\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 151, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/151\hfil Hyperbolic equations in cusp domains] {$L_p$-Regularity of solutions to first initial-boundary value problem for hyperbolic equations in cusp domains} \author[N. M. Hung, V. T. Luong\hfil EJDE-2009/151\hfilneg] {Nguyen Manh Hung, Vu Trong Luong} % in alphabetical order \address{Nguyen Manh Hung \newline Faculty of Mathematics, Hanoi National University of Education, Vietnam} \email{hungnmmath@hnue.edu.vn} \address{Vu Trong Luong \newline Department of Mathematics, Taybac University, Sonla City, Vietnam} \email{vutrongluong@gmail.com} \thanks{Submitted July 24, 2009. Published November 25, 2009.} \subjclass[2000]{35D05, 35D10, 35L30, 35L35, 35G15} \keywords{Hyperbolic equations; first initial-boundary value problem; \hfill\break\indent $L_p$-regularity; cusp domain} \begin{abstract} In this article, we establish well-posedness and $L_p$-regularity of solutions to the first initial-boundary value problem for general higher order hyperbolic equations in cylinders whose base is a cusp domain. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction}\label{sec1} Initial boundary-value problems for hyperbolic and parabolic type equations in a cylinder with base containing conical point have been studied by many authors \cite{H1, H2, H3, H4, KK}. The main results are about the uniqueness and existence of the solutions, and asymptotic expansions of the solution near a neighborhood of conical point. Those results are mainly based on Galerkin's approximate method and $L_2$-theory. Boundary-value problems for elliptic type equations and systems have also well studied. The main results, presented in \cite{K, M, M1, S2}, established estimates in $L_p$ for solutions of elliptic boundary value problems in domains with singular points on the boundary. The question is whether similar results can be obtained based on these results for initial boundary-value problems for non-stationary equations. In this paper, we find the answer for this question. Firstly, we show the existence of a sequence of smooth domains $\{\Omega^\epsilon\}_{\epsilon>0}$ such that $\Omega^\epsilon\subset \Omega$ and $\lim_{\epsilon\to0}\Omega^\epsilon=\Omega$. Furthermore, we proved existence, uniqueness and smoothness, with respect to time variable, of the generalized solution by approximating boundary method, which can be applied for non-linear equations. Next, by modifying the arguments in \cite{M1}, we take the term containing the derivative in time of the unknown function to the right-hand side of the equation, such that the problem can be considered as an elliptic problem. With the help of some auxiliary results, we apply the estimates in $L_p$ for solution of the elliptic boundary value problem and our previous estimates to deal with the $L_p$-regularity with respect to both of time and spatial variables of the solution. Finally, in order to illustrate the results above we show an example for the Cauchy-Dirichlet problem for the beam equation in cylinder with base containing a cuspidal point. \section{Preliminaries}\label{sec2} Let $\Omega$ be bounded domain in $\mathbb{R}^n$, $n\geq 2$, with boundary $\partial\Omega$. Let $p, q$ be real numbers with $10$, such that $$\label{GD.1} \sup \{\big|B_1[u,\eta]\big|:\eta\in {\mathaccent"7017 W}_q^{m,1 }(Q_T), \|\eta\|_{m,1;q}\leq 1\}\geq \gamma_2\|u\|_{m,1;p},\;$$ for all $u\in {\mathaccent"7017 W}_p^{m,1}(Q_T)$. \end{lemma} \begin{proof} We prove this result with $u\in C_0^\infty(Q_T)$. Suppose that there is no $\gamma_2>0$ such that \eqref{GD.1} holds. Then there is a sequence $\{u_k\}\subset C_0^\infty(Q_T)$ with $\|u_k\|_{m,1;p}=1$ and $$\label{eq.6} \sup \{\big|B_1[u_k,\eta]\big|:\eta\in {\mathaccent"7017 W}_q^{m,1 }(Q_T), \|\eta\|_{m,1;q}\leq 1\}\leq \frac{1}{k}, \quad \text{for every } k\geq 1.$$ Using Garding' inequality \eqref{GD}, we obtain $$\label{eq.7} \big|B_1[u_k,u_k]\big|\geq \gamma_0\|u_k\|^2_{m,0;2} +\int_{Q_T}|u_{kt}|^2\,dx\,dt \geq c_1 \|u\|^2_{m,1;2}.$$ On the other hand, by using H\"{o}lder's inequality with $10$. Combining (\ref{eq.7}) and (\ref{eq.8}), we obtain $$\big|B_1[u_k,u_k]\big|\geq C\|u_k\|^2_{m,1;p},$$ where $c$ is a constant independent of $k$. From the above inequality and (\ref{eq.6}), we have $$\|u_k\|^2_{m,1;p}\leq \frac{1}{k\,C},\quad \text{for } k= 1, 2, \dots.$$ which contradicts $\|u_k\|_{m,1;p}=1$. Therefore, there is a constant $\gamma_2>0$ such that \eqref{GD.1} holds. Since $u\in C_0^\infty(Q_T)$ which is dense in ${\mathaccent"7017 W}_p^{m,1}(Q_T)$, this completes the proof. \end{proof} Lemma \ref{lem2.1} implies the uniqueness of generalized $L_p$-solution, according to the following theorem. \begin{theorem}\label{Theo1} Assume that coefficients of operator \eqref{PDO} satisfy \eqref{GD} and $f\in W^{-m,-1}_p(Q_T)$. Then \eqref{eq.2} -\eqref{eq.4} has at most one generalized $L_p$-solution. \end{theorem} \begin{proof} Firstly, we prove the theorem in the case $12$. Since $p>2$, and $Q_T$ is bounded, we have ${\mathaccent"7017 W}_p^{m,1 }(Q_T)\hookrightarrow {\mathaccent"7017 W}_2^{m,1 } (Q_T)$. Therefore, if $u$ is a generalized $L_p$-solution, and then $u$ is a generalized $L_2$-solution. We obtain the uniqueness of a generalized $L_p$-solution from the uniqueness of a generalized $L_2$-solution. Hence, $u\equiv 0$ in $Q_T$. This completes the proof of theorem. \end{proof} Next, we prove the approximate boundary lemma, which is the essential tool in solving \eqref{eq.2} -\eqref{eq.4}. \begin{lemma}[\cite{L3}] \label{A} Let $\Omega$ be a bounded domain in $\mathbb{R}^n$; then there exists a sequence of smooth domains $\{\Omega^\varepsilon\}$ such that $\Omega^\varepsilon\subset\Omega$ and $\lim_{\varepsilon\to 0} \Omega^\varepsilon =\Omega$. \end{lemma} \begin{proof} For $\varepsilon>0$ arbitrary, set $S^\varepsilon =\{x\in\Omega: \text{dist}(x, \partial \Omega)\leq \varepsilon\}, \Omega^\varepsilon=\Omega\setminus S^\varepsilon$ and $\partial\Omega^\varepsilon$ is the boundary of $\Omega^\varepsilon$. Denote by $J(x)$ the characteristic function of $\Omega^\varepsilon$ and by $J_h(x)$ the mollification of $J(x)$; i.e., $$J_h(x)= \int_{\mathbb{R}^n}\theta_h(x-y)J(y)dy,$$ where $\theta_h$ is a mollifier. If $h<\varepsilon/2$, then $J_h(x)$ has following properties: \begin{itemize} \item [(1)] $J_h(x)=0$ if $x\notin\Omega^{\frac{\varepsilon}{2}}$; \item [(2)] $0\leq J_h(x)\leq1, \forall x\in \Omega$; \item [(3)] $J_h(x)=1$ in $\Omega^{2\varepsilon}$; \item [(4)] $J_h\in C_0^\infty(\mathbb{R}^n)$. \end{itemize} We now fix a constant $c\in(0,1)$, set $\Omega_c^\varepsilon=\{x\in\Omega: J_h(x)>c\}$. It is obvious that $\Omega^{\frac{\varepsilon}{2}}\supset\Omega^\varepsilon_c \supset\Omega^{2\varepsilon}$. Therefore, $\Omega_c^\varepsilon\subset\Omega$ and $\lim_{\varepsilon\to 0}\Omega_c^\varepsilon=\Omega$. Assume that $K$ is the critical set of $J_h$, i.e. $K$ consisting of all point $x$, such that the gradient of $J_h$ at $x$ vanishes. A number $c\in\mathbb{R}$ such that $J_h^{-1}(c)$ contains at least one $x\in K$ is called a critical value. By Sard's theorem then the set of critical values of $J_h$ is of measure zero (see\cite[Theorem 1.30]{TA}), it implies that there exists a constant $c_0 \in( 0,1)$ such that $c_0$ is not a critical value of $J_h$. Denote $\Omega^\varepsilon_{c_0}=\{x\in\Omega: J_h(x)>c_0\}$ and $F(x)=J_h(x)-c_0$. For all $x^0\in \partial\Omega_{c_0}^\varepsilon$, then $F(x^0)=J_h(x^0)-c_0=0$ and vector $\text{grad}J_h(x^0)\ne 0$. This implies that there exists a $\frac{\partial J_h}{\partial x_i}(x^0)\ne 0$, without loss of generality we can suppose that $\frac{\partial J_h}{\partial x_n}(x^0)\ne 0$. Using the implicit function theorem, we obtain that there exists a neighborhood $\mathcal{W}$ of $(x^0_1, \dots, x^0_{n-1})$ in $\mathbb{R}^{n-1}$ a neighborhood $\mathcal{V}$ of $x_n^0$ in $\mathbb{R}$ and an infinitely differentiable function $z:\mathcal{W}\to \mathbb{R}$ such that $x\in \mathcal{U}_{x^0}\cap \partial\Omega_{c_0}^\varepsilon$, where $\partial\Omega_c^\varepsilon=\{x\in\Omega: J_h(x)=c\}$, $\mathcal{U}_{x^0}=\mathcal{W}\times \mathcal{V}$, if and only if $x=(x_1, \dots, x_n)\in U_{x^0}$, $x_n=z(x_1, \dots, x_{n-1})$. Hence, $\Omega_{c_0}^\varepsilon$ is smooth and $\lim_{\varepsilon\to 0}\Omega_{c_0}^\varepsilon=\Omega$. The lemma proved. \end{proof} Suppose that $\{\Omega^\epsilon\}$ is a smooth domain subsequence and $\lim_{\varepsilon\to 0}\Omega^\varepsilon=\Omega$. Set $Q^\epsilon_T=\Omega^\epsilon\times(0,T), S^\epsilon_T=\partial\Omega^\epsilon\times(0,T)$. It is known that the problem \begin{gather*} Lu-u_{tt}=f \text{ in } Q^\epsilon_T, \\ u =0, u_{t} =0\text{ on }\Omega^\epsilon, \\ \partial^j_\nu u =0\text{ on }S^\epsilon_T, j=0,1, \dots, m-1, \end{gather*} has a unique function $u^\epsilon(x,t)\in C^\infty(\overline{Q^\epsilon_T})$; if $f\in C^\infty(\overline{Q^\epsilon_T}), f_{ t^k}\big|_{t=0}=0$, for $k=0,1,\dots$. Moreover, $u^\epsilon(.,t) \in {\mathaccent"7017 W}_2^{m}(\Omega^\epsilon)$, for all $t\in [0,T]$, (see\cite{F, LD, KR}). \section{Main results}\label{sec3} \subsection{Existence of generalized $L_p$-solutions} In this subsection, we prove the existence of generalized $L_p$-solution. Firstly, we prove the needed following propositions: \begin{proposition}\label{pro.1a} Suppose that $10}$ is uniformly bounded in the space ${\mathaccent"7017 W}_p^{m,1}(Q_T)$. So we can take a subsequence, denoted also by $\widetilde{u}^{\varepsilon}$ for convenience, which converges weakly to a function $u \in {\mathaccent"7017 W}_p^{m,1 }(Q_T)$. We will show that $u$ is a generalized $L_p$-solution of \eqref{eq.2}-\eqref{eq.4} in cylinder $Q_T$. In fact for all $\eta\in {\mathaccent"7017 W}_q^{m,1 }(Q_T),$ there exists $\eta_\delta\in C_0^\infty(Q_T)$ such that $\eta_\delta\equiv 0$ in $Q_T\setminus Q^\varepsilon_T,$ and $\|\eta_\delta-\eta\|_{m,1;q}\to 0$ when $\delta\to 0$. Since $\widetilde{u}^{\varepsilon}$ is a generalized solution of \eqref{eq.2}-\eqref{eq.4} in the smooth cylinder $Q^\varepsilon_T$, we have $$B_1[\widetilde{u}^\varepsilon, \eta_{\delta}] =\langle f,\eta_{\delta} \rangle$$ Passing to the limit when $\varepsilon\to 0,\delta\to0$ for the weakly convergent sequence, we get $$B_1[u, \eta ]=\langle f,\eta \rangle$$ Since ${\mathaccent"7017 W}_p^{m,1 }(Q_T)$ is imbedded continuously into $L_p(\Omega)$, the trace sequence $\{\widetilde{u}^{\varepsilon}(x,0)\}$ of $\{\widetilde{u}^\varepsilon(x,t)\}$ converges weakly to the trace $u(x,0)$ of $u(x,t)$ in $L_p(\Omega )$. On the other hand, $\widetilde{u}^{\varepsilon}(x,0)=0$, so that $u(x,0)=0$;by analogous arguments, we have $u_t(x,0)=0$. Hence, $u(x,t)$ is a generalized $L_p$-solution of \eqref{eq.2}-\eqref{eq.4}. Moreover, from (\ref{eq.5c}) we have $$\|u\|_{m,1;p} \leq\varliminf_{\varepsilon\to0} \|\widetilde{u}^{\varepsilon}\|_{m,1;p}\leq C \|f \|_{-m,-1;p}.$$ \end{proof} Proposition \ref{pro.1b} stated the existence of generalized $L_p$-solutions of \eqref{eq.2}-\eqref{eq.4} in ${\mathaccent"7017 W}_p^{m,1 }(Q_T)$ when $f\in C^\infty(\overline{Q}_T)$ and $f_{ t^k}\big|_{t=0}=0$, for $k=0,1,\dots$. We now establish the problem when $f\in W^{-m,-1}_p(Q_T )$. \begin{theorem}\label{Theo2} Suppose that $f\in W^{-m,-1}_p(Q_T ), p\in(1,+\infty)$, then \eqref{eq.2}-\eqref{eq.4} has a generalized $L_p$-solution $u\in{\mathaccent"7017 W}_p^{m,1 }(Q_T)$, and $$\| u\|^p_ {m,1;p} \leq C \|f\|^p_{0,p},$$ where $C$ is a constant independent of $u$ and $f$. \end{theorem} \begin{proof} We start by studying the case 1h \\ 0, & t\leq h \end{cases} $$for all h>0. We denote by g_{\frac{h}{2}} the mollification of f_h. Then g_{\frac{h}{2}}\in C_0^\infty(Q_T), g_{\frac{h}{2}}\equiv 0, t<\frac{h}{2} and g_{\frac{h}{2}} \to f in W^{-m,-1}_p(Q_T ) . By Proposition \ref{pro.1b}, problem \eqref{eq.2}-\eqref{eq.4} has a generalized L_p-solution u_h\in\overset{\circ}{W}_p^{m,1 }(Q_T) with replacing f by g_{\frac{h}{2}}, and the following estimates holds $$\label{eq.6a} \| u_h\|_ {m,1;p} \leq C \| g_{\frac{h}{2}} \|_{-m,-1;p}$$ where C is a constant independent of h, u and f. Since \{g_{\frac{h}{2}} \} is a Cauchy sequence in L_p(Q_T) and inequality (\ref{eq.6a}), it follows that \{u_h\} is a Cauchy sequence in \overset{\circ}{W}_p^{m,1 }(Q_T). Hence, u_h\to u\in \overset{\circ}{W}_p^{m,1 }(Q_T), then u is a generalized L_p-solutions of \eqref{eq.2}-\eqref{eq.4} and satisfies$$ \| u\|_ {m,1;p} \leq C \|f\|_{-m,-1;p}\,. $$Thus, the theorem is proved in the case 12. It is clear that q=\frac{p}{p-1}\in(1,2); by the proof above, for any g\in W_q^{-m,-1 }(Q_T) there exists a solution v\in {\mathaccent"7017 W}_q^{m,1 }(Q_T) of the adjoint problem $$\label{eq.6b} B_1[v,u]=\langle g,u\rangle$$ for all u\in {\mathaccent"7017 W}_p^{m,1 }(Q_T), and$$ \| v\|_ {m,1;q} \leq C \| g \|_{-m,-1;q}. We suppose that f\in C^\infty(\overline{Q_T}), f_{t^k}(x,0)=0, k=0,1, \dots and for u=u^\epsilon in (\ref{eq.6b}). Then, by (\ref{eq.6b}), we have \begin{align*} |\langle g, u^\epsilon\rangle| &=|B_1[v,u^\epsilon]|=|\overline{B_1[u^\epsilon,v]}| =|\langle f, v\rangle|\\ &\leq \| f \|_{-m,-1;p}\| v\|_ {m,1;q}\\ &\leq C\| f \|_{-m,-1;p}\| g \|_{-m,-1;q} \end{align*} for any g\in W_q^{-m,-1 }(Q_T). This implies \|u^\epsilon\|_{m,1;p} =\sup\big\{\frac{|\langle g, u^\epsilon\rangle|}{\| g \|_{-m,-1;q}}: \;0\ne g\in W_q^{-m,-1 }(Q_T)\big\}\leq C\| f \|_{-m,-1;p}. $$From this inequality and arguments analogous to proofs above, we get the proof of the theorem in this case. The proof is complete. \end{proof} We should remark that by replacing the condition f\in W_p^{-m,-1 }(Q_T) by condition f\in L_p(Q_T), and noting that$$ \|f\|_{W_p^{-m,-1 }(Q_T)}\leq \|f\|_{L_p(Q_T)}, $$we obtain the following theorem. \begin{theorem}\label{Theo3} If f\in L_p(Q_T), p\in(1,+\infty), then \eqref{eq.2}-\eqref{eq.4}, in the cylinder Q_T, has a generalized L_p-solution u\in \overset{\circ}{W_p^{m,1 }}(Q_T) which satisfies \begin{equation*} \| u\|_ {m,1;p} \leq C \|f \|_{L_p(Q_T)} \end{equation*} where C is a constant independent of u and f. \end{theorem} \subsection{Smoothness of the generalized L_p-solution with respect to time} The following theorem shows that the generalized L_p-solution u\in {\mathaccent"7017 W}_p^{m,1}(Q_T) of problem \eqref{eq.2}-\eqref{eq.4} is smooth with respect to time variable t if right hand-side f and coefficients of operator \eqref{PDO} are smooth enough with respect to t. \begin{theorem}\label{Theo4} Let h be a positive integer, and assume that \begin{itemize} \item[(1)] f_{t^k} \in L_p(Q_T), k\leq h, \item[(2)] f_{t^k}\big|_{t=0}=0,x\in \Omega, k\leq h-1, \item[(3)] \sup\big\{\big| \frac{\partial^k a_{\alpha\beta}}{\partial t^k}\big|, k2. It is easy to recognize that q=\frac{p}{p-1}\in(1,2); by Theorem \ref{Theo2}, for any g\in W_q^{-m,-1 }(Q_T) there exists a solution v\in {\mathaccent"7017 W}_q^{m,1 }(Q_T) of the adjoint problem $$\label{eq.7d} B_1[v,u]=\langle g,u\rangle$$ which for all u\in {\mathaccent"7017 W}_p^{m,1 }(Q_T), and$$ \| v\|_ {m,1;q} \leq C \| g \|_{-m,-1;q}. We assume that f\in C^\infty(\overline{Q_T}), f_{t^k}(x,0)=0, k=0,1, \dots and for u=u^\epsilon_{t^h} in (\ref{eq.7d}). Then, by (\ref{eq.7d}) and (\ref{eq.7c}), we have \begin{align*} |\langle g, u^\epsilon_{t^h}\rangle| &=|B_1[v,u^\epsilon_{t^h}]|=|\overline{B_1[u^\epsilon_{t^h},v]}|\\ &\leq C(\|f_{t^h}\|_{L_p(Q_T)} +\sum_{k=0}^{h-1}\|u^\epsilon_{t^k}\|_{m,1;p}) \|v\|_{m,1;q}\\ &\leq C(\|f_{t^h}\|_{L_p(Q_T)} +\sum_{k=0}^{h-1}\|u^\epsilon_{t^k}\|_{m,1;p}) \| g \|_{-m,-1;q} \end{align*} for any g\in W_q^{-m,-1 }(Q_T). Hence, \begin{align*} \|u^\epsilon_{t^h}\|_{m,1;p} &=\sup\big\{\frac{|\langle g, u^\epsilon_{t^h}\rangle|} {\| g \|_{-m,-1;q}}: \;0\ne g\in W_q^{-m,-1 }(Q_T)\big\}\\ &\leq C(\|f_{t^h}\|_{L_p(Q_T)} +\sum_{k=0}^{h-1}\|u^\epsilon_{t^k}\|_{m,1;p}). \end{align*} From this inequality and induction assumption, we have the proof of this case, and complete the proof. \end{proof} \subsection{Regularity of the generalized L_p-solution}\label{sec4} In this section, we consider problem \eqref{eq.2}-\eqref{eq.4} in cylinders Q_T=\Omega\times (0,T), where its base \Omega is described as follows: Let \varphi be an infinitely differentiable positive function on the interval (0,1] satisfying the conditions \begin{itemize} \item[(i)] \lim_{\tau\to0}\varphi(\tau)^{k-1}\varphi(\tau)^{(k)}<\infty for k=1,2, \dots; \item[(ii)] \int_0^1\frac{d\tau}{\varphi(\tau)}=+\infty \end{itemize} These conditions are satisfied, for example, by the function \varphi(\tau)=\tau^\alpha if \alpha\geq1. Obviously, conditions (i) and (ii) imply \varphi(0)=0. Suppose that \Omega is a bounded domain in \mathbb{R}^n (n\geq 2),\partial \Omega\backslash\{\mathcal{O}\} is smooth, and \{x\in\Omega:\;0< x_n<1\}=\{x\in\mathbb{R}^n: x_n<1,\; x'\in\varphi(x_n)\omega\}, $$where x'=(x_1,\dots,x_{n-1}), \omega is a smooth domain in \mathbb{R}^{n-1}. Then the mapping $$\label{map} y_j=\frac{x_j}{\varphi(x_n)}, \text{ if } j=1,\dots,n-1,\quad \text{and}\quad y_n= \int_{x_n}^1\frac{d\tau}{\varphi(\tau)}$$ takes the set \{x\in\Omega:\; 00\}=\omega\times (0,+\infty). Moreover, it follows that$$ \det \Big(\frac{\partial y_j}{\partial x_k}\Big)_{j,k=1,\dots,n}=\varphi(x_n)^{-n}. $$It is known that the function \varphi can be extended to an infinitely differentiable positive function on the interval (0,+\infty). To consider the problem, we need to introduce some weighted Sobolev spaces. The space W^l_{p,\beta,\gamma}(\Omega) can be defined as the closure of the set C_0^\infty(\overline{\Omega}\backslash\{\mathcal{O}\}) with respect to the norm$$ \|u\|_{W^l_{p,\beta,\gamma}(\Omega)} =\Big(\int_{\Omega} \sum_{|\alpha|\leq l}e^{p\beta y_n(x_n)}\varphi(x_n)^{p(\gamma-l+|\alpha|)} |D^\alpha u|^p\,dx\Big)^{1/p}. $$Let X, Y be Banach spaces, we denote by L_p(0,T;X) the spaces consisting of all measurable functions u:(0,T)\to X with norm$$ \|u\|_{L_p(0,T;X)}=\Big(\int^T_0\|u(t)\|^p_X\,dt\Big)^{1/p}, $$and by W_p^k(0,T;X,Y), k=1,2, the spaces consisting of functions u\in L_p(0,T;X) such that generalized derivatives u_{t^k}=u^{(k)} exist and belong to L_p(0,T;Y), (see \cite{EV}), with norm$$ \|u\|_{W_p^k(0,T;X,Y)}=\Big( \|u\|^2_{L_p(0,T;X)} +\sum_{j=1}^k\|u_{t^j}\|^p_{L_p(0,T;Y)}\Big)^{1/p}. $$For short notation, we set \begin{gather*} V_p^l(\Omega)=W_{p,0,0}^l(\Omega), \quad V_p^{l,k}(Q_T)=W_p^k(0,T; V_p^l(\Omega) ,L_p(\Omega)),\\ W_{p,\beta,\gamma}^{l,k}(Q_T)= W_p^k(0,T; W_{p,\beta,\gamma}^l(\Omega), L_p(\Omega)). \end{gather*} Finally, we define the weighted Sobolev space W_{p,\beta,\gamma}^l(Q_T) as the set of functions defined in Q_T such that$$ \|u\|_{W_{p,\beta,\gamma}^l(Q_T)} =\Big(\int_{Q_T}\sum_{|\alpha|+k\leq l} e^{2\beta y(x_n)}\varphi(x_n)^{p(|\gamma-l+\alpha|+k)} |D^\alpha u_{t^k}|^p\,dx\,dt\Big)^{1/p}<+\infty. $$To simplify notation, we continue to write V_p^l(Q_T) instead of W_{p,0,0}^l(Q_T). Moreover, we assume that the functions $$\widehat{a}_{\alpha\beta}(y,.)=\varphi(x(y))^{2m-|\alpha| -|\beta|}a_{\alpha\beta}(x(y),.)$$ satisfy the condition of stabilization for y_n\to +\infty for a.e. t in (0,T)(see\cite[Sec.9]{M1}). Then the coefficients of the operators \widehat{L}(y,t;\,D_y), which arises from the operators \varphi(x_n)^{2m}L(x,t;D_x) via the coordinate change x\to y, stabilize for y_n\to +\infty. If we replace the coefficients of the differential operator \widehat{L}(y,t;\,D_y) by their limits for y_n\to +\infty, we get differential operator (denote also by \widehat{L}(y',t;\,D_{y'}, D_{y_n}) for convenience) which has coefficients depending only on y' and t. By the following proposition, we can apply the results of the Dirichlet problem to elliptic equations in domains with cuspidal points on boundary. \begin{proposition} \label{pro.1} Suppose that u=u(x,t) is a generalized solution of problem \eqref{eq.2} -\eqref{eq.4} and u_{tt}\in L_p(Q_T). Then for a.e. t\in(0,T),\,u(t)=u(.,t) is a generalized solution in {\mathaccent"7017 W}_p^m(\Omega) of the Dirichlet problem for elliptic equation $$L(.,t;D_x)u=f_1(.,t) \label{eq4.1}$$ where f_1=u_{tt}+f. \end{proposition} \begin{proof} For any \psi \in {\mathaccent"7017 W}_q^m(\Omega), \theta\in C^\infty_0(0,T) and setting v(x,t)=\psi(x)\theta(t), we substitute the function v(x,t) into (\ref{eq.5}), we conclude that $$\label{eq4.2} \int_{Q_T} \Big[\sum_{|\alpha|,|\beta|=0}^m a_{\alpha\beta}D_x^\beta\, u\overline{D_x^\alpha \psi} -(u_{tt}\,\overline{\psi}+f\overline{\psi})\Big]\overline{\theta(t)} \,dx\,dt= 0.$$ We will denote by$$ \xi(t)= \int_{\Omega} \Big[\sum_{|\alpha|,|\beta|=0}^m a_{\alpha\beta}D_x^\beta\, u\overline{D_x^\alpha \psi } - (u_{tt}\,\overline{\psi}+f\overline{\psi })\Big]\,dx, $$then \xi(t)\in L_p(0,T). Noting that \theta\in C^\infty_0(0,T), which dense in L_q(0,T) and using Fubini's theorem, we obtain from (\ref{eq4.2}) that $$\label{eq4.3} \int_0^T\xi(t)\overline{\theta}(t)\,dt= 0, \quad\text{for any } \theta\in L_q(0,T), (1/p+1/q=1).$$ Therefore,$$ \|\xi\|_{L_p(0,T)}=\sup\big\{\int_0^T\xi(t)\overline{\theta}(t)\,dt: \theta\in L_q(0,T), \|\theta\|_{L_q(0,T)}=1\big\}=0. $$This implies \xi=0 for a.e. t\in(0,T) . Hence,$$ \int_{\Omega} \sum_{|\alpha|,|\beta|=0}^m a_{\alpha\beta}D_x^\beta\, u\overline{D_x^\alpha \psi}\,dx =\int_{\Omega}(u_{tt} +f)\overline{\psi}\,dx $$for all \psi\in {\mathaccent"7017 W}_q^{m}(\Omega), for a.e. t\in (0,T). It follows that u(t) is a generalized solution in {\mathaccent"7017 W}_p^m(\Omega) of the Dirichlet problem for elliptic equation \eqref{eq4.1}, for a.e. t\in (0,T). \end{proof} In this section, we present the main results which is based on our previous subsection and the results for elliptic equations in cusp domains (cf. \cite{M1}). For the start of this section, we denote by \mathcal{U}(\lambda,t) (\lambda\in \mathbb{C},t\in(0,T)) the operator corresponding to the parameter-depending boundary-value problem $$\label{eq4.4} \begin{gathered} \widehat{L}(y',t;\,D_{y'}, \lambda)v=0 \quad \text{in } \omega; \\ \partial^j_\nu v=0\quad \text{on } \partial \omega,\; j=1,\dots, m-1. \end{gathered}$$ Where \widehat{L}(y',t;\,D_{y'},\,\lambda ) is the Fourier transformation y_n\to \lambda of \widehat{L}(y',t;\,D_{y'},\,D_{y_n}). For each t\in(0,T), the operator pencil \mathcal{U}(\lambda,t) is Fredholm, and its spectrum consists of a countable numbers of isolated eigenvalues. The similarly, to Theorem 9.1 in \cite{M1}, we have the following lemma. \begin{lemma}\label{lem.1} Assume that f_1\in W^k_{p,\beta,\gamma}(\Omega), where \beta, \gamma are real numbers. Suppose further that no eigenvalues of \;\mathcal{U}(\lambda,t), t\in(0,T)) line in strip$$ \mathop{\rm Im}\lambda_{-}\leq \mathop{\rm Im}\lambda \leq\mathop{\rm Im}\lambda_{+};\quad \mathop{\rm Im}\lambda_{-}<\beta<\text{ Im}\lambda_{+} $$where \lambda_{+},\,\lambda_{-} are eigenvalues of \mathcal{U}(\lambda,t), and \mathop{\rm Im}\lambda_{-}<0<\mathop{\rm Im}\lambda_{+}. Then the generalized solution u of the Dirichlet problem for the elliptic equation \eqref{eq4.1}, u\equiv 0 if x_n>1, belongs to the space W^{2m+k}_{p,\beta,\gamma}(\Omega) and satisfies the inequality $$\label{eq4.5} \|u\|^2_{W^{2m+k}_{p,\beta,\gamma}(\Omega)} \leq C\|f_1\|^2_{W^k_{p,\beta,\gamma}(\Omega)}$$ where the constant C is independent of f_1. \end{lemma} \begin{proof} Setting \omega_\tau=\varphi(\tau)\omega by the Friederichs inequality, we have$$ \int_{\omega_\tau}|u|^pdx'\leq C \varphi(\tau)^{pk} \sum_{|\gamma|=k}\int_{\omega_\tau}|D^\gamma_{x'}u|^pdx'\,; $$therefore,$$ \varphi(x_n)^{p(|\gamma|-m)}\int_{\omega_{x_n}}|D^\gamma_{x'}u |^pdx' \leq C \sum_{|\alpha|=m}\int_{\omega_{x_n}}|D^\alpha_{x'}u|^pdx' $$for all |\gamma|\leq m. Hence, $$\label{eq4.6} \sum_{|\gamma|\leq m}\int_{\Omega}\varphi(x_n)^{p(|\gamma|-m)} |D_x^\gamma u |^pdx\leq C \sum_{|\alpha|\leq m} \int_{\Omega}|D_x^\alpha u|^pdx$$ Let v=v(y) be the function that arises from \varphi(x_n)^{m-\frac{n}{p}}u(x) via the coordinate change x\to y. We set \overline{\varphi}(y_n)=\varphi(x_n), from the properties of the mapping (\ref{map}) and from inequality (\ref{eq4.6}), it follows that (\overline{\varphi})^{-m+\frac{n}{p}}v\in {\mathaccent"7017 W}_p^m(\mathcal{C_+}). Since (\overline{\varphi})^{-m+\frac{n}{p}}v is the solution of an elliptic equation in \mathcal{C_+} with coefficients which stabilize for y_n\to+\infty, i.e.$$ \widehat{L}(\overline{\varphi})^{-m+\frac{n}{p}}v=\widehat{f_1} $$where \widehat{f_1}=(\overline{\varphi})^{2m }f_1, we obtain (\overline{\varphi})^{-m+\frac{n}{p}}v\in{\mathaccent"7017 W}_p^{2m+k}(\mathcal{C_+}) (cf. \cite[Theorem 8.1, 8.2]{M1}). This implies u\in W^{2m+k}_{p,0,m+k}(\Omega). Using the fact that$$ \varphi(x_n)^{\gamma-m+k}e^{-\epsilon y_n(x_n)}\to 0 $$as x_n\to 0, if 0<\epsilon<\beta, we conclude that u\in W^{2m+k}_{p,-\epsilon,\gamma}(\Omega). In a similar manner, Theorem 8.2 in \cite{M1} it follows that u\in W^{2m+k}_{p,\beta,\gamma}(\Omega). Furthermore, \eqref{eq4.5} is valid. \end{proof} \begin{lemma}\label{lem.2} Suppose that f,f_t\in L_p(Q_T), f(x,0)=0, and the strip \mathop{\rm Im}\lambda_{-}\leq \mathop{\rm Im}\lambda \leq\mathop{\rm Im}\lambda_{+} does not contain eigenvalues of \mathcal{U}(\lambda,t), t\in(0,T)). Then the generalized solution u of problem \eqref{eq.2}-\eqref{eq.4}, u\equiv 0 if x_n>1, belongs to the V_p^{2m,2}(Q_T) and satisfies the inequality $$\label{eq.9} \|u\|_{V_p^{2m,2}(Q_T)}\leq C[\|f\|_{L_p(Q_T) }+\|f_t\|_{L_p(Q_T)}],$$ where the constant C is independent of f. \end{lemma} \begin{proof} Using the smoothness of the generalized solution of \eqref{eq.2}-\eqref{eq.4} with respect to t in Theorem \ref{Theo4} and Proposition \ref{pro.1}, we can see that for a.e. t\in (0,T), u\in{\mathaccent"7017 W}_p^m(\Omega) is the generalized solution of Dirichlet problem for equation \eqref{eq4.1} with compact support, where f_1=u_{tt}+f\in L_p(\Omega)=W^0_{p,0,0}(\Omega)=V_p^0(\Omega). From Lemma \ref{lem.1}, it implies that u\in V_p^{2m}(\Omega) for a.e. t\in(0,T) and satisfies the inequality$$ \|u\|_{V_p^{2m}(\Omega)}\leq C_1\|f_1\|_{L_p(\Omega)} \leq C\left(\|f\|_{L_p(\Omega)}+\|u_{tt}\|_{L_p(\Omega)}\right) . $$By integrating the inequality above with respect to t from 0 to T, and using the estimates for derivatives of u with respect to t again, we obtain u\in V_p^{2m,2}(Q_T), which satisfies (\ref{eq.9}). \end{proof} \begin{theorem} \label{Theo5} Let the assumptions of Lemma \ref{lem.2} be satisfied, and f_{t^k}\in L_p(Q_T), k\leq 2m, f_{t^k}(x,0)=0, for k=0,1,\dots,2m-1. Then the generalized solution u of problem \eqref{eq.2}-\eqref{eq.4}, u\equiv 0 if x_n>1, belongs to the V_p^{2m}(Q_T) and satisfies the inequality $$\label{eq.9a} \|u\|_{V_p^{2m}(Q_T)}\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{L_p(Q_T)}$$ where the constant C is independent of f. \end{theorem} \begin{proof} Let us first prove that u_{t^s} belongs to V_p^{2m,0}(Q_T) for s=0,\dots,2m-1 and satisfy $$\label{eq.10} \|u_{t^s}\|_{V_p^{2m,0}(Q_T)}\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{L_p(Q_T)}.$$ The proof is by done induction on s. According to Lemma \ref{lem.2}, it is valid for s=0. Now let this assertion be true for s-1, we will prove that this also holds for s. Due to Lemma \ref{lem.2} then u satisfies \eqref{eq.2}, by differentiating both sides of \eqref{eq.2} with respect to t, s times, we obtain $$\label{eq.11} Lu_{t^s}=f_{t^s}+u_{t^{s+2}}-\sum_{k=1}^{s}\binom{s}{k}L_{t^k}u_{t^{s-k}}$$ where$$ L_{t^k}=L_{t^k}(x,t;D_x)=\sum_{\alpha,\beta=0}^{m} D_x^\alpha\Big(\frac{\partial^ka_{\alpha\beta}(x,t) } {\partial t^k}\,D_x^\beta\Big). $$By the supposition of the theorem and the inductive assumption, the right-hand side of (\ref{eq.11}) belongs to L_p(Q_T). By the arguments analogous to the proof of Lemma \ref{lem.2}, we get u_{t^s}\in V_p^{2m,0}(Q_T) and $$\label{eq.12} \|u_{t^s}\|_{V_p^{2m,0}(Q_T)}\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{L_p(Q_T)}$$ where C is a constant independent of u,\,f, and s\leq m-1. Using (\ref{eq.12}) and estimates for derivatives of u with respect to t in Theorem \ref{Theo3}, we have $\|u\|_{V_p^{2m}(Q_T)} \leq \sum_{k=0}^{2m-1} \|u_{t^k}\|_{V_p^{2m,0}(Q_T)}+\|u_{t^{2m}}\|_{L_p(Q_T)} \leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{L_p(Q_T)}.$ \end{proof} \noindent\textbf{Remark.} Let \beta be a sufficiently small positive number. Suppose that$$ e^{\beta y_n(x_n)}f\in L_p(Q_T), \quad \mathop{\rm Im}\lambda_{-}< \beta<\mathop{\rm Im}\lambda_{+} $$and the strip$$ \mathop{\rm Im}\lambda_{-}\leq \mathop{\rm Im}\lambda \leq\mathop{\rm Im}\lambda_{+} contains no eigenvalues of \mathcal{U}(\lambda,t), t\in(0,T)); then the generalized solution u of \eqref{eq.2} -\eqref{eq.4}, u\equiv 0 if x_n>1, belongs to the W_{p,\beta,0}^{2m}(Q_T). In fact that, setting u=e^{-\beta y_n(x_n)}U, we obtain the first initial boundary value problem which differs little from \eqref{eq.2}-\eqref{eq.4}. Therefore, U\in V_p^{2m}(Q_T), and then u\in W_{p,\beta,0}^{2m}(Q_T). Using the remark above and Lemma \ref{lem.1}, we obtain the following theorem. \begin{theorem}\label{Theo6} Let the assumptions of Lemma \ref{lem.1} be satisfied. Furthermore, we assume that f_{t^k}\in W^0_{p,\beta,\gamma}(Q_T), k\leq 2m and f_{t^k}(x,0)=0, for k=0,1,\dots,2m-1. Then the generalized solution u of \eqref{eq.2}-\eqref{eq.4}, such that u\equiv 0 if x_n>1, belongs to the W_{p,\beta,\gamma}^{2m}(Q_T) and satisfies the inequality $$\label{eq.12a} \|u\|_{W_{p,\beta,\gamma}^{2m}(Q_T)} \leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W^0_{p,\beta,\gamma}(Q_T)}$$ where the constant C is independent of f. \end{theorem} This theorem is proved by arguments analogous to those proofs of Lemma \ref{lem.2} and Theorem \ref{Theo4}. Next, we will prove the regularity of the generalized solution of problem \eqref{eq.2}-\eqref{eq.4}. \begin{theorem}\label{Theo6b} Let the assumptions of Lemma \ref{lem.1} be satisfied. Furthermore, we assume that f_{t^k}\in W_{p,\beta,\gamma}^{h}(Q_T), k\leq 2m+h and f_{t^k}(x,0)=0, for k=0,1,\dots,2m+h-1,\; h\in \mathbb{N}. Then the generalized solution u of \eqref{eq.2}-\eqref{eq.4}, such that u\equiv 0 if x_n>1, belongs to W_{p,\beta,\gamma}^{2m+h}(Q_T) and satisfies the inequality $$\label{eq.13} \|u\|_{W_{p,\beta,\gamma}^{2m+h}(Q_T)} \leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}$$ where the constant C is independent of u and f. \end{theorem} \begin{proof} The theorem is proved by induction on h. Thanks to Theorem \ref{Theo5}, this theorem is obviously valid for h=0. Assume that the theorem is true for h-1, we will prove that it also holds for h. It is only needed to show that $$\label{eq.14} \begin{gathered} u_{t^s}\in W_{p,\beta,\gamma}^{{2m+h-s},0}(Q_T) \quad\text{for } s=h,h-1\dots, 0;\\ \|u_{t^s}\|_{W_{p,\beta,\gamma}^{2m+h-s}(Q_T)} \leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}. \end{gathered}$$ Differentiating both sides of \eqref{eq.2} again with respect to t, h times, we obtain $$\label{eq.14a} Lu_{t^h}=f_{t^h}+u_{t^{h+2}}-\sum_{k=1}^{h}\binom{h}{k}L_{t^k}u_{t^{h-k}}$$ By the supposition of the theorem and the inductive assumption, the right-hand side of (\ref{eq.14a}) belongs to W^0_{p,\beta,\gamma}(\Omega) for a.e. t\in (0,T). Using Lemma \ref{lem.1}, we conclude that u_{t^h}\in W_{p,\beta,\gamma}^{{2m},0}(Q_T). It implies that (\ref{eq.14}) holds for s=h. Suppose that (\ref{eq.14}) is true for s=h, h-1,\dots, j+1 and set v=u_{t^j}, we obtain $$\label{eq.15} Lv=F_j,$$ where F_j=f_{t^j}+v_{tt}-\sum_{k=1}^{j}\binom{j}{k}L_{t^k}u_{t^{j-k}}. By the inductive assumption with respect to s, v_{tt} belongs to W_{p,\beta,\gamma}^{{h-j}}(\Omega) for a.e. t\in(0,T). Thus, the right-hand side of (\ref{eq.15}) belongs to W_{p,\beta,\gamma}^{h-j}(\Omega). Applying Lemma \ref{lem.1} again for k=h-j, we get that v=u_{t^j}\in W_{p,\beta,\gamma}^{2m+h-j}(\Omega) for a.e. t\in(0,T). It means that v=u_{t^j} belongs to W_{p,\beta,\gamma}^{{2m+h-j},0}(Q_T). Furthermore, we have $$\|v\|_{W_{p,\beta,\gamma}^{{2m+h-j},0}(Q_T)} \leq C\|F_j\|_{W^{h-j,0}_{p,\beta,\gamma}(Q_T)} \leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}.$$ Therefore, \begin{align*} \|u_{t^j}\|_{W_{p,\beta,\gamma}^{2m+h-j}(Q_T)} &\leq\|u_{t^{j+1}}\|_{W_{p,\beta,\gamma}^{2m+h-j-1}(Q_T)} +\|u_{t^j}\|_{W_{p,\beta,\gamma}^{{2m+h-j},0}(Q_T)}\\ &\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}. \end{align*} It implies that (\ref{eq.14}) holds for s=j. The proof is complete. \end{proof} Now we prove the global regularity of the solution. \begin{theorem}\label{Theo7} Let the hypotheses of Lemma \ref{lem.1} be satisfied. Furthermore, assume that f_{t^k}\in W_{p,\beta,\gamma}^{h}(Q_T), k\leq 2m+h and f_{t^k}(x,0)=0, for k=0,1,\dots,2m+h-1, h\in \mathbb{N}. Then the generalized solution u of \eqref{eq.2}-\eqref{eq.4} belongs to W_{p,\beta,\gamma}^{2m+h}(Q_T) and satisfies the inequality $$\label{eq.15a} \|u\|_{W_{p,\beta,\gamma}^{2m+h}(Q_T)} \leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}$$ where the constant C is independent of u and f. \end{theorem} \begin{proof} We denote by B the unit ball and suppose that \zeta\in C^\infty_0(B), and \zeta\equiv 1 in the neighborhood of the origin \mathcal{O}. We have L(\zeta u)-(\zeta u)_{tt}=\zeta f+L_1u $$where L_1 is a differential operator, whose coefficients have compact support in a neighborhood of the origin. By Theorem \ref{Theo6}, we obtain$$ \|\zeta u\|_{W_{p,\beta,\gamma}^{2m+h}(Q_T)} \leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}. $$Setting \zeta_1u=(1-\zeta)u, then \zeta_1u\equiv 0 in a neighborhood of the origin and u=\zeta u+(1-\zeta)u, and using the smoothness of the solution of this problem in domain with smooth boundary, we get$$ \|\zeta_1u\|_{W_p^{2m+h}(Q_T)}\sim \|\zeta_1u\|_{W_{p,\beta,\gamma}^{2m+h}(Q_T)} \leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}. $$The proof is complete . \end{proof} \section{An example}\label{sec5} In this section, we apply the results of the previous section to the Cauchy-Dirichlet problem for the beam equation. Suppose that \Omega is a bounded domain in \mathbb{R}^n,\partial \Omega\backslash\{\mathcal{O}\} is smooth, and$$ \{x\in\Omega: 00 such that $$B[u,u;\,t]=\int_{\Omega}\left( |D^2_xu|^2+|D_x u|^2\right)dx \geq \gamma_0|u|^2_{W{}^2_2(\Omega)}.$$ Hence, \eqref{GD} is satisfied for all $u\in {\mathaccent"7017 W}^2_2(\Omega)$, for all $t\in(0,T)$. For simplicity, we consider \eqref{eq.2a}-\eqref{eq.4a} in the two-dimensional case $(n=2)$, and let $\varphi(\tau)=\tau^2$; Then $\Omega$ is a bounded domain in $\mathbb{R}^2$, $\partial \Omega\backslash\{\mathcal{O}\}$ is smooth, and $$\{(x,y)\in\Omega: 00, \,\eta\in(-1,1)\}. \] With the notation v(\xi,\eta,t)=u(x,y,t), we have$$ u(x,y,t)=v( x^{-1}-1,yx^{-2},t) and \begin{align*} \partial_y u&=x^{-2} \partial_\eta v, \\ \partial_xu &=-x^{-2}\partial_\xi v -2yx^{-3}\partial_\eta v,\\ \partial^2_{yy}u&=x^{-4}\partial_{\eta\eta}^2v\\ \partial_{xx}^2u &=x^{-3}\partial_\xi v +6yx^{-4}\partial_\eta v +x^{-4}\partial_{\xi\xi}^2v +4yx^{-5}\partial_{\xi\eta}^2v+4y^2x^{-6} \partial^2_{\eta\eta} v\\ \\ &=x^{-4}\left[x\partial_\xi v +6y\partial_\eta v +\partial_{\xi\xi}^2v +4yx^{-1}\partial_{\xi\eta}^2v+4y^2x^{-3}\partial^2_{\eta\eta} v\right]\\ \\ &=x^{-4}\big[\partial_{\xi\xi}^2v +4\eta(\xi+1)^{-1} \partial_{\xi\eta}^2v+4\eta^2(\xi+1)^{-2}\partial^2_{\eta\eta} v\\ \\ &\quad+(\xi+1)^{-1}\partial_\xi v +6\eta(\xi+1)^{-2}\partial_\eta v \big]. \end{align*} Hence, the differential operator \widehat{\Delta}, which arises from the differential operator x^{8}\Delta u, (\varphi(x)=x^2, 2m=4)  via the coordinate change (x,y)\to (\xi,\eta), turns out to be \begin{align*} \widehat{\Delta}v&=(\xi+1)^{-4}\left(\partial_{\xi\xi}^2v+\partial_{\eta\eta}^2v\right) +4\eta(\xi+1)^{-5}\partial_{\xi\eta}^2v+4\eta^2(\xi+1)^{-6}\partial^2_{\eta\eta} v\\ \\ &\quad+(\xi+1)^{-5}\partial_\xi v +6\eta(\xi+1)^{-6}\partial_\eta v; \end{align*} the similar calculation for \widehat{\Delta^2} . Clearly, coefficients of differential operator  \widehat{L}=\widehat{\Delta^2} -\widehat{\Delta} stabilize for \xi\to+\infty and the limit differential operator of  \widehat{L}  (denote by \widehat{L} for convenience) is \widehat{L}=\widehat{\Delta^2}v =\partial_{\xi^4}^4v+2\partial_{\eta^2\xi^2}^4v+\partial_{\eta^4}^4v. $$We denote also by \mathcal{U}(\lambda) (\lambda\in \mathbb{C}) the operator corresponding to the parameter-depending boundary value problem \begin{gather*} \frac{d^4v}{d\eta^4}-2\lambda^2\frac{d^2v}{d\eta^2}+\lambda^4v=0,\\ v(-1)=v(1)=0,\\ v'(-1)=v'(1)=0. \end{gather*} It is easy to see that \mathcal{U}(\lambda) is invertible for all \lambda\in \mathbb{C}. From arguments above in combination with Theorem \ref{Theo6} and Theorem \ref{Theo7}, we obtain the following results. \begin{theorem} \label{thm4.1} Suppose that e^{ \beta(\frac{1}{x}-1)}x^{2\gamma} f_{t^k}\in L_p(Q_T), k\leq 2, \beta, \gamma are real numbers and f_{t^k}(x,0)=0, for k=0,1. Then \eqref{eq.2a}-\eqref{eq.4a} has a unique solution u in W^2_{p,\beta,\gamma}(Q_T) and$$ \|u\|_{W^2_{p,\beta,\gamma}(Q_T)}\leq C\sum_{k=0}^2\|e^{ \beta(\frac{1}{x}-1)}x^{2\gamma} f_{t^k}\|_{L_p(Q_T)}. $$Moreover, if f_{t^k}\in W^h_{p,\beta,\gamma}(Q_T), k\leq 2+h, and f_{t^k}(x,0)=0 for k=0,1,\dots, 1+h, then u\in W^{2+h}_{p,\beta,\gamma}(Q_T) and satisfies$$ \|u\|_{W^{2+h}_{p,\beta,\gamma}(Q_T)} \leq C\sum_{k=0}^2 \|f_{t^k}\|_{W^h_{p,\beta,\gamma}(Q_T)}.  \end{theorem} In case boundary when $\Omega$ has cuspidal points, then by arguments analogous to Section \ref{sec3}, we obtain the similar results. \subsection*{Acknowledgments} This work was supported by National Foundation for Science and Technology Development (NAFOSTED), Vietnam. \begin{thebibliography}{99} \bibitem{A} Adams, R. A.; \emph{Sobolev spaces}, Academic Press, New York- San Francisco- London 1975. \bibitem{TA} Aubin, T.; \emph{A course in Differential Geometry.} Grad. Stud. Math., vol. 27., Amer. Math. 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