore, Lemma 3.4 shows that

${\displaystyle {\begin{array}{l}\displaystyle \vert 2\left\langle \partial _{t}u_{n}\right.\\\left.\displaystyle Au_{n}\right\rangle \vert \geq -2\vert \int _{Q_{n}}{\frac {\varphi ^{'}}{\varphi }}(\sum _{j=1}^{2}x_{j}\partial _{x_{j}}^{m}u_{n})Au_{n}.\omega (t)dtdx_{1}dx_{2}\vert +\\\displaystyle +\int _{\Gamma _{T}}\sum _{j=1}^{2}(\partial _{x_{j}}^{m}u_{n})^{2}(T,x_{1}\\\displaystyle x_{2}).\omega (T)dx_{1}dx_{2}.\geq -4C\epsilon \Vert Au_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}{\mbox{.}}\end{array}}}$

Hence

${\displaystyle {\begin{array}{l}\displaystyle \Vert f_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}=\Vert \partial _{t}u_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}+\Vert Au_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}+2\left\langle \partial _{t}u_{n}\right.\\\left.\displaystyle Au_{n}\right\rangle \geq \Vert \partial _{t}u_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}+(1-4C\epsilon )\Vert Au_{n}\Vert _{L_{\omega }^{2}(Q_{n})}^{2}{\mbox{.}}\end{array}}}$

Then, it is sufficient to choose ${\textstyle \epsilon }$ such that ${\textstyle 1-4C\epsilon >0}$ to get a constant ${\textstyle K_{0}>0}$ independent of ${\textstyle n}$ such that

${\displaystyle \Vert f_{n}\Vert _{L_{\omega }^{2}(Q_{n})}\geq K_{0}\Vert u_{n}\Vert _{H_{\omega }^{1,2m}(Q_{n})}{\mbox{,}}}$

and since

${\displaystyle \Vert f_{n}\Vert _{L_{\omega }^{2}(Q_{n})}\leq \Vert f\Vert _{L_{\omega }^{2}(Q_{n})}{\mbox{,}}}$

there exists a constant ${\textstyle K_{1}>0}$, independent of ${\textstyle n}$ satisfying

${\displaystyle \Vert u_{n}\Vert _{H_{\omega }^{1,2m}(Q_{n})}\leq K_{1}\Vert f_{n}\Vert _{L_{\omega }^{2}(Q_{n})}\leq K_{1}\Vert f\Vert _{L_{\omega }^{2}(Q)}{\mbox{.}}}$

This completes the proof of Proposition 3.2.

3.3. Passage to the limit

Choose a sequence ${\textstyle (Q_{n})_{n\in N^{_{\ast }}}}$ of the domains defined above (see Section  3.1), such that ${\textstyle Q_{n}\subseteq Q}$. Then, we have ${\textstyle Q_{n}\rightarrow Q}$, as ${\textstyle n\rightarrow \infty }$. Consider the solution ${\textstyle u_{n}\in H_{\omega }^{1,2m}(Q_{n})}$ of the Cauchy–Dirichlet problem

${\displaystyle \lbrace {\begin{array}{c}\partial _{t}u_{n}+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}u_{n}=f_{n}\in L_{\omega }^{2}(Q_{n}){\mbox{,}}\\\partial _{x_{j}}^{k}u_{n}\vert _{\partial Q_{n}\setminus \varsupsetneq \Gamma _{T}}=0{\mbox{,}}\quad k=0,1,\ldots ,m-1;{\mbox{   }}j=1,2{\mbox{,}}\end{array}}}$

where ${\textstyle f_{n}=f\vert _{Q_{n}}}$. Such a solution ${\textstyle u_{n}}$ exists by Theorem 3.1. Let ${\textstyle {\tilde {u_{n}}}}$ be the 0-extension of ${\textstyle u_{n}}$ to ${\textstyle Q}$. In virtue of Proposition 3.2 for ${\textstyle T}$ small enough, we know that there exists a constant ${\textstyle C}$ such that

${\displaystyle {\begin{array}{l}\displaystyle \Vert {\tilde {u_{n}}}\Vert _{L_{\omega }^{2}(Q)}+\Vert {\tilde {\partial _{t}u_{n}}}\Vert _{L_{\omega }^{2}(Q)}+\\\displaystyle +\sum _{1\leq \vert \alpha \vert \leq 2m}\Vert {\tilde {\partial ^{\alpha }u_{n}}}\Vert _{{\underset {\omega }{\overset {2}{L}}}(Q)}\leq C\Vert f\Vert _{{\underset {\omega }{\overset {2}{L}}}(Q)}{\mbox{.}}\end{array}}}$

This means that ${\textstyle {\tilde {u_{n}}}}$, ${\textstyle {\tilde {\partial _{t}u_{n}}},{\tilde {\partial ^{\alpha }u_{n}}}}$ for ${\textstyle 1\leq \vert \alpha \vert \leq 2m}$ are bounded functions in ${\textstyle L_{\omega }^{2}(Q)}$. So, for a suitable increasing sequence of integers ${\textstyle n_{k}}$, ${\textstyle k=1,2,\ldots }$, there exist functions

${\displaystyle u{\mbox{,  }}v{\mbox{  and  }}v_{\alpha }\quad 1\leq \vert \alpha \vert \leq 2m}$

in ${\textstyle L_{\omega }^{2}(Q)}$ such that

${\displaystyle {\begin{array}{ccccc}{\tilde {u_{n_{k}}}}&\rightharpoonup &u&{\mbox{weakly in  }}L_{\omega }^{2}(Q){\mbox{,}}&k\rightarrow \infty \\{\tilde {\partial _{t}u_{n_{k}}}}&\rightharpoonup &v&{\mbox{weakly in  }}L_{\omega }^{2}(Q){\mbox{,}}&k\rightarrow \infty \\{\tilde {\partial ^{\alpha }u_{n_{k}}}}&\rightharpoonup &v_{\alpha }\quad &{\mbox{weakly in  }}L_{\omega }^{2}(Q){\mbox{,}}&k\rightarrow \infty {\mbox{,}}\end{array}}}$

${\textstyle 1\leq \vert \alpha \vert \leq 2m}$. Clearly,

${\displaystyle v=\partial _{t}u{\mbox{,}}\quad v_{\alpha }=\partial ^{\alpha }u{\mbox{,}}\quad 1\leq \vert \alpha \vert \leq 2m}$

in the sense of distributions in ${\textstyle Q}$ and so in ${\textstyle L_{\omega }^{2}(Q)}$. So, ${\textstyle u\in H_{\omega }^{1,2m}(Q)}$ and

${\displaystyle \partial _{t}u+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}u=f\quad {\mbox{in  }}Q{\mbox{.}}}$

On the other hand, the solution ${\textstyle u}$ satisfies the boundary conditions

${\displaystyle \partial _{x_{j}}^{k}u_{n}\vert _{\partial Q_{n}\setminus \varsupsetneq \Gamma _{T}}=0{\mbox{,}}\quad k=0,1,\ldots ,m-1;\quad j=1,2{\mbox{,}}}$

since

${\displaystyle \forall n\in N^{_{\ast }}{\mbox{,}}\quad u\vert _{Q_{n}}=u_{n}{\mbox{.}}}$

This proves the existence of a solution to Problem (1.3). This ends the proof of Theorem 1.1 in the case of ${\textstyle T}$ small enough.

3.4. The general case

Assume that ${\textstyle Q}$ satisfies (1.1). In the case where ${\textstyle T}$ is not small enough, we set ${\textstyle Q=D_{1}\cup D_{2}\cup \Gamma _{T_{1}}}$ where

${\displaystyle D_{1}=\lbrace (t,x_{1},x_{2})\in Q:0<t<T_{1}\rbrace }$

with ${\textstyle T_{1}}$ small enough. In the sequel, ${\textstyle f}$ stands for an arbitrary fixed element of ${\textstyle L_{\omega }^{2}(Q)}$ and ${\textstyle f_{i}=f\vert _{D_{i}}}$, ${\textstyle i=1}$, 2. We know that (see Section  3.3) there exists a unique solution ${\textstyle w_{1}\in H_{\omega }^{1,2m}(D_{1})}$ of the problem

${\displaystyle \lbrace {\begin{array}{c}\partial _{t}w_{1}+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}w_{1}=f_{1}\in L_{\omega }^{2}(D_{1}){\mbox{,}}\\\partial _{x_{j}}^{k}w_{1}\vert _{\partial D_{1}\setminus \varsupsetneq \Gamma _{T_{1}}}=0{\mbox{,}}\quad k=0,\ldots ,m-1;{\mbox{   }}j=1,2{\mbox{.}}\end{array}}}$

(3.5)

Hereafter, we denote the trace ${\textstyle w_{1}\vert _{\Gamma _{T_{1}}}}$ by ${\textstyle \psi }$ which is in the Sobolev space ${\textstyle H_{\omega }^{m}(\Gamma _{T_{1}})}$ because ${\textstyle w_{1}\in H_{\omega }^{1,2m}(D_{1})}$ (see  [12]). Now, consider the following problem in ${\textstyle D_{2}}$

${\displaystyle \lbrace {\begin{array}{c}\partial _{t}w_{2}+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}w_{2}=f_{2}\quad \in L_{\omega }^{2}(D_{2}){\mbox{,}}\\w_{2}\vert _{\Gamma _{T_{1}}}=\psi {\mbox{,  }}\\\partial _{x_{j}}^{k}w_{2}\vert _{\partial D_{2}\setminus \varsupsetneq (\Gamma _{T_{1}}\cup \Gamma _{T})}=0{\mbox{,  }}\quad k=0,\ldots ,m-1;\quad j=1,2{\mbox{.}}\end{array}}}$

(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  ${\textstyle R}$be the cylinder  ${\textstyle ]0,T[\quad \times \quad B(0,1)}$where  ${\textstyle B(0,1)}$is the unit disk of  ${\textstyle R^{2}}$,  ${\textstyle f\in L_{\omega }^{2}(R)}$and  ${\textstyle u_{0}\in H_{\omega }^{m}(\gamma _{0})}$. Then, the problem

${\displaystyle \lbrace {\begin{array}{c}\partial _{t}u+(-1)^{m}\sum _{j=1}^{2}\partial _{x_{j}}^{2m}u=f\quad {\mbox{in   }}R{\mbox{,}}\\u\vert _{\gamma _{0}}=u_{0}{\mbox{,}}\\\partial _{x_{j}}^{k}u\vert _{\gamma _{1}}=0{\mbox{,   }}\quad k=0,\ldots ,m-1;\quad j=1,2{\mbox{,}}\end{array}}}$

where  ${\textstyle \gamma _{0}=\lbrace 0\rbrace \times B(0,1)}$,  ${\textstyle \gamma _{1}=]0,T[\quad \times \quad \partial B(0,1)}$, admits a (unique) solution  ${\textstyle u\in H_{\omega }^{1,2m}(R)}$if and only if the following compatibility conditions are fulfilled

${\displaystyle \partial _{x_{j}}^{k}u_{0}\vert _{\partial \gamma _{0}}=0{\mbox{,   }}\quad k=0,\ldots ,m-1;\quad j=1,2{\mbox{.}}}$

Thanks to the transformation

${\displaystyle (t,x_{1},x_{2})\longmapsto (t,y_{1},y_{2})=(t,\varphi (t)x_{1},\varphi (t)x_{2}){\mbox{,}}}$

we deduce the following result:

Proposition 3.4.

Problem   (3.6)   admits a (unique) solution  ${\textstyle w_{2}\in H_{\omega }^{1,2m}(D_{2})}$if and only if the following compatibility conditions are fulfilled

${\displaystyle \partial _{x_{j}}^{k}\psi \vert _{\partial \Gamma _{T_{1}}}=0{\mbox{,   }}\quad k=0,\ldots ,m-1;\quad j=1,2{\mbox{.}}}$

Remark 3.4.

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

${\displaystyle w_{1}\vert _{\Gamma _{T_{1}}}=w_{2}\vert _{\Gamma _{T_{1}}}}$

and ${\textstyle \partial _{x_{j}}^{k}w_{i}\vert _{\Gamma _{T_{1}}}\in H_{\omega }^{1-{\frac {1}{2m}}}(\Gamma _{T_{1}});\quad i,j=1,2}$. Then the compatibility conditions

${\displaystyle \partial _{x_{j}}^{k}\psi \vert _{\partial \Gamma _{T_{1}}}=0{\mbox{,  }}\quad k=0,\ldots ,m-1;\quad j=1,2}$

are satisfied since ${\textstyle w_{1}\vert _{\Gamma _{T_{1}}}=\psi }$.

Now, define the function ${\textstyle u}$ in ${\textstyle Q}$ by

${\displaystyle u:=\lbrace {\begin{array}{c}w_{1}\quad {\mbox{in  }}D_{1}\\w_{2}\quad {\mbox{in  }}D_{2}\end{array}}}$

where ${\textstyle w_{1}}$ and ${\textstyle w_{2}}$ are the solutions of Problem (3.5) and Problem (3.6) respectively. Observe that ${\textstyle w_{1}\vert _{\Gamma _{T_{1}}}=w_{2}\vert _{\Gamma _{T_{1}}}}$, see Remark 3.4, so

${\displaystyle \partial _{x_{j}}^{k}w_{1}\vert _{\Gamma _{T_{1}}}=\partial _{x_{j}}^{k}w_{2}\vert _{\Gamma _{T_{1}}}{\mbox{,}}\quad k=0,\ldots ,m-1;\quad j=1,2{\mbox{.}}}$

This implies that ${\textstyle u\in H_{\omega }^{1,2m}(Q)}$ and ${\textstyle u}$ is the (unique) solution of Problem (1.3) for an arbitrary ${\textstyle T}$. 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.

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