\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 116, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/116\hfil Resolutions of parabolic equations] {Resolutions of parabolic equations in non-symmetric conical domains} \author[A. Kheloufi\hfil EJDE-2012/116\hfilneg] {Arezki Kheloufi} % in alphabetical order \address{Arezki Kheloufi \newline Department of Technology, Faculty of Technology\\ B\'ejaia University, 6000 B\'ejaia, Algeria} \email{arezkinet2000@yahoo.fr} \thanks{Submitted May 31, 2012. Published July 9, 2012.} \subjclass[2000]{35K05, 35K20} \keywords{Parabolic equation; conical domain; anisotropic Sobolev space} \begin{abstract} This article is devoted to the analysis of a two-space dimensional linear parabolic equation, subject to Cauchy-Dirichlet boundary conditions. The problem is set in a conical type domain and the right hand side term of the equation is taken in a Lebesgue space. One of the main issues of this work is that the domain can possibly be non regular. This work is an extension of the symmetric case studied in Sadallah \cite{Sadallah2011}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Let $Q$ be an open set of $\mathbb{R}^3$ defined by $Q=\{ (t,x_1,x_2) \in \mathbb{R} ^3:(x_1,x_2) \in \Omega_t,00$ for every $t\in$ $] 0,T]$. $h$ is a Lipschitz continuous real-valued function defined on $[0,T]$, such that $$0<\delta \leq h(t)\leq \beta \label{HCondition}$$ for every $t\in$ $[0,T]$, where $\delta$ and $\beta$ are positive constants. In $Q$, we consider the boundary-value problem $$\begin{gathered} \partial_tu-\partial_{x_1}^2u-\partial_{x_2}^2u=f\in L^2(Q) , \\ u\big|_{\partial Q-\Gamma_T}=0, \end{gathered} \label{leProbleme}$$ where $L^2(Q)$ is the usual Lebesgue space on $Q$, $\partial Q$ is the boundary of $Q$ and $\Gamma_T$ is the part of the boundary of $Q$ where $t=T$. The difficulty related to this kind of problems comes from this singular situation for evolution problems; i.e., $\varphi$ is allowed to vanish for $t=0$, which prevents the domain $Q$ from being transformed into a regular domain without the appearance of some degenerate terms in the parabolic equation, see for example Sadallah \cite{Sadallah1983}. In order to overcome this difficulty, we impose a sufficient condition on the function $\varphi$; that is, $$\varphi'(t) \varphi (t) \to 0 \quad \text{as }t\to 0, \label{PhiCond1}$$ and we obtain existence and regularity results for Problem \eqref{leProbleme} by using the domain decomposition method. More precisely, we will prove that Problem \eqref{leProbleme} has a solution with optimal regularity, that is a solution $u$ belonging to the anisotropic Sobolev space $H_0^{1,2}(Q) :=\{ u\in H^{1,2}(Q) :u\big|_{\partial Q-\Gamma_T}=0\} ,$ with $H^{1,2}(Q) =\{ u\in L^2(Q) :\partial _tu,\partial_{x_1}^{j}u,\partial_{x_2}^{j}u,\partial _{x_1}\partial_{x_2} u\in L^2(Q) ,j=1,2\}.$ In Sadallah \cite{Sadallah2011} the same problem has been studied in the case of a symmetric conical domain; i.e., in the case where $h=1$. Further references on the analysis of parabolic problems in non-cylindrical domains are: Alkhutov \cite{Alkhutov2002,Alkhutov2007}, Degtyarev \cite{Degtyarev2010}, Labbas, Medeghri and Sadallah \cite{Labbas2002,Labbas2005}, Sadallah \cite{Sadallah1983}. There are many other works concerning boundary-value problems in non-smooth domains (see, for example, Grisvard \cite{Grisvard1985} and the references therein). The organization of this article is as follows. In Section 2, first we prove an uniqueness result for Problem \eqref{leProbleme}, then we derive some technical lemmas which will allow us to prove an uniform estimate (in a sense to be defined later). In Section 3, there are two main steps. First, we prove that Problem \eqref{leProbleme} admits a (unique) solution in the case of a domain which can be transformed into a cylinder. Secondly, for $T$ small enough, we prove that the result holds true in the case of a conical domain under the above mentioned assumptions on functions $\varphi$ and $h$. The method used here is based on the approximation of the conical domain by a sequence of subdomains $(Q_n)_n$ which can be transformed into regular domains (cylinders). We establish an uniform estimate of the type $\| u_n\|_{H^{1,2}(Q_n) }\leq K\| f\|_{L^2(Q_n) },$ where $u_n$ is the solution of Problem \eqref{leProbleme} in $Q_n$ and $K$ is a constant independent of $n$. This allows us to pass to the limit. Finally, in Section 4 we complete the proof of our main result (Theorem \ref{thm3}). \section{Preliminaries} \begin{proposition}\label{Proposition4.1} Problem \eqref{leProbleme} is uniquely solvable. \end{proposition} \begin{proof} Let us consider $u\in H_0^{1,2}(\Omega )$ a solution of Problem \eqref{leProbleme} with a null right-hand side term. So, $\partial_tu-\partial_{x_1}^2u-\partial_{x_2}^2u=0\quad\text{in }Q.$ In addition $u$ fulfils the boundary conditions $u\big|_{\partial Q-\Gamma_T}=0.$ Using Green formula, we have \begin{align*} \int_Q(\partial_tu-\partial_{x_1}^2u-\partial _{x_2}^2u) u\,dt\,dx_1\,dx_2 &=\int_{\partial Q}(\frac{1}{2}| u| ^2\nu _t-\partial_{x_1}u.u\nu_{x_1}-\partial_{x_2}u.u\nu _{x_2}) d\sigma \\ &\quad +\int_Q(| \partial_{x_1}u|^2+|\partial_{x_2}u| ^2) dt\,dx_1\,dx_2 \end{align*} where $\nu_t$, $\nu_{x_1}$, $\nu_{x_2}$ are the components of the unit outward normal vector at $\partial Q$. Taking into account the boundary conditions, all the boundary integrals vanish except $\int_{\partial Q}| u| ^2\nu_t\,d\sigma$. We have $\int_{\partial Q}| u| ^2\nu_td\sigma =\int_{\Gamma_T}| u| ^2\,dx_1\,dx_2.$ Then \begin{align*} &\int_Q(\partial_tu-\partial_{x_1}^2u-\partial _{x_2}^2u) u\,dt\,dx_1\,dx_2 \\ &=\int_{\Gamma_T}\frac{1}{2}| u| ^2\,dx_1\,dx_2 +\int_Q(| \partial_{x_1}u|^2+| \partial_{x_2}u| ^2) dt\,dx_1\,dx_2. \end{align*} Consequently, $\int_Q(\partial_tu-\partial_{x_1}^2u-\partial _{x_2}^2u) u\,dt\,dx_1\,dx_2=0$ yields $\int_Q(| \partial_{x_1}u| ^2+| \partial_{x_2}u| ^2) dt\,dx_1\,dx_2= 0,$ because $\frac{1}{2}\int_{\Gamma_T}| u| ^2\,dx_1\,dx_2\geq 0 .$ This implies $| \partial_{x_1}u|^2+| \partial_{x_2}u| ^2=0$ and consequently $\partial_{x_1}^2u=\partial_{x_2}^2u=0$. Then, the hypothesis $\partial_tu-\partial_{x_1}^2u-\partial_{x_2}^2u=0$ gives $\partial_tu=0$. Thus, $u$ is constant. The boundary conditions imply that $u=0$ in $Q$. This proves the uniqueness of the solution of Problem \eqref{leProbleme}. \end{proof} \begin{remark} \rm In the sequel, we will be interested only by the question of the existence of the solution of Problem \eqref{leProbleme}. \end{remark} The following result is well known (see, for example, \cite{Lions1968}) \begin{lemma}\label{Lemma4.1} Let $D(0,1)$ be the unit disc of $\mathbb{R} ^2$. Then, the Laplace operator $\Delta :H^2( D(0,1) ) \cap H_0^{1}(D(0,1)) \to L^2(D(0,1) )$ is an isomorphism. Moreover, there exists a constant $C>0$ such that $\| v\|_{H^2(D(0,1) )}\leq C\| \Delta v\|_{L^2(D( 0,1) ) } \text{, }\forall v\in H^2(D(0,1) ) .$ \end{lemma} In the above lemma, $H^2$ and $H_0^{1}$ are the usual Sobolev spaces defined, for instance, in Lions-Magenes \cite{Lions1968}. In section 3, we will need the following result. \begin{lemma} \label{lem5}Let $t\in] \alpha_n,T[$, where $( \alpha_n)_n$ is a decreasing sequence to zero. Then, there exists a constant $C>0$ independent of $n$ such that for each $u_n\in H^2(\Omega_t)$, we have \begin{itemize} \item[(a)] $\| \partial_{x_1}u_n\|_{L^2( \Omega_t) }^2\leq C\varphi ^2(t) \| \Delta u_n\|_{L^2(\Omega_t) }^2$, \item[(b)] $\| \partial_{x_2}u_n\|_{L^2( \Omega_t) }^2\leq C\varphi ^2(t) \| \Delta u_n\|_{L^2(\Omega_t) }^2$. \end{itemize} \end{lemma} \begin{proof} It is a direct consequence of Lemma \ref{Lemma4.1}. Indeed, let $t\in] \alpha_n,T[$ and define the following change of variables $\begin{array}{lll} D(0,1) & \to & \Omega_t \\ (x_1,x_2) & \mapsto & (\varphi ( t) x_1,h(t) \varphi (t) x_2) =(x_1',x_2'). \end{array}$ Set $v(x_1,x_2) =u_n(\varphi (t) x_1,h(t) \varphi (t) x_2) ,$ then if $v\in H^2(D(0,1) )$, $u_n$ belongs to $H^2(\Omega_t)$. (a) We have \begin{align*} \| \partial_{x_1}v\|_{L^2(D( 0,1) ) }^2 &= \int_{D(0,1) }( \partial_{x_1}v) ^2(x_1,x_2) \,dx_1\,dx_2 \\ &= \int_{\Omega_t}(\partial_{x_1'}u_n) ^2(x_1',x_2') \varphi ^2(t) \frac{1}{h(t)\varphi ^2(t) }dx_1'dx_2' \\ &= \frac{1}{h(t)}\int_{\Omega_t}(\partial_{x_1'}u_n) ^2(x_1',x_2') \,dx_1'\,dx_2' \\ &= \frac{1}{h(t)}\| \partial_{x_1'}u_n\|_{L^2(\Omega_t) }^2. \end{align*} On the other hand, \begin{align*} \| \Delta v\|_{L^2(D(0,1)) }^2 &= \int_{D(0,1) }[(\partial_{x_1}^2v+\partial_{x_2}^2v) (x_1,x_2) ] ^2\,dx_1\,dx_2 \\ &= \int_{\Omega_t}(\varphi ^2(t) \partial _{x_1'}^2u_n+(h\varphi)^2(t) \partial_{x_2'}^2u_n) ^2(x_1',x_2') \text{ }\frac{dx_1'dx_2'}{(h\varphi^2)( t) } \\ &= \frac{\varphi ^2(t) }{h(t)}\int_{\Omega_t}( \partial_{x_1'}^2u_n+h^2(t)\partial_{x_2'}^2u_n) ^2(x_1',x_2') \,dx_1'dx_2' \\ & \leq \frac{1}{\delta}\varphi ^2(t) \| \Delta u_n\|_{L^2(\Omega_t)}^2, \end{align*} where $\delta$ is the constant which appears in \eqref{HCondition}. Using Lemma \ref{Lemma4.1} and the condition \eqref{HCondition}, we obtain the desired inequality. (b) We have \begin{align*} \| \partial_{x_2}v\|_{L^2(D(0,1) ) }^2 &= \int_{D(0,1) }(\partial_{x_2}v) ^2(x_1,x_2) \,dx_1\,dx_2 \\ &= \int_{\Omega_t}(\partial_{x_2'}u_n) ^2(x_1',x_2') h^2(t)\varphi ^2(t) \frac{1}{h(t)\varphi ^2( t) }dx_1'dx_2' \\ &= h(t)\int_{\Omega_t}(\partial_{x_2'}u_n) ^2(x_1',x_2') \,dx_1'\,dx_2' \\ &= h(t)\| \partial_{x_2'}u_n\|_{L^2(\Omega_t) }^2. \end{align*} On the other hand, $\| \Delta v\|_{L^2(D(0,1)) }^2\leq \frac{1}{\delta}\varphi ^2(t) \| \Delta u_n\|_{L^2(\Omega_t)}^2.$ Using the inequality $\| \partial_{x_2}v\|_{L^2(D(0,1) ) }^2 \leq C\| \Delta v\|_{L^2(D(0,1) ) }^2$ of Lemma \ref{Lemma4.1} and condition \eqref{HCondition}, we obtain the desired inequality $\| \partial_{x_2'}u_n\|_{L^2(\Omega_t) }^2 \leq C\varphi ^2(t) \| \Delta u_n\|_{L^2(\Omega_t) }^2.$ \end{proof} \section{Local in time result} \subsection{Case of a truncated domain $Q_{\alpha }$} In this subsection, we replace $Q$ by $Q_{\alpha }$ $Q_{\alpha }=\{ (t,x_1,x_2) \in \mathbb{R}^3:\frac{1}{\alpha} 0. \begin{theorem}\label{thm1} The problem $$\begin{gathered} \partial_tu-\partial_{x_1}^2u-\partial_{x_2}^2u=f\in L^2(Q_{\alpha }) \text{,\medskip \medskip } \\ u\big|_{\partial Q_{\alpha }-\Gamma_T}=0, \end{gathered} \label{leProbleme2}$$ admits a unique solution u\in H^{1,2}(Q_{\alpha }) . \end{theorem} \begin{proof} The change of variables \[ (t,x_1,x_2) \mapsto (t,y_1,y_2) =(t,\frac{x_1}{\varphi (t) },\frac{x_2}{h(t)\varphi (t) })$ transforms $Q_{\alpha }$ into the cylinder $P_{\alpha }=]\frac{1}{\alpha} ,T [ \times D(\frac{1}{\alpha} ,1)$, where $D(\frac{1}{\alpha} ,1)$ is the unit disk centered on $(\frac{1}{\alpha} ,0,0)$. Putting $u(t,x_1,x_2) =v(t,y_1,y_2)$ and $f(t,x_1,x_2) =g(t,y_1,y_2)$, then Problem \eqref{leProbleme2} is transformed, in $P_{\alpha }$ into the variable-coefficient parabolic problem \begin{gather*} \partial_tv-\frac{1}{\varphi ^2(t) }\partial_{y_1}^2v- \frac{1}{h^2(t)\varphi ^2(t) }\partial _{y_2}^2v-\frac{ \varphi'(t) y_1}{\varphi (t) }\partial_{y_1}v-\frac{( h\varphi )'(t) y_2}{ h(t)\varphi (t) }\partial_{y_2}v=g\medskip \\ v\big|_{\partial P_{\alpha }-\Gamma_T}=0. \end{gather*} This change of variables conserves the spaces $H^{1,2}$ and $L^2$. In other words \begin{gather*} f\in L^2(Q_{\alpha }) \Rightarrow g\in L^2(P_{\alpha }) \\ u\in H^{1,2}(Q_{\alpha }) \Rightarrow v\in H^{1,2}(P_{\alpha }) . \end{gather*} \end{proof} \begin{proposition} \label{Proposition2} The operator $-\big[\frac{\varphi'(t) y_1}{\varphi (t) }\partial_{y_1}+\frac{(h\varphi ) '(t) y_2}{h(t)\varphi (t) }\partial_{y_2}\big] :H_0^{1,2}(P_{\alpha }) \to L^2(P_{\alpha })$ is compact. \end{proposition} \begin{proof} $P_{\alpha }$ has the horn property of Besov (see \cite{Besov1967}). So, for $j=1,2$ $\begin{array}{llll} \partial_{y_j} & H_0^{1,2}(P_{\alpha }) & \to & H^{\frac{1}{2},1}(P_{\alpha }) \\ & v & \mapsto & \partial_{y_j}v, \end{array}$ is continuous. Since $P_{\alpha }$ is bounded, the canonical injection is compact from $H^{\frac{1}{2},1}(P_{\alpha})$ into $L^2(P_{\alpha })$ (see for instance \cite{Besov1967}), where $H^{1/2,1}(P_{\alpha }) =L^2\Big( \frac{1}{\alpha} ,T;H^{1}\big(D(\frac{1}{\alpha} ,1) \big) \Big) \cap H^{1/2}\Big(\frac{1}{\alpha} ,T;L^2\big(D(\frac{1}{\alpha} ,1) \big) \Big).$ For the complete definitions of the $H^{r,s}$ Hilbertian Sobolev spaces see for instance \cite{Lions1968}. Consider the composition $\begin{array}{cccccc} \partial_{y_j}: & H_0^{1,2}(P_{\alpha }) & \to & H^{\frac{1}{2},1}(P_{\alpha }) & \to & L^2(P_{\alpha }) \\ & v & \mapsto & \partial_{y_j}v & \mapsto & \partial_{y_j}v, \end{array}$ then $\partial_{y_j}$ is a compact operator from $H_0^{1,2}(P_{\alpha })$ into $L^2(P_{\alpha })$. Since $-\frac{ \varphi'(t) }{\varphi (t) }$, $-\frac{ (h\varphi ) '(t) }{h(t)\varphi (t) }$ are bounded functions, the operators $-\frac{\varphi'(t) y_1}{\varphi (t) }\partial_{y_1}$, $-\frac{(h\varphi )'(t) y_2}{h(t)\varphi (t) }\partial_{y_2}$ are also compact from $H_0^{1,2}(P_{\alpha })$ into $L^2(P_{\alpha })$. Consequently, $-\big[\frac{\varphi'(t) y_1}{\varphi (t) }\partial_{y_1}+\frac{(h\varphi ) '(t) y_2}{h(t)\varphi (t) }\partial_{y_2}\big]$ is compact from $H_0^{1,2}(P_{\alpha })$ to $L^2(P_{\alpha })$. \end{proof} So, to complete the proof of Theorem \ref{thm1}, it is sufficient to show that the operator $\partial_t-\frac{1}{\varphi ^2(t) }\partial_{y_1}^2- \frac{1}{h^2(t)\varphi ^2(t) }\partial_{y_2}^2$ is an isomorphism from $H_0^{1,2}(P_{\alpha })$ into $L^2(P_{\alpha })$. \begin{lemma} The operator $\partial_t-\frac{1}{\varphi ^2(t) }\partial_{y_1}^2- \frac{1}{h^2(t)\varphi ^2(t) }\partial_{y_2}^2$ is an isomorphism from $H_0^{1,2}(P_{\alpha })$ to $L^2(P_{\alpha })$. \end{lemma} \begin{proof} Since the coefficients $\frac{1}{\varphi ^2(t) }$ and $\frac{ 1}{h^2(t)\varphi ^2(t) }$ are bounded in $\overline{ P_{\alpha }}$, the optimal regularity is given by Ladyzhenskaya-Solonnikov-Ural'tseva \cite{Ladyzhenskaya}. \end{proof} We shall need the following result to justify the calculus of this section. \begin{lemma}\label{lem4} The space $\{ u\in H^{4}(P_{\alpha }) :u\big|_{\partial_p P_{\alpha }}=0\}$ is dense in the space $\{ u\in H^{1,2}(P_{\alpha }) :u\big|_{\partial_p P_{\alpha }}=0\} .$ Here, $\partial_{p}P_{\alpha }$ is the parabolic boundary of $P_{\alpha }$ and $H^{4}$ stands for the usual Sobolev space defined, for instance, in Lions-Magenes \cite{Lions1968}. \end{lemma} The proof of the above lemma can be found in \cite{Kheloufi2010}. \begin{remark} \label{remq1}\rm In Lemma \ref{lem4}, we can replace $P_{\alpha }$ by $Q_{\alpha }$ with the help of the change of variables defined above. \end{remark} \subsection{Case of a conical type domain} In this case, we define $Q$ by $Q=\{ (t,x_1,x_2) \in \mathbb{R} ^3:00,\quad t\in]0,T] . \label{RaccordCondition} We assume that the functions h and \varphi  satisfy conditions \eqref{HCondition} and \eqref{PhiCond1}. For each n\in \mathbb{N}^*, we define Q_n by \[ Q_n=\{ (t,x_1,x_2) \in \mathbb{R}^3:\frac{1}{n}0 independent of n such that \[ \| \partial_{x_1}^2u_n\|_{L^2(Q_n) }^2+\| \partial_{x_2}^2u_n\| _{L^2(Q_n) }^2+\| \partial_{x_1x_2}^2u_n\|_{L^2(Q_n)}^2 \leq C\| \Delta u_n\|_{L^2(Q_n) }^2.$ \end{lemma} \subsection*{Proof of Proposition \ref{Prop}} Let us denote the inner product in $L^2(Q_n)$ by $\langle \cdot,\cdot\rangle$, then we have \begin{align*} \| f_n\|_{L^2(Q_n) }^2 &= \langle \partial_tu_n-\Delta u_n,\partial_tu_n-\Delta u_n\rangle \\ &= \| \partial_tu_n\|_{L^2(Q_n) }^2+\| \Delta u_n\|_{L^2(Q_n) }^2-2\langle \partial_tu_n,\Delta u_n\rangle \end{align*} \noindent\textbf{Estimation of $-2\langle \partial_tu_n,\Delta u_n\rangle$:} We have $\partial_tu_n.\Delta u_n=\partial_{x_1}(\partial _tu_n\partial_{x_1}u_n) +\partial_{x_2}( \partial_tu_n\partial_{x_2}u_n) -\frac{1}{2}\partial _t[(\partial_{x_1}u_n) ^2+(\partial _{x_2}u_n) ^2] .$ Then \begin{align*} -2\langle \partial_tu_n,\Delta u_n\rangle &= -2\int_{Q_n} \partial_tu_n.\Delta u_ndt\,dx_1\,dx_2 \\ &= -2\int_{Q_n}[\partial_{x_1}(\partial _tu_n\partial_{x_1}u_n) +\partial_{x_2}( \partial _tu_n\partial_{x_2}u_n) ] dt\,dx_1\,dx_2 \\ &\quad +\int_{Q_n}\partial_t[(\partial_{x_1}u_n) ^2+(\partial_{x_2}u_n) ^2] dt\,dx_1\,dx_2 \\ &= \int_{\partial Q_n}[| \nabla u_n| ^2\nu_t-2\partial_tu_n(\partial_{x_1}u_n\nu _{x_1}+\partial_{x_2}u_n\nu_{x_2}) ] d\sigma \end{align*} where $\nu_t,\nu_{x_1},\nu_{x_2}$ are the components of the unit outward normal vector at $\partial Q_n$. We shall rewrite the boundary integral making use of the boundary conditions. On the part of the boundary of $Q_n$\ where $t=\frac{1}{n}$, we have $u_n=0$ and consequently $\partial_{x_1}u_n=\partial_{x_2}u_n=0$. The corresponding boundary integral vanishes. On the part of the boundary\ where $t=T$, we have $\nu_{x_1}=0$, $\nu_{x_2}=0$ and $\nu_t=1$. Accordingly the corresponding boundary integral $A=\int_{\Gamma_T}| \nabla u_n|^2\,dx_1\,dx_2$ is nonnegative. On the part of the boundary\ where $\frac{x_1^2}{ \varphi ^2(t) }+\frac{x_2^2}{h^2(t)\varphi ^2(t) }=1$, we have $\nu_{x_1}=\frac{h(t)\cos \theta }{\sqrt{(\varphi'(t) h(t)\cos ^2\theta +(h\varphi )'( t) \sin^2 \theta ) ^2+(h(t)\cos \theta ) ^2+\sin ^2\theta }} ,$ $\nu_{x_2}=\frac{\sin \theta }{\sqrt{(\varphi'(t) h(t)\cos ^2\theta +(h\varphi )'( t) \sin^2 \theta ) ^2+(h(t)\cos \theta ) ^2+\sin ^2\theta }} ,$ $\nu_t=\frac{-(\varphi'(t) h(t)\cos ^2\theta +(h\varphi )'(t) \sin^2 \theta ) }{\sqrt{(\varphi'(t) h(t)\cos ^2\theta +(h\varphi )'(t) \sin^2 \theta ) ^2+(h(t)\cos \theta ) ^2+\sin ^2\theta }}$ and $u_n(t,\varphi (t) \cos \theta ,h(t)\varphi (t) \sin \theta ) =0$. Differentiating with respect to $t$ then with respect\ to $\theta$ we obtain \begin{gather*} \partial_tu_n=-\varphi'(t) \cos \theta .\partial _{x_1}u_n-(h\varphi )'(t) \sin \theta.\partial_{x_2}u_n, \\ \sin \theta .\partial_{x_1}u_n=h(t)\cos \theta .\partial_{x_2}u_n . \end{gather*} Consequently the corresponding boundary integral is \begin{align*} J_n &= -2\int_0^{2\pi }\int_{1/n}^T\partial _tu_n.(h\varphi \cos \theta .\partial _{x_1}u_n+h\varphi \sin \theta .\partial _{x_2}u_n) \,dt\,d\theta \\ &\quad -\int_0^{2\pi }\int_{1/n}^T| \nabla u_n| ^2((h\varphi )'\varphi \sin ^2\theta +\varphi'(h\varphi )\cos ^2\theta ) \,dt\,d\theta \\ &= 2\int_0^{2\pi }\int_{1/n}^T\{(\varphi '\cos \theta .\partial_{x_1}u_n+(h\varphi )'\sin \theta .\partial_{x_2}u_n)\\ &\quad \times (h\varphi \cos \theta .\partial_{x_1}u_n+h\varphi \sin \theta .\partial_{x_2}u_n) \}\,dt\,d\theta \\ & \quad -\int_0^{2\pi }\int_{1/n}^T| \nabla u_n| ^2((h\varphi )'\varphi \sin ^2\theta +\varphi'h\varphi \cos ^2\theta ) \,dt\,d\theta \\ &= 2\int_0^{2\pi }\int_{1/n}^T| \nabla u_n| ^2((h\varphi )'\varphi \sin ^2\theta +\varphi'h\varphi \cos ^2\theta ) \,dt\,d\theta \\ &\quad -\int_0^{2\pi }\int_{1/n}^T| \nabla u_n| ^2((h\varphi )'\varphi \sin ^2\theta +\varphi'h\varphi \cos ^2\theta ) \,dt\,d\theta \\ &= \int_0^{2\pi }\int_{1/n}^T| \nabla u_n| ^2((h\varphi )'\varphi \sin ^2\theta +\varphi'h\varphi \cos ^2\theta ) \,dt\,d\theta . \end{align*} Finally, $$\begin{split} -2\langle \partial_tu_n,\Delta u_n\rangle &= \int_0^{2\pi }\int_{1/n}^T| \nabla u_n| ^2((h\varphi )'\varphi \sin^2\theta +\varphi'h\varphi \cos ^2\theta ) \,dt\,d\theta \\ &\quad +\int_{\Gamma_T}| \nabla u_n|^2(T,x_1,x_2) \,dx_1\,dx_2. \end{split}\label{3}$$ \begin{lemma} \label{lem6} One has \begin{align*} -2\langle \partial_tu_n,\Delta u_n\rangle &= 2\int_{Q_n}(\frac{\varphi'}{\varphi }x_1\partial_{x_1}u_n+\frac{ (h\varphi )'}{h\varphi }x_2\partial_{x_2}u_n) \Delta u_ndt\,dx_1\,dx_2 \\ & \quad +\int_{\Gamma_T}| \nabla u_n| ^2(T,x_1,x_2) \,dx_1\,dx_2. \end{align*} \end{lemma} \begin{proof} For $\frac{1}{n}0$ to get a constant $K_0>0$ independent of $n$ such that $\| f_n\|_{L^2(Q_n) }\geq K_0\| u_n\|_{H^{1,2}(Q_n)},$ and since $\| f_n\|_{L^2(Q_n) }\leq \| f\|_{L^2(Q_n) },$ there exists a constant $K_1>0$, independent of $n$ satisfying $\| u_n\|_{H^{1,2}(Q_n) }\leq K_1\| f_n\|_{L^2(Q_n) }\leq K_1\| f\|_{L^2(Q) }.$ This completes the proof of Proposition \ref{Prop}. \subsection*{Passage to the limit} We are now in position to prove the main result of this work. \begin{theorem} \label{thm2} Assume that the functions $h$ and $\varphi$ verify the conditions \eqref{HCondition}, \eqref{PhiCond1} and \eqref{RaccordCondition}. Then, for $T$ small enough, Problem \eqref{leProbleme} admits a unique solution $u\in H^{1,2}(Q)$. \end{theorem} \begin{proof} Choose a sequence $Q_n$ $n=1,2, \dots$, of truncated conical domains (see subsection 3.2) such that $Q_n\subseteq Q$. Then we have $Q_n\to Q$, as $n\to \infty$. Consider the solution $u_n\in H^{1,2}(Q_n)$ of the Cauchy-Dirichlet problem \begin{gather*} \partial_tu_n-\partial_{x_1}^2u_n-\partial_{x_2}^2u_n=f\quad \text{in }Q_n \\ u_n\big|_{\partial Q_n-\Gamma_T}=0, \end{gather*} where $\Gamma_T$ is the part of the boundary of $Q_n$ where $t=T$. Such a solution $u_n$ exists by Theorem \ref{thm1}. Let $\widetilde{u}_n$ the 0-extension of $u_n$ to $Q$. By Proposition \ref{Prop}, we know that there exists a constant $C$ such that $\| \widetilde{u}_n\|_{L^2(Q)}+\| \partial_t\widetilde{u}_n\|_{L^2( Q) }+\sum_{ i,j=0,\, 1\leq i+j\leq2} ^2 \| \partial_{x_1}^{j}\partial_{x_2}^{j}\widetilde{u}_n\|_{L^2(Q) }\leq C\| f\|_{L^2(Q) }.$ This means that $\widetilde{u}_n$, $\partial_t\widetilde{u}_n$, $\partial_{x_1}^{j}\partial_{x_2}^{j}\widetilde{u}_n$ for $1\leq i+j\leq 2$ are bounded functions in $L^2(Q)$. So for a suitable increasing sequence of integers $n_k$, $k=1,2,\dots$, there exist functions $u, v, v_{i,j}$, $1\leq i+j\leq 2$ in $L^2(Q)$ such that \begin{gather*} \widetilde{u}_{n_k} \rightharpoonup u \quad \text{weakly in $L^2(Q)$ as $k\to \infty$} \\ \partial_t\widetilde{u}_{n_k} \rightharpoonup v \quad \text{weakly in $L^2(Q)$ as $k\to \infty$} \\ \partial_{x_1}^{j}\partial_{x_2}^{j}\widetilde{u}_{n_k} \rightharpoonup v_{i,j} \quad \text{weakly in $L^2(Q)$ as $k\to \infty$, $1\leq i+j\leq 2$.} \end{gather*} Clearly, $v=\partial_tu,\quad v_{i,j}=\partial_{x_1}^{i}\partial _{x_2}^{j}u , \quad 1\leq i+j\leq 2$ in the sense of distributions in $Q$ and so in $L^2(Q)$. So, $u\in H^{1,2}(Q)$ and $\partial_tu-\partial_{x_1}^2u-\partial_{x_2}^2u=f\quad \text{in }Q.$ On the other hand, the solution $u$ satisfies the boundary conditions $u\big|_{\partial Q-\Gamma_T}=0$ since $u\big|_{Q_n}=u_n$ for all $n\in \mathbb{N}^*$. This proves the existence of a solution to Problem \eqref{leProbleme}. \end{proof} \section{Global in time result} Assume that $Q$ satisfies \eqref{RaccordCondition}. In the case where $T$ is not in the neighborhood of zero, we set $Q=D_1\cup D_2\cup \Gamma_{T_1}$ where \begin{gather*} D_1=\{ (t,x_1,x_2) \in \mathbb{R} ^3:0