\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 36, pp. 1--16.\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-2009/36\hfil Asymptotic behavior of solutions] {Asymptotic behavior of solutions to Cauchy-Dirichlet problems for second-order hyperbolic equations in cylinder with non-smooth base} \author[N. M. Hung, B. T. Kim\hfil EJDE-2009/36\hfilneg] {Nguyen Manh Hung, Bui Trong Kim} % in alphabetical order \address{Nguyen Manh Hung \newline Department of Mathematics, Hanoi National University of Education, Vietnam} \email{hungnmmath@hnue.edu.vn} \address{Bui Trong Kim \newline Department of Mathematics, Hanam College of Education, Vietnam} \email{buitrongkim@gmail.com} \thanks{Submitted August 18, 2008. Published March 3, 2009.} \subjclass[2000]{35L05, 35L15, 35L20} \keywords{Generalized solution; asymptotic; conical point on the boundary; \hfill\break\indent non-smooth domains} \begin{abstract} This paper concerns a Cauchy-Dirichlet problem for second-order hyperbolic equations in infinite cylinders with the base containing conical points. Some results on the asymptotical expansions of generalized solutions of this problem are given. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} Boundary-value problems for partial differential equations and systems in domains with smooth boundary have been nearly completely studied in the works \cite{a1,d1}. General boundary-value problems for elliptic equations and systems in domains with conical points were considered by Kondratiev \cite{k1}, Nazarov and Plamenevsky \cite{n1}. The initial boundary-value problems for non-stationary equations and systems on non-smooth domains have been studied by many authors \cite{h1,h2,h3,k2,s1}. The Neumann problem for hyperbolic systems in domains with conical point was described in \cite{k2} and the same problem for the classical heat equation in a dihedral angle was investigated in \cite{s1}. The first initial boundary-value problems for strongly hyperbolic systems in an cylinder with conical point on the boundary of base have been investigated in \cite{h1}, where the problem was only investigated in the finite cylinder. In this paper we consider a Cauchy-Dirichlet problem for second-order hyperbolic equations in infinite cylinders with non-smooth base. First, we study the existence, uniqueness and smoothness with respect to time variable of a generalized solution in the Sobolev space by Galerkin's approximate method. After that, 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 one. We can apply the results of elliptic boundary-value problems to deal with the asymptotic of the solutions. The main goal of this paper is obtaining asymptotical expansions of solutions of the problem. In section 2 we introduce some notations and the formulation of the problem. We receive results on the unique existence and the smoothness with respect to time variable of solutions in section 3 and the asymptotical expansions of the solutions in section 4. Finally, in the last section we apply the results of section 4 to the problems of mathematical physics. \section{Formulation of the problem} Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with the boundary $\partial{\Omega}$. Set $\Omega_t =\Omega\times(0,t)$ for each $t\in{(0,\infty)}$, $\Omega_{\infty} =\Omega\times(0,\infty)$, $S_t=\partial{\Omega}\times(0,t)$ and $S_\infty=\partial{\Omega}\times(0,\infty)$. We use the following notation: For each multi-index $\alpha=(\alpha_1,\dots .\alpha_n)\in{N^n}$, $|\alpha|=\alpha_1 +\dots +\alpha_n$ and $D^{\alpha}u=\frac{\partial^{|\alpha|}u} {\partial{^{\alpha_1}_{x_1}}\dots \partial{^{\alpha_n}_{x_n}}}$ $=u_{x^{\alpha_1}_{1}\dots x^{\alpha_n}_{n}}$ is the generalized derivative up to order $\alpha$ with respect to $x = (x_1,\dots ,x_n)$ ; $u_{t^k} =\frac{\partial{^k}u}{\partial{t^k}}$ is the generalized derivative up to order $k$ with respect to $t$. We begin by recalling some functional spaces which will be used frequently in this paper. $W^{l}(\Omega)$ is the space consisting of all functions $u(x)$, $x\in{\Omega}$, with the norm $$\|u\|_{W^l(\Omega)} = \Big(\sum_{|\alpha| =0}^{l}\int_{\Omega}|D^{\alpha}u|^2 dx\Big)^{1/2}.$$ ${\mathaccent"7017 W}^l(\Omega)$ is the completion of ${\mathaccent"7017 C} C^\infty (\Omega)$ in the norm of the space $W^l(\Omega)$. \\ $W_\beta^l (\Omega)$ is the space consisting of all functions $u(x) = (u_1 (x), \ldots, u_s (x))$ which have generalized derivatives $D^\alpha u_i$, $| \alpha | \le l$, $1 \le i \le s$, satisfying $$\| u\|^2_{W_\beta^l (\Omega)} =\sum_{| \alpha | =0}^l \int_{\Omega} r^{2(\beta +| \alpha | -l)} | D^\alpha u|^2 dx < +\infty.$$ $W^{l,k}(e^{-\gamma t}, \Omega_\infty)$ is the space consisting of functions $u(x,t)$, $(x,t)\in\Omega_\infty$, with the norm $$\|u\|_{W^{l,k}(e^{-\gamma t}, \Omega_\infty)} = \Big(\int_{\Omega_\infty}\big(\sum_{|\alpha|=0}^{l}|D^{\alpha} u|^{2} +\sum_{j=1}^{k} |u_{t^j}|^{2}\big)e^{-2\gamma t}\,dx\,dt\Big)^{1/2},\quad k\geq 1.$$ $W^{l,0}(e^{-\gamma t}, \Omega_\infty)$ is the space consisting of functions $u(x,t)$, $(x,t)\in\Omega_\infty$, with the norm $$\|u\|_{W^{l,0}(e^{-\gamma t}, \Omega_\infty)} = \Big(\int_{\Omega_\infty}\big(\sum_{|\alpha|=0}^{l}|D^{\alpha} u|^{2} \,dx\,dt\Big)^{1/2}.$$ ${\mathaccent"7017 W}^{l,k} (e^{-\gamma t},\Omega_\infty)$ is the closure in $W^{l,k}(e^{-\gamma t},\Omega_\infty)$ of the set consisting of infinite differentiable in $\Omega_\infty$ functions which belong to $W^{l,k}(e^{-\gamma t},\Omega_\infty)$ and vanish near $S_\infty$. \\ $W^{l,k}_{\beta}(e^{-\gamma t},\Omega_\infty)$ is the space consisting of functions $u(x,t)$ satisfying $$\|u\|^{2}_{W^{l,k}_{\beta}(e^{-\gamma t},\Omega_\infty)} = \int_{\Omega_\infty}\big(\sum_{|\alpha|=0}^{l} r^{2(\beta+|\alpha|-l)}|D^{\alpha}u|^{2} + \sum_{j=1}^{k}|u_{t^j}|^{2}\big)e^{-2\gamma t}\,dx\,dt <\infty.$$ Denote by $L^\infty (0, \infty; X)$ the space consisting of measurable functions $u :(0, \infty) \to X, t\mapsto u (x, t)$ satisfying $$\| u\|_{L^\infty (0, \infty; X)} = \mathop{\rm ess\,sup}_{t>0} \big\| u (x, t)\big\|_X < + \infty.$$ Let $L(x,t,D)$ be a differential operator $$L(x,t,D) \equiv \sum_{i,j=1}^{n}\frac{\partial}{\partial x_i} \Big(a_{ij}\frac {\partial}{\partial x_j}\Big)+a, \label{e2.1}$$ where $a_{ij} \equiv a_{ij}(x,t), i,j=1,\dots ,n$ are infinitely differentiable bounded complex-valued functions on $\overline\Omega_{\infty}$, $a_{ij} = \overline a_{ji}$, $a\equiv a(x,t)$ are infinitely differentiable bounded real-valued functions on $\overline\Omega_{\infty}$. Suppose that $a_{ij}, i,j = 1,\dots ,n$, are continuous in $x \in \overline \Omega$ uniformly with respect to $t \in [ 0,\infty)$ and $$\sum_{i,j=1}^{n} a_{ij}(x,t)\xi_i \xi_j \geq \mu_{0}|\xi|^2 \label{e2.2}$$ for all $\xi \in \mathbb{R}^{n}\backslash \{0\}$ and $(x,t) \in \overline\Omega_{\infty}$, where $\mu_0$ is a positive constant. We consider the following problem in the infinite cylinder $\Omega_{\infty}$: \begin{gather} L(x,t,D)u - u_{tt} = f(x,t), \label{e2.3}\\ u|_{t=0} = u_{t}|_{t=0} =0, \label{e2.4} \\ u|_{S_{\infty}} = 0. \label{e2.5} \end{gather} A function $u(x,t)$ is called a generalized solution of the problem \eqref{e2.3}--\eqref{e2.5} in $W^{1,1}(e^{- \gamma t}, \Omega_\infty)$ if $u(x,t)\in {\mathaccent"7017 W}{^{1,1}}(e^{- \gamma t}, \Omega_\infty)$, $u(x,0)=0$ and for each $T>0$ the following equality holds: $$\int_{\Omega_\infty} u_t \overline\eta_{t} \,dx\,dt - \int_{\Omega_\infty}\Big(\sum_{i,j=1}^{n}a_{ij}u_{x_j}\overline \eta_{x_i} -au\overline\eta\Big)\,dx\,dt = \int_{\Omega_\infty}f\ \overline\eta \,dx\,dt \label{e2.6}$$ for all test functions $\eta = \eta (x,t)\in {\mathaccent"7017 W}{^{1,1}}(e^{- \gamma t}, \Omega_\infty)$ such that $\eta(x,t)=0$ with $t\in [T,\infty)$. Set $$B[u,v](t)=\sum_{i,j=1}^ n\int_ {\Omega} a_{ij} u_{x_j} \overline v_{x_i}dx.$$ The following lemma can be proved similarly to Garding's inequality. \begin{lemma} \label{lem2.1} Assume that coefficients $a_{ij}=a_{ij}(x,t)$, $i,j=1\dots n$, $a=a(x,t)$ of the operator $L(x,t,D)$ satisfy condition \eqref{e2.2} and $a_{ij}(x,t)$ are continuous in $x\in \overline\Omega$ uniformly with respect to $t \in [0,\infty)$. Then there exist two constants $\mu_{0} >0$, $\lambda_{0}\geq 0$ such that $$B[u,u](t) \geq \mu_0 \|u(x,t)\|^2_{W^1(\Omega)} - \lambda_0 \|u(x,t)\|^2 _{L_2(\Omega)}$$ for all functions $u=u(x,t)\in {\mathaccent"7017 W}{^{1,0}}(e^{-\gamma t}, \Omega_\infty)$. \end{lemma} \noindent \textbf{Remark.} It follows from the above lemma that if the function $a(x,t)$ in \eqref{e2.1} satisfies $$a(x,t)\le -\lambda_0, {\rm for~ all}~(x,t)\in\Omega_\infty,$$ then $$B_1[u,u](x,t)\equiv B[u,u](t)- \int_\Omega a(x,t)|u(x,t)|^2 dx\geq \mu_0 \|u(x,t)\|^2_{W^1(\Omega)} \label{e2.7}$$ for all functions $u=u(x,t)\in {\mathaccent"7017 W}{^{1,0}}(e^{-\gamma t}, \Omega_\infty)$. \section{Solvability of the problem} In this section we investigate the smoothness of generalized solutions with respect to time. We begin by studying uniqueness of the problem. \begin{theorem} \label{thm3.1} Assume that for a positive constant $\mu$, $$\sup\Big\{\big|\frac{\partial a_{ij}}{\partial t}\big|, \big|\frac{\partial a}{\partial t}\big|: (x,t) \in\Omega_\infty,\; i,j=1\dots ,n\Big\} \leq \mu.$$ In addition, suppose $a(x,t)\le -\lambda_0$, for all $(x,t)\in\Omega_\infty$. Then \eqref{e2.3}-\eqref{e2.5} has at most one generalized solution in $W^{1,1}(e^{-\gamma t},\Omega_\infty)$ for $\gamma>0$ arbitrary. \end{theorem} \begin{proof} Suppose that there are two solutions $u_1,u_2$ in ${\mathaccent"7017 W}{^{1,1}}(e^{-\gamma t},\Omega_\infty)$. Putting $u=u_1-u_2$, so for each $T>0$ the following equality holds: $$\int_{\Omega_\infty} u_t \overline\eta_{t} \,dx\,dt - \int_{\Omega_\infty}\Big(\sum_{i,j=1}^{n}a_{ij}u_{x_j}\overline \eta_{x_i} -au\overline\eta\Big)\,dx\,dt = 0$$ for all test functions $\eta = \eta (x,t)\in {\mathaccent"7017 W}{^{1,1}}(e^{- \gamma t}, \Omega_\infty)$ such that $\eta(x,t)=0$ with $t\in [T,\infty)$. For $b$ with 0b.\end{cases} It is easy to check that \eta(x,t) \in {\mathaccent"7017 W}{^{1,1}}(e^{- \gamma t}, \Omega_\infty), \eta(x,t)=0 with t\in [T,\infty) and \eta_t(x,t)=u(x,t). We have \begin{align*} \int_{\Omega_\infty}\eta_{tt}\overline\eta_{t} \,dx\,dt - \int_{\Omega_\infty}\Big(\sum_{i,j=1}^{n}a_{ij}\eta_{tx_j}\overline \eta_{x_i} -a\eta_t\overline\eta\Big)\,dx\,dt = 0.\label{e3.1} \end{align*} Adding this equality and its complex conjugate, using a_{ij} = \overline a_{ji}, i,j = 1,\dots n and integrating by parts with respect to t, we obtain $$\int_{\Omega}|\eta_{t}(x,b)|^2dx +B_1[\eta,\eta](x,0) +\int_{\Omega_\infty}\Big(\sum_{i,j=1}^n \frac {\partial a_{ij}}{ \partial t}\eta _{x_j} \overline {\eta_{x_i}}-\frac{\partial a}{\partial t}\eta\overline\eta\Big) \,dx\,dt = 0.\label{e3.2}$$ Putting v_i(x,t)=\int_t^0u_{x_i}(x,s)ds, i=1,\dots ,n, v_0(x,t)=\int_t^0u(x,s)ds, we can write \begin{gather*} \eta_{x_i}(x,t)=\int_b^tu_{x_i}(x,s)ds =v_i(x,b)-v_i(x,t),\quad \eta_{x_i}(x,0)=v_i(x,b).\\ \eta(x,t)=\int_b^tu(x,s)ds =v_0(x,b)-v_0(x,t), \eta(x,0)=v_0(x,b). \end{gather*} Substituting those into \eqref{e3.2}, then using the Cauchy's inequality and \eqref{e2.7}, we obtain \begin{aligned} &\int_{\Omega}|\eta_{t}(x,b)|^2dx+ \mu_0\sum_{i=0}^n\int_\Omega\ |v_i(x,b)|^2dx\\ &\le C_1b\ \sum_{i=0}^n\int_\Omega\ |v_i(x,b)|^2dx+ C_2\int_0^b\Big(\sum_{i=0}^n\int_\Omega |v_i(x,t)|^2dx\Big)dt\\ &\le C_1b\ \sum_{i=0}^n\int_\Omega\ |v_i(x,b)|^2dx+ C_2\int_0^b\Big(\int_{\Omega}|\eta_{t}(x,t)|^2dx +\sum_{i=0}^n\int_\Omega |v_i(x,t)|^2dx\Big)dt\label{e3.3} \end{aligned} where C_1, C_2 are positive constants. Put J(t)=\int_{\Omega}|\eta_{t}(x,b)|^2dx+ \sum_{i=0}^n\int_\Omega\ |v_i(x,b)|^2dx. $$From \eqref{e3.3} we get$$ J(b)\le C \int_0^bJ(t)dt $$for all b\in [0,\mu_0/2C_1], where C is a positive constant. This implies that J(t)\equiv 0 on [0,\mu_0/2C_1] by Gronwall-Bellman's inequality. It follows u_1\equiv u_2 on [0,\mu_0/2C_1], where C_1 does not depend on b. By similar arguments for two functions u_1, u_2 on [\mu_0/2C_1,\tau], we can show that after finite steps u_1\equiv u_2 on [\mu_0/2C_1,\tau]. Since \tau>0 is arbitrary, so u_1=u_2 in W^{1,1}(e^{-\gamma t},\Omega_\infty). The proof is complete. \end{proof} Now, we establish the existence of generalized solution of the mentioned problem by Galerkin's approximate method. We use the notation:$$ \gamma_0=\frac{n\mu}{2\mu_0}, $$where n is dimensional number of the space \mathbb{R}^n, \mu is the constant in theorem 3.1 and \mu_0 is the constant in lemma 2.1. We have following theorem. \begin{theorem} \label{thm3.2} Assume that a(x,t)\le -\lambda_0, for all (x,t)\in\Omega_\infty and the following conditions are fulfilled: \begin{itemize} \item[(ii)] \sup\big\{\big|\frac{\partial^k a_{ij}}{\partial t^k}\big|, \big|\frac{\partial^k a}{\partial t^k}\big|\big\}: (x,t) \in \Omega_\infty,\; i,j=1,\dots ,n;\; k\le h-1\big\} \leq \mu, h\geq 1, \item[(iii)] f_{t^k}\in L^{\infty}(0,\infty; L_{2}(\Omega)),k\leq h, \item[(iv)] f_{t^k}(x,0) = 0, k\leq{h-1}. \end{itemize} Then \eqref{e2.3}--\eqref{e2.5} has a unique generalized solution u(x,t) in W^{1,1}(e^{-\gamma t},\Omega_{\infty}) for every \gamma>\gamma_0. Moreover, u(x,t) has derivatives with respect to t up to order h belonging to {\mathaccent"7017 W}{^{1,1}}(e^{-(2h+1)\gamma t},\Omega_\infty) and the following inequality holds:$$ \|u_{t^h}\|^2 _{W^{1,1}(e^{-(2h+1)\gamma t},\Omega_\infty)} \leq C \sum_{k=0}^h \|f_{t^k}\|^{2}_{L^\infty (0,\infty ; L^2 (\Omega))}, where C is a positive constant independent of u and f. \end{theorem} \begin{proof} The uniqueness follows from theorem 3.1. The existence is obtained using Galerkin's method. Let \{ \varphi_k \}^\infty _{k=1} \subset {\mathaccent"7017 C}^{\infty} (\Omega) be an orthogonal system in L_2 (\Omega) such that its linear closure in W^1(\Omega) is the space {\mathaccent"7017 W}^1(\Omega). For each integer N we consider the function u^N (x,t) = \sum_{k=1}^N C^N _k (t) \varphi_k (x), where (C^N _1 (t),\dots ,C^N _N (t)) is the solution of the ordinary differential system \begin{gather} \int_ {\Omega} u^N _{tt} \overline \varphi_l dx + \int_ {\Omega}(\sum_{i,j=1}^{n} a_{ij} u^N _{x_j} \overline\varphi_{l x_i}- a u^N \overline\varphi_l )dx = - \int_{\Omega} f \overline\varphi_l dx; \quad l=1,\dots ,N, \label{e3.4}\\ C^N _k (0) =\frac {d} {dt} C^N _k (0) =0, \quad k=1,\dots N.\label{e3.5} \end{gather} Let us multiply \eqref{e3.4} by d C^N _k (t) / dt and take the sum with respect to l from 1 to N. Then we integrate the equality obtained with respect to t from 0 to t and add this equality to its complex conjugate. Finally, integrating by part and applying condition \eqref{e3.5}, we obtain \begin{align*} &\int_{\Omega} |u^N _t (x,t)|^2 dx + \int_{\Omega} \big( \sum_{i,j=1}^{n} a_{ij} u^N _{x_j} \overline {u^N _{x_i}} -a u^N \overline {u^N}\big)|_{t=t}dx\\ &=\int_{\Omega_t}\Big(\sum_{i,j=1}^n \frac {\partial a_{ij}}{ \partial t} u^N _{x_j} \overline {u^N _{x_i}} - \frac {\partial a}{ \partial t} u^N \overline {u^N}\Big)\,dx\,dt- 2\mathop{\rm Re} \int_{\Omega_t} f \overline {u^N _t} \,dx\,dt. \end{align*} Using \eqref{e2.7} and Cauchy's inequality, one has \begin{aligned} &\|u^N _t (x,t)\|^2 _{L_2 (\Omega)} + \mu _0 \|u^N (x,t)\|^2 _{W^1 (\Omega)}\\ & \leq \int_{\Omega_t} \big(n\mu \sum_{i=1}^n |u_{x_i}|^2 + \mu |u^N|^2+\delta |u^N_t|^2\big)\,dx\,dt + \frac{t}{\delta} \|f\|^2 _{L^\infty (0, \infty ; L_2 ( \Omega))}\\ &\leq \delta\int _{0}^{t} \big( \|u^N _t \|^2 _{L_2 (\Omega)} + \frac {n\mu}{\delta} \|u^N (x,t)\| ^2 _{W^1 ( \Omega)}\big)dt+ \frac{t}{\delta} \|f\|^2 _{L^\infty (0, \infty ; L_2 ( \Omega))}, \end{aligned}\label{e3.6} where \delta is a positive constant. Choosing \delta=\frac{n\mu}{\mu_0} and putting J^N (t) =\|u^N _t (x,t) \|^2 _{L_2 (\Omega)} + \mu _0 \|u^N (x,t) \|^2 _{ W^1 ( \Omega)}, $$from inequality \eqref{e3.6} we obtain$$ J^N (t) \leq \frac {n \mu}{\mu _0}\int _{0}^{t}J^N (\tau)d \tau + t \Big(\frac { \mu_0}{n\mu} \|f\|^2 _{L_{\infty} (0, \infty ; L_2 (\Omega))}\Big). $$From this inequality and Gronwall-Bellman's inequality it follows that$$ J^N (t) \leq C_1 e^{\frac{n\mu}{\mu_0}t} \|f\|^2 _{L^\infty(0, \infty ; L_2 (\Omega))}, where C_1 is a positive constant independent of N and f. Since e^{-\gamma t}\leq 1 with \gamma>0 and t\geq 0, putting C_0=\min\{\mu_0,1\} we have \begin{aligned} C_0\|u^N \|^2 _{W^{1,1} (e^{- \gamma t}, \Omega_ \infty)} &\leq \|u^N _t (x,t)\|^2 _{ L_2 (\Omega)} + \mu _0 \|u^N (x,t)\|^2 _{W^1 (\Omega)}\\ & \leq C_1 e^{\frac{n\mu}{\mu_0}t }\|f\|^2 _{L^{\infty} (0, \infty ; L_2 (\Omega))}. \end{aligned}\label{e3.8} Let \gamma be a positive constant such that \gamma >\gamma_0=\frac{n\mu}{2\mu_0}. Multiplying both sides of \eqref{e3.8} by e^{-2 \gamma t} and integrating with respect to t from 0 to \infty, we get $$\|u^N \|^2 _{W^{1,1} (e^{- \gamma t}, \Omega_ \infty)} \leq C_2 \|f\|^2 _{L^{\infty} (0, \infty ; L_2 (\Omega))} ,\label{e3.9}$$ where C_2 is a positive constant independent of N and f. From \eqref{e3.9} it follows that there exists a subsequence of the sequence \{u^N\} with converges weakly to a function u(x,t) in the space W^{1,1}(e^{-\gamma t} , \Omega_ \infty). We can check that u(x,t) is a generalized solution of the problem. Now we prove the smoothness with respect to time variable of the generalized solution. We use the induction to show that the following inequalities are true for all integer g \geq 0: \begin{gather} \|u^N _{t^g} (x,t)\|^2 _{W^1 (\Omega)} \leq C_3 e^{\frac{(2g+1) n \mu+\epsilon}{\mu_0-\epsilon} t} \sum_ {k=0}^g \|f_{t^k} \|^2 _{L^{\infty} (0, \infty ; L_2 (\Omega))}, \label{e3.10}\\ \|u^N _{t^g} (x,t)\|^2 _{W^{1,1} (e^{-(2g+1) \gamma_t}, \Omega_{\infty})} \leq C_4 \sum_ {k=0}^ g \|f_{t^k} \|^2 _{L^{\infty} (0, \infty ; L_2 (\Omega))}, \label{e3.11} \end{gather} where 0<\epsilon<\mu_0, C_i is a positive constant independent of N and f; i=3,4. For g=0 inequalities \eqref{e3.10} and \eqref{e3.11} are true according to relations \eqref{e3.8} and \eqref{e3.9}. Assume that s \geq 1 and inequalities \eqref{e3.10} and \eqref{e3.11} hold for all g \leq {s-1}. From identity \eqref{e3.4} we have \int_{\Omega} u^N _{t^{s +2}} \overline \varphi_l dx + \int_{\Omega} \Big( \sum_{i,j=1}^{n} \frac {\partial^s}{\partial t^s} \big(a_{ij} u^N _{x_{j}}\overline \varphi_{l x_i} - au \overline\varphi_l\big)\Big)dx = - \int_{\Omega} {f_{t^s} \overline \varphi_l} dx . Multiplying both sides of this identity by d^{s+1}C^N_l/d t^{s+1} and taking the sum with respect to l from 1 to N and integrating the result over (0,t). Then adding this identity with its complex conjugate, we get \begin{aligned} &\int_{\Omega_t}\frac{\partial}{\partial t}\big(u^N_{t^{s+1}}\bar u^N_{t^{s+1}}\big) \,dx\,dt\\ &+2\mathop{\rm Re} \int_{\Omega_t}\Big(\sum_{i, j=1}^n\big(a_{ij} u^N_{x_j}\big)_{t^s}\big(\bar u^N_{x_i}\big)_{t^{s+1}} -\big(au^N\big)_{t^s}\bar u^N_{t^{s+1}}\Big) \,dx\,dt\\ &= -2\mathop{\rm Re} \int_{\Omega_t}f_{t^s} \bar u^N_{t^{s+1}}\,dx\,dt. \end{aligned}\label{e3.12} Denote \binom{k}{s}=\frac{s!}{k!(s-k)!}. Since a_{ij}=\bar a_{ji} with i, j=1, 2, \dots, n, we have \begin{aligned} &2\mathop{\rm Re}\sum_{i, j=1}^n(a_{ij} u^N_{x_j})_{t^s}(\bar u^N_{x_i})_{t^{s+1}} -(au^N)_{t^s}\bar u^N_{t^{s+1}}\\ &=\frac{\partial}{\partial t}\Big(\sum_{i, j=1}^na_{ij}( u^N_{x_j})_{t^s}(\bar u^N_{x_i})_{t^{s}} -au^N_{t^s}\bar u^N_{t^s}\Big) -\mathop{\rm Re}\sum_{i, j =1}^m\frac{\partial a_{ij}}{\partial t}(u^N_{x_j})_{t^s}(\bar u^N_{x_i})_{t^s} + \frac{\partial a}{\partial t} u^N_{t^s}\bar u^N_{t^s} \\ &+{2 \rm Re}\Big(\sum_{i, j=1}^n\sum_{k=1}^s \binom{k}{s}\frac{\partial}{\partial t}\big(\frac{\partial ^k a_{ij}}{\partial t^{k}}(u^N_{x_j})_{t^{s-k}}(\bar u^N_{x_i})_{t^{s}}\big)-\sum_{k=1}^s\binom{k}{s}\frac{\partial}{\partial t}\big(\frac{\partial^k a}{\partial t^k} u^N_{t^{s-k}}\bar u^N_{t^s}\big)\Big)\\ &-2\mathop{\rm Re}\Big(\sum_{i, j=1}^n\sum_{k=1}^s \binom{k}{s}\big(\frac{\partial ^{k+1} a_{ij}}{\partial t^{k+1}}(u^N_{x_j})_{t^{s-k}}(\bar u^N_{x_i})_{t^{s}}\big)-\sum_{k=1}^s\binom{k}{s}\frac{\partial^{k+1}a}{\partial t^{k+1} }u^N_{t^{s-k}}\bar u^N_{t^s}\Big) \\ & -2\mathop{\rm Re}\Big(\sum_{i, j=1}^n\sum_{k=1}^s \binom{k}{s}\frac{\partial ^k a_{ij}}{\partial t^k}(u^N_{x_j})_{t^{s-k+1}}(\bar u^N_{x_i})_{t^s} - \sum_{k=1}^s\binom{k}{s}\frac{\partial^k a}{\partial t^k} u^N_{t^{s-k+1}} \bar u^N_{t^{s}}\Big). \end{aligned} \label{e3.13} From the condition u^N (x,0) = 0 we get \begin{align*} &\int_{\Omega_t}\frac{\partial}{\partial t}\Big(\sum_{i, j=1}^n a_{ij}( u^N_{x_j})_{t^s}(\bar u^N_{x_i})_{t^{s}} - a u^N_{t^s}\bar u^N_{t^s}\Big)\,dx\,dt\\ &=\int_\Omega\Big(\sum_{i, j=1}^n a_{ij}( u^N_{x_j})_{t^s}(\bar u^N_{x_i})_{t^{s}} - a u^N_{t^s}\bar u^N_{t^s}\Big)\Big|_{t=0}^{t=t}dx =B_1(u^N_{t^s}, u^N_{t^s})(x,t). \end{align*} Therefore, from identities \eqref{e3.12} and \eqref{e3.13} it follows that \begin{align*} & \int_ {\Omega} |u^N _{t^{s+1}} (x,t)|^2 dx + B_1(u^N _{t^s} , u^N _{t^s})(x,t)\\ & = -2\mathop{\rm Re} \sum _{k=1}^{s} \binom{k}{s}\int_ {\Omega}\Big( \sum _{i,j=1}^{n}\frac{\partial ^k a_{ij}}{\partial t^k}(u^N _{x_j})_{t^{s-k}} (\bar {u^N _{x_i}})_{t^s} - \frac{\partial ^k a}{\partial t^k} u^N _{t^{s-k}} \bar {u^N _{t^s}}\Big)|_{t=t} dx \\ &\quad + 2 \mathop{\rm Re}\sum _{k=1}^{s} \binom{k}{s}\int_ {\Omega _t}\Big( \sum _{i,j=1}^{n} \frac{\partial ^{k+1} a_{ij}}{\partial t^{k+1}}(u^N _{x_j})_{t^{s-k}}(\bar {u^N _{x_i})}_{t^s} - \frac {\partial ^{k+1} a}{\partial t^{k+1}} u^N _{t^{s-k }}\bar {u^N _{t^s}}\Big) \,dx\,dt \\ &\quad + 2 \mathop{\rm Re}\sum _{k=1}^{s}\binom{k}{s}\int_ {\Omega _t} \Big(\sum _{i,j=1}^{n}\frac{\partial ^{k} a_{ij}}{\partial t^k}(u^N _{x_j})_{t^{s-k+1}}( \bar {u^N _{x_i})}_{t^s} - \frac{\partial ^k a}{\partial t^k}u^N _ {t^{s-k+1}}\bar {u^N _{t^s}}\Big) \,dx\,dt \\ &\quad +\mathop{\rm Re}\int_ {\Omega _t}\Big(\sum _{i,j=1}^{n}\frac{\partial a_{ij}}{\partial t}(u^N _{x_j})_{t^{s}}\bar { (u^N _{x_i})}_{t^s} + \frac{\partial a}{\partial t} u^N _{t^s} \bar {u^N _{t^s}}\Big)\,dx\,dt -2\mathop{\rm Re} \int_{\Omega _t} f_{t^s} \bar { u^N _{t^{s+1}}} \,dx\,dt. \end{align*} %\label{e3.14} From this equality, \eqref{e2.7} and Cauchy's inequality, we obtain \begin{align*} & \|u^N _{t^{s+1}} (x,t) \|^2 _{L_2 (\Omega)} + \mu_0 \|u^N _{t^{s}} (x,t) \|^2 _{W^1 ( \Omega )} \\ &\leq \epsilon \|u^N _{t^{s}} (x,t) \|^2 _{W^1 (\Omega)} + C_1( \epsilon ) \sum_{k=0}^{s-1} \|u^N _{t^k} (x,t) \|^2 _{W^1 (\Omega)}\\ &\quad +C_2(\epsilon)\sum_{k=0}^{s-1} \int_{0}^{t} \|u^N _{t^k} (x,t) \|^2 _{W^1 (\Omega)}dt\\ &\quad + \int_{\Omega_t}\Big( \big( n \mu (2s+1)+\epsilon\big)\sum_{i=1}^n|(u_{x_i})_{t^s}|^2+\big(\mu (2s+1)+\epsilon\big)|u_{t^s}|^2\Big)\,dx\,dt\\ &\quad +\int_{\Omega_t}\delta|u_{t^{s+1}}|^2\,dx\,dt+\int_{\Omega_t}\frac{1}{\delta}|f_{t^s}|^2\,dx\,dt\\ &\leq \epsilon \|u^N _{t^{s}} (x,t) \|^2 _{W^1 (\Omega)} + C_1( \epsilon ) \sum_{k=0}^{s-1} \|u^N _{t^k} (x,t) \|^2 _{W^1 (\Omega)} \\ &\quad +C_2(\epsilon)\sum_{k=0}^{s-1} \int_{0}^{t} \|u^N _{t^k} (x,t) \|^2 _{W^1 (\Omega)}\,dx\,dt + \int_{0}^{t}\big( n \mu (2s+1)+\epsilon\big)\|u_{t^s}^N(x,t) \|^2 _{W^1 (\Omega)}dt\\ &\quad +\int_0^t\delta\|u_{t^{s+1}}\|^2_{L_2(\Omega)}dt+\int_{\Omega_t}\frac{1}{\delta}|f_{t^s}|^2\,dx\,dt, \end{align*} where \epsilon is a positive constant and C_i(\epsilon) is positive constant that depends \epsilon, i=1,2. From this inequality we obtain \begin{aligned} &\|u^N _{t^{s+1}} (x,t) \|^2 _{L_2 (\Omega)} + (\mu_0-\epsilon)\|u^N _{t^{s}} (x,t) \|^2 _{W^1 ( \Omega )} \\ &\leq C_1( \epsilon ) \sum_{k=0}^{s-1} \|u^N _{t^k} (x,t) \|^2 _{W^1 (\Omega)}\\ &\quad +\delta\int_0^t\frac{(n\mu(2s+1) +\epsilon)}{\delta}\|u_{t^s}(x,t)\|^2_{W^1(\Omega)} +\|u_{t^{s+1}}(x,t)\|^2_{L_2(\Omega)}\Big)dt\\ &\quad +\int_{\Omega_t}\frac{1}{\delta}|f_{t^s}|^2\,dx\,dt + C_2(\epsilon) \sum_{k=0}^{s-1} \int_{0}^{t} \|u_{t^k}^N(x,t) \|^2 _{W^1 (\Omega)} dt, \end{aligned} \label{e3.15} Substituting \delta=\frac{(n\mu(2s+1) +\epsilon)}{\mu_0-\epsilon} into \eqref{e3.15}, one can see that \begin{align*} & \|u^N _{t^{s+1}} (x,t) \|^2 _{L_2 (\Omega) } +(\mu_0 - \epsilon)\|u^N _{t^{s}} (x,t) \|^2 _{W^1 (\Omega) }\\ & \leq \frac{n \mu (2s+1)+\epsilon}{ \mu_0 - \epsilon} \int_0 ^t \Big( \|u^N _{t^{s+1}} (x,t) \|^2 _{L_2 (\Omega) }+ ( \mu_0 - \epsilon)\|u^N _{t^{s}} (x,t) \|^2 _{W^1 (\Omega) }\Big)dt\\ &\quad + C\Big (\sum_{k=0}^{s-1} \|u^N _{t^{k}} (x,t) \|^2 _{W^1 (\Omega) } + \sum_{k=0}^{s-1} \int_0 ^t \|u^N _{t^{k}} (x,t) \|^2 _{W^1 (\Omega) }dt + \int_{\Omega_t}|f_{t^s}|^2\,dx\,dt \Big),\\ \end{align*} where C is a positive constant independent of N and f. From this inequality and by the inductive hypothesis for \eqref{e3.10}, we get \begin{aligned} & \|u^N _{t^{s+1}} (x,t) \|^2 _{L_2 (\Omega) } + ( \mu_0 - \epsilon)\|u^N _{t^{s}} (x,t) \|^2 _{W^1 (\Omega) }\\ & \leq \frac{n \mu (2s+1)+\epsilon}{ \mu_0 - \epsilon }\int_0 ^t \Big( \|u^N _{t^{s+1}} (x,t) \|^2 _{L_2 (\Omega) }+( \mu_0 - \epsilon)\|u^N _{t^{s}} (x,t) \|^2 _{W^1 (\Omega) }\Big)dt\\ &\quad + C_1(1+t) e^{\frac{n\mu (2s-1)+\epsilon}{\mu_0-\epsilon} t }\sum_{k=0}^{s-1} \|f_{t^k} \|^2 _{L^{\infty} ( 0, \infty ; L_2 (\Omega))} +C_2t \|f_{t^s} \|^2 _{L^{\infty} ( 0, \infty ; L_2 (\Omega))},\\ \end{aligned} \label{e3.16} where C_i are positive constants independent of N and f; i = 1,2. Put \begin{gather*} J^N _s (t) = \|u^N _{t^{s+1}} (x,t) \|^2 _{L_2 (\Omega) } + ( \mu_0 - \epsilon)\|u^N _{t^{s}} (x,t) \|^2 _{W^1 (\Omega) },\\ \phi(t)=C_1(1+t) e^{\frac{n\mu (2s-1)+\epsilon}{\mu_0-\epsilon} t } \sum_{k=0}^{s-1} \|f_{t^k} \|^2 _{L^{\infty} ( 0, \infty ; L_2 (\Omega))} +C_2t \|f_{t^s} \|^2 _{L^{\infty} ( 0, \infty ; L_2 (\Omega))}. \end{gather*} From \eqref{e3.16} we have J^N _s (t)\leq \frac{n \mu (2s+1)+\epsilon}{ \mu_0 - \epsilon }\int_0 ^t J^N _s (\tau) d\tau +\phi(t). \label{e3.17} $$From this inequality and Gronwall-Bellman inequality we obtain$$ J^N _s (t)\leq C e^{\frac{n \mu (2s+1)+\epsilon}{ \mu_0 - \epsilon }t}\sum _{k=0}^{s} \|f_{t^k}\|^2 _ {L^{\infty} ( 0, \infty ; L_2 (\Omega))}, where C is a positive constant is independent of N and f. Therefore, \begin{aligned} &\|u^N _{t^{s+1}} (x,t) \|^2 _{L_2 (\Omega) } +(\mu_0 - \epsilon )\|u^N _{t^{s}} (x,t) \|^2 _{W^1 (\Omega) }\\ &\leq Ce^{\frac{n \mu (2s+1)+\epsilon}{ \mu_0 - \epsilon }t }\sum _{k=0}^{s} \|f_{t^k}\|^2 _ {L^{\infty} ( 0, \infty ; L_2 (\Omega))},\label{e3.18} \end{aligned} where C is positive constant independent of N and f. Therefore, \begin{aligned} \|u^N _{t^s} (x,t)\|^2 _{W^1 (\Omega)} \leq C_1 e^{\frac{(2s+1) n \mu+\epsilon}{\mu_0-\epsilon} t} \sum_ {k=0}^s\|f_{t^k} \|^2 _{L^{\infty} (0, \infty ; L_2 (\Omega))}, \end{aligned}\label{e3.19} where C_1 is a positive constant independent of N and f. Hence \eqref{e3.10} holds for s. Since \gamma>\gamma_0=\frac{n\mu}{2\mu_0}, there exists a constant \epsilon such that $$2(2s+1)\gamma_0=\frac{n\mu(2s+1)}{\mu_0}<\frac{n\mu(2s+1) +\epsilon}{\mu_0-\epsilon}<2(2s+1)\gamma. \label{e3.20}$$ Multiplying both sides of \eqref{e3.19} by e^{ -2(2s+1)\gamma t} and integrating it with respect to t from 0 to \infty. Then applying \eqref{e3.20}, we obtain \| u^N _{t^s} (x,t)\|^2 _{W^{1,1} ( e^{-(2s+1) \gamma t} , \Omega_\infty)} \leq C_2 \sum_{k=0}^s \|f_{t^k}\|^2 _{L^{\infty} ( 0, \infty ; L_2 (\Omega))}, $$where C_2 is a positive constant independent of N and f. Hence \eqref{e3.11} holds for s and \eqref{e3.11} is proved. Since C_2 from inequality \eqref{e3.11} are independent of N, relation \eqref{e3.11} yields the inequality. The proof is complete. \end{proof} \section{Asymptotical expansions of solutions} Let \Omega be a bounded domain in \mathbb{R}^n (n \geq 2) with the boundary \partial \Omega. We suppose that \partial \Omega\setminus\{0\} is a smooth manifold and \Omega in a neighborhood of the origin 0 coincides with the cone K=\{x: x/|x| \in G\}, where G is a smooth domain on the unit sphere S^{n-1} in \mathbb{R}^n. Set Q_\infty=\Omega\times (0,\infty) and S_\infty=\partial \Omega\times (0,+\infty). We will use notations: D^\alpha=\partial^{|\alpha|}/\partial^{\alpha_1}_{x_1}\ldots \partial^{\alpha_n}_{x_n} for each multi-index \alpha=(\alpha_1, \ldots, \alpha_n)\in\mathbb{N}^n, |\alpha|=\alpha_1+ \dots+ \alpha_n, u_{t^k}=\partial^ku/\partial t^k, r=|x|=\big(\sum_{k=1}^nx^2_k\big)^\frac{1}{2}. Suppose that w = (w_1,\dots ,w_{n-1}) is a local coordinate system on the unit sphere S^{n-1}. Let L_{0} (0,t,D) be the principal part of the operator L(x,t,D) at the coordinate origin. We can write L_{0} (0,t,D) in the form$$L_{0} (0,t,D)= r^{-2} Q(w,t,D_{w},rD_{r}),$$where Q(w,t,D_{w},rD_{r}) is the linear operator with smooth coefficients, D_{r} =i\partial/\partial_r D_{w}=\partial/\partial {w_1}\dots \partial{ w_{n-1}}. Consider the spectral problem \begin{gather} Q(\omega,t,\lambda,D_w)v(w)=0, w\in G, \label{e4.1}\\ v|_{\partial G} = 0. \label{e4.2} \end{gather} It is well known that for every t\in [0,\infty) its spectrum is discrete \cite{a1}. In the cone K we consider Dirichlet problem for the equation $$L_0(0,t,D)u=r^{-i\lambda(t)-2}\sum_{s=0}^M \ln ^srf_s(\omega,t),\label{e4.3}$$ The following lemma can be found in \cite{n1}. \begin{lemma} \label{lem4.1} Assume that f_s(\omega,t), s=0,\dots ,M are infinitely differentiable functions with respect to \omega. Then there exists the solution of the Dirichlet problem for \eqref{e4.3} in the form $$u(x,t)=r^{-i\lambda(t)}\sum_{s=0}^{M+\mu} \ln ^s rg_s(\omega,t),\label{e4.4}$$ where g_s, s=0,\dots M+\mu, are infinitely differentiable functions with respect to \omega, \mu=1 if \lambda_0 is simple eigenvalue of problem \eqref{e4.1}-\eqref{e4.2}, and \mu=0 if \lambda_0 is not a eigenvalue of this problem. \end{lemma} Now we will study the asymptotical expansions of solutions of problem \eqref{e2.3}--\eqref{e2.5}. Denote by K_\infty a infinite cylinder with base K:~K_\infty=K\times (0,\infty). Rewrite the equation \eqref{e2.3} in the form $$L_0(0,t,D)u=F(x,t)\label{e4.5}$$ where F(x,t)=(u_{tt}+f)+[L_0(0,t,D)-L(x,t,D)]u. \begin{lemma} \label{lem4.2} Assume that u(x,t) is a generalized solution of problem \eqref{e2.3}--\eqref{e2.5} in the space W^{1,1}(e^{- \gamma t}, K_\infty) such that u\equiv 0 whenever |x|>R, R a positive constant, and u_{t^k}\in W^{2+l,0}_{\beta}(e^{-(2k+1)\gamma t},K_\infty), F_{t^k} \in W^{l,0}_{\beta'}(e^{-(2k+1)\gamma t},K_\infty) for k\leq h, \beta'<\beta\leq l+1. In addition, suppose that the straight lines$$ \mathop{\rm Im} \lambda= -\beta +l+2 - \frac {n}{2},\quad \mathop{\rm Im} \lambda= -\beta' +l+2 - \frac {n}{2} $$do not contain points of spectrum of problem \eqref{e4.1}-\eqref{e4.2} for every t\in [0,\infty), and in the strip$$ -\beta +l+2 - \frac {n}{2}< {\rm Im \lambda} < -\beta' +l+2 - \frac {n}{2} $$there exists only simple eigenvalue \lambda(t) of problem \eqref{e4.1}-\eqref{e4.2}. Then the following representation holds$$ u(x,t)=c(t)r^{-i\lambda(t)}\phi(\omega,t)+u_1(x,t), $$where \phi(x,t) is an infinitely differentiable function of (\omega,t), c_{t^k}\in L_{2, \rm loc}(0,\infty), and (u_1)_{t^k}\in W^{l+2,0}_{\beta'}(e^{-(k+1)\gamma t},K_\infty) for k\leq h. \end{lemma} \begin{proof} From the result of \cite{n1} it follows that for almost every t\in(0,\infty) we have $$u(x,t)=c(t)r^{-i\lambda(t)}\phi(\omega,t)+u_1(x,t)\label{e4.6},$$ where \phi(\omega,t) is the energy function of the problem \eqref{e4.1}-\eqref{e4.2} which corresponds to the eigenvalue \lambda(t), u_1\in W^{2+l,0}_{\beta'}(e^{-(k+1)\gamma t},K_\infty) and$$ c(t)=i\int_K F(x,t)r^{-i\overline{\lambda(t)}+2-n}\psi(x,t)dx, where \psi(x,t) is the energy function of the problem conjugating to the problem \eqref{e4.1}-\eqref{e4.2} which corresponds to the eigenvalue \overline {\lambda(t)}. Since \mathop{\rm Im} \overline{\lambda(t)}> \beta' -l-2 +\frac{n}{2}, from F(x,t) \in W^{l,0}_{\beta'}(e^{-(2k+1)\gamma t},K_\infty) it follows that c(t)\in L_{2, \rm loc}(0,\infty). Hence the assertion is proved for h=0. Assume that the assertion is true for 0,1,\dots, h -1. Denoting u_{t^h} by v. From \eqref{e2.3} and \eqref{e4.5} we obtain $$L_0(0,t,D)v = F_{t^h} -\sum_{k=1}^h \begin{pmatrix} h\cr k\end{pmatrix} L_{0t^k}(0,t,D) u_{t^{h-k}}, \label{e4.7}$$ where $L_{0t^k} = \sum_{i,j=1}^n \frac{\partial^k a_{pq}(0,t)}{\partial t^k}\frac{\partial}{\partial x_j} \frac{\partial}{\partial x_i}.$ Putting S_0(\omega,t) = r^{-i \lambda(t)}\phi(\omega,t). From \eqref{e4.7} it follows that \begin{align*} &\sum_{k=1}^h \begin{pmatrix}h\cr k\end{pmatrix} L_{0t^k}(0,t,D)u_{t^{h-k}}\\ &= \sum_{k=1}^h \begin{pmatrix} h\cr k\end{pmatrix}L_{0t^k}(0,t,D) \big[(cS_0)_{t^{h-k}}\big] + \sum_{k=1}^h \begin{pmatrix} h\cr k\end{pmatrix} L_{0t^k}(0,t,D)(u_1)_{t^{h-k}}. \end{align*} Using the inductive hypothesis and by arguments used in the proof of case h=0 we find that $u_{t^h} = v = \sum_{k=1}^h \begin{pmatrix}h\cr k\end{pmatrix}c_{t^{h-k}}(S_0)_{t^k}+d(t)S_0 + u_2,$ where d(t) \in L_{2,\rm loc}(0,\infty), u_2 \in W^{2+l,0}_{\beta'} (e^{-(2h+1)\gamma t},K_\infty). Putting S_1 = S_0^{-1}(u_1)_{t^{h-1}}, S_2 = S_0^{-1}u_2 - S_0^{-2}(S_0)_t (u_1)_{t^{h-1}}. Since (u_1)_{t^{h-1}}, u_2\in W^{l,0}_{\beta'}(e^{-(2h+1)\gamma t},K_\infty) so S_1, S_2 \in W^{0,0}_{-\frac{n}{2}} (e^{-(2h+1)\gamma t}, K_\infty). Therefore, I(t)=c_{t^{h-1}}(t)-c_{t^{h-1}}(0)-\int_0^td(\tau)d\tau =\int_0^tS_2(x,\tau)d\tau-S_1(x,t)+S_1(x,0)\in W^0_{-\frac{n}{2}}(K). Hence I(t) \equiv 0 and c_{t^h} = d \in L_{2, \rm loc}[0,\infty), (u_1)_{t^h} = u_2 \in W^{2m+l,0}_{\beta'} (e^{-(h+1)\gamma t},K_\infty). This completes the proof. \end{proof} \begin{theorem} Let u(x,t) be a generalized solution of \eqref{e2.3}-\eqref{e2.5} in the space W^{1, 1} (e^{-\gamma t}, K_\infty) such that u \equiv 0 whenever |x| > R , and f_{t^k} \in L^\infty(0, \infty; W^{l}_0 (K)) for k\leq 2l+h+1, f_{t^k}(x,0) = 0 for k \leq 2l+h. Assume that the straight lines \begin{align*} \mathop{\rm Im} \lambda = 1 - \frac{n}{2}, \quad \mathop{\rm Im} \lambda = 2 + l - \frac{n}{2} \end{align*} do not contain points of spectrum of \eqref{e4.1}-\eqref{e4.2} for every t \in [0,\infty), and in the strip \begin{align*} 1 - \frac{n}{2} < \mathop{\rm Im} \lambda < 2+l - \frac{n}{2} \end{align*} there exists only one simple eigenvalue \lambda(t) of \eqref{e4.1}-\eqref{e4.2}. Then the following representation holds $$u(x,t) = \sum_{s=0}^{l} c_s(t)r^{- i \lambda(t) +s} P_{3l ,s}(\ln r) + u_1(x,t), \label{e4.8}$$ where P_{3l,s} is a polynomial of order less than 3l +1 and coefficients infinitely differentiable functions of (\omega,t), (c_s)_{t^k} \in L_{2,\rm loc}(0,\infty), (u_1)_{t^k} \in W^{2+l,0}_0 (e^{-(2k+1)\gamma t}, K_\infty) for k \leq h+l. \end{theorem} \begin{proof} We will use induction on l. If l = 0 the statement follows from Lemma 4.2 with \beta=1, \beta'=0 and theorem 3.2. Let the statement be true for j\leq (l-1). We distinguish the following cases: \noindent\textbf{Case 1:} 1 - \frac{n}{2} < \mathop{\rm Im} \lambda(t) < 2+j-\frac{n}{2}. From the induction hypothesis, $$u(x,t) = \sum_{s=0}^j c_s(t)r^{- i \lambda(t)+s}P_{3j,s} (\ln r) + u_1(x,t), \label{e4.9}$$ where P_{3j,s} is a polynomial of less than 3j+1 and coefficients infinitely differentiable functions of (\omega,t), (c_s)_{t^k} \in L_{2, \rm loc}(0,\infty), (u_1)_{t^k} \in W^{2+j,0}_0 (e^{- (2k+1)\gamma t}, K_\infty) for k \leq h+j. Therefore $L_0(0,t,D) u_1 = F_1 -LS +S_{tt},$ where F_1 = (u_1)_{tt} + f + [L_0(0,t,D)-L(x,t,D)]u_1, S = \sum_{s=0}^{j} c_s(t) r^{-i \lambda(t)+s}P_{3j,s}(\ln r). Since f_{t^k} \in L^\infty (0, \infty; W^{j+1}_0(K)) for k \leq 2(j+1)+h+1 and f_{t^k}(x,0)=0 for k \leq 2j+h+1, so f_{t^k} \in L^\infty(0,\infty ; W^j_0(K)), k \leq 2j+(h+2)+1, and f_{t^k}(x,0) = 0, k \leq 2j+h+1. Therefore, (c_s)_{t^k} \in L_{2, \rm loc}(0,\infty) and (u_1)_{t^k} \in W^{j+2,0}_0(e^{- (2k+1)\gamma t}, K_\infty) for k \leq h + j + 2. Hence it follows that (F_1)_{t^k} \in W^{j+1,0}_0(e^{-(2k+1)\gamma t},K_\infty) for k \leq j+h+1. On the other hand $-LS +S_{tt} = F_2 + \sum_{s=0}^{j+1} \widetilde c_s(t) r^{-i \lambda(t)-2+s} \widetilde P_{3j+2,s}(\ln r),$ where \widetilde P_{3j+2,s} is a polynomial having order less than 3j+3 and its coefficients are infinitely differentiable functions of (\omega,t), (F_2)_{t^k} \in W^{j+1,0}_0(e^{-(2k+1)\gamma t},K_\infty), and (\widetilde c_s)_{t^k} \in L_{2, \rm loc}(0,\infty) for k \leq h+j+1. Therefore we obtain $L_0(0,t,D) u_1 = F_3 + \sum_{s=0}^{j+1} \widetilde c_s(t)r^{-i \lambda(t)-2+s} \widetilde P_{3j+2,s}(\ln r),$ where F_3 = F_1 + F_2 \in W^{j+1,0}_0(e^{-(2k+1)\gamma t},K_\infty) \subseteq H^{j,0}_{-1}(e^{-(2k+1)\gamma t},K_\infty). By Lemma 3.1 we find $u_1(x,t) = \sum_{s=0}^{j+1} \widetilde c_s(t)r^{-i \lambda(t)+s} \widetilde P_{3j+3,s}(\ln r) + u_2(x,t),$ where \widetilde P_{3j+3,s} is a polynomial having order less than 3j+4 and its coefficients are infinitely differentiable functions of (\omega,t), (u_2)_{t^k} \in W^{2+j,0}_{-1}(e^{-(2k+1)\gamma t}, K_\infty) for k \leq h+j+1. Therefore (u_2)_{t^k} \in W^{j+3,0}_0 (e^{-(2k+1)\gamma t},K_\infty) for k \leq h+j+1. Hence and from \eqref{e4.9} it follows that $u(x,t) = \sum_{s=0}^{j+1} c_s(t)r^{-i \lambda(t)+s} P_{3j+3,s}(\ln r) + u_2(x,t),$ where P_{3j+3,s} is a polynomial having order less than 3j+4 and its coefficients are infinitely differentiable functions of (\omega,t), (c_s)_{t^k} \in L_{2, \rm loc}(0,\infty), and (u_2)_{t^k} \in W^{j+3,0}_0(e^{- (2k+1)\gamma t}, K_\infty) for k \leq h+j+1. \noindent\textbf{Case 2:} 2+j-\frac{n}{2} < \mathop{\rm Im} \lambda(t) < 3+j-\frac{n}{2}. From theorem 3.2 we have u_{t^k}\in W^{1,1}(e^{-(2k+1)\gamma t},K_\infty). Hence (see \cite{h1}) u_{t^k} \in W^{2,0}_1(e^{-(2k+1)\gamma t}, K_\infty) for k\leq h+2l. On the other hand, the strip 1-\frac{n}{2} \leq \mathop{\rm Im}\lambda \leq 2-\frac{n}{2} does not contain points of spectrum of the problem \eqref{e4.1}-\eqref{e4.2} for every t \in (0, \infty). Hence and from theorems on the smoothness of solutions of elliptic problems in domains with conical points (see \cite{n1}) it follows that u_{t^k} \in W^{2,0}_0(e^{- (2k+1)\gamma t},K_\infty) for k \leq h +2l. We will prove that if f_{t^k} \in L^\infty (0, \infty; W^{j}_0(K)) for k \leq 2j+h+1 and f_{t^k}(x,0)=0 for k \leq 2j+h, then u_{t^k} \in W^{2+j,0}_0(e^{-(2k+1)\gamma t}, K_\infty), \ k \leq h + 2l -j. This assertion was proved for j = 0. Assume that it is true for j-1. Since f_{t^k} \in L^\infty (0, \infty; W^{j-1}_0(K)) for k \leq 2(j-1)+(h+2)+1 and f_{t^k}(x,0) = 0 for k \leq 2(j-1)+h+2, then from inductive hypothesis it follows that u_{t^k} \in W^{j+1,0}_0 (e^{-(2k+1)\gamma t}, K_\infty), k \leq h + 2l -j+3. Therefore, u_{t^{k+2}}\in W^{j-1,-1}_{-1} (e^{-(2k+3)\gamma t}, K_\infty) for k \leq h + 2l -j. Hence and from the fact that the strip \begin{align*} j+1- \frac{n}{2} < \mathop{\rm Im}\lambda < j+2 - \frac{n}{2} \end{align*} does not contain points of spectrum of \eqref{e4.1}-\eqref{e4.2} for every t \in [0, \infty), we obtain u_{t^k} \in W^{j+1,0}_{-1} (e^{-(2k+1)\gamma t}, K_\infty), k \leq h+2l -j. Hence u_{t^k} \in W^{j+2,0}_0 (e^{- (2k+1)\gamma t}, K_\infty) for k \leq h + 2l -j. By Lemma 4.2 and from above arguments we obtain \begin{align*} u(x,t) = c(t) r^{-i\lambda(t)} \varphi(\omega,t) + u_1(x,t), \end{align*} where \varphi is an infinitely differentiable function of (\omega,t) which does not depend on the solution, c_{t^k} \in L_{2, \rm loc}(0, \infty), and (u_1)_{t^k} \in W^{2+l,0}_0(e^{-(k+1)\gamma t}, K_\infty) for k \leq h+l. \noindent\textbf{case 3:} There exists t_0 such that \mathop{\rm Im} \lambda (t_0) = 2+j - \frac{n}{2}. We can assume that 2+j-\epsilon - \frac{n}{2}<\mathop{\rm Im} \lambda (t)<3+j-\epsilon - \frac{n}{2}, 0<\epsilon<1. By arguments used in case 1 and 2 we obtain \eqref{e4.8}. The proof is complete. \end{proof} \begin{theorem} \label{thm4.2} Let u(x,t) be a generalized solution of \eqref{e2.3}-\eqref{e2.5} in the space W^{1, 1} (e^{-\gamma t}, \Omega_\infty), and f_{t^k} \in L^\infty(0, \infty;W^{l}_0 (\Omega)) for k\leq2l+h+1, f_{t^k}(x,0) = 0 for k \leq 2l+h. Assume that the straight lines $\mathop{\rm Im} \lambda = 1 - \frac{n}{2}, \quad \mathop{\rm Im} \lambda = 2 + l - \frac{n}{2}$ do not contain points of spectrum of \eqref{e4.1}-\eqref{e4.2} for every t \in [0,\infty), and in the strip $1 - \frac{n}{2} < \mathop{\rm Im} \lambda < 2+l - \frac{n}{2}$ there exists only one simple eigenvalue \lambda(t) of \eqref{e4.1}-\eqref{e4.2}. Then the following representation holds $$u(x,t) = \sum_{s=0}^{l} c_s(t)r^{- i \lambda(t) +s} P_{3l ,s}(\ln r) + u_1(x,t), \label{e4.10}$$ where P_{3l,s} is a polynomial of order less than 3l +1 and coefficients infinitely differentiable functions of (\omega,t), (c_s)_{t^k} \in L_{2, \rm loc}(0,\infty), (u_1)_{t^k} \in W^{2+l,0}_0 (e^{-(2k+1)\gamma t}, \Omega_\infty) for k \leq h+l. \end{theorem} \begin{proof} Surrounding the point 0 by a neighbourhood U_0 with small diameter that the intersection of \Omega and U_0 coincides with K. Consider a function u_0 = \varphi_0 u, where \varphi_0 \in {\mathaccent"7017 C}^\infty (U_0) and \varphi_0 \equiv 1 in some neighbourhood of 0. The function u_0 satisfies the system L(x, t, D) u_0 -(u_0)_{tt} =\varphi_0 f + L' (x, t, D)u, $$where L' (x, t, D) is a linear differential operator having order less than 2. Coefficients of this operator depend on the choice of the function \varphi_0 and equal to 0 outside U_0. Hence and from arguments analogous to the proof of Theorem 4.1, we obtain $$\varphi_0u(x,t) =\sum_{s=0}^l c_s(t) r^{- i \lambda(t)+s} P_{3l,s}(\ln r) + u_2(x,t), \label{e4.11}$$ where P_{3l,s} is a polynomial of order less than 3l +1 and coefficients infinitely differentiable functions of (\omega,t), (c_s)_{t^k} \in L_{2, \rm loc}(0,\infty), (u_2)_{t^k} \in W^{2+l,0}_0(e^{- (k+1)\gamma t}, \Omega_\infty) for k \leq h+l. The function \varphi_1 u = (1- \varphi_0) u equals to 0 in some neighbourhood of the conical point. We can apply the known theorem on the smoothness of solutions of elliptic problems in a smooth domain to this function and obtain \varphi_1 u \in W^{2+ l}_0 (\Omega) for almost every t \in (0, \infty). Hence we have (\varphi_1 u)_{t^k} \in W^{2 + l,0}_0 (e^{-(k+1)\gamma t}, \Omega_\infty) for k \leq h+l. Since u = \varphi_0 u + \varphi_1 u so from \eqref{e4.11} we obtain \eqref{e4.10}. The proof is complete. \end{proof} \section{An example} In this section we apply the previous results to the Cauchy-Dirichlet problem for the wave equation. Let \Omega be a bounded domain in \mathbb{R}^2. It is shown that the asymptotic of the generalized solution of the problem depends on the structure of the boundary of the domain, and the right-hand side. We consider the Cauchy-Dirichlet problem for wave equation in \Omega_\infty : $$\Delta u - u_{tt} = f(x,t) \label{e5.1}$$ with initial conditions $$u|_{t=0} = u_t |_{t=0} =0 \label{e5.2}$$ and boundary condition $$u|_{S_\infty } =0, \label{e5.3}$$ where \Delta is the Laplace operator. Assume that in a neighborhood of the origin of coordinates, the boundary \partial \Omega coincides with a rectilinear angle having measure w_0 . Then spectral problem \eqref{e4.1}-\eqref{e4.2} is Sturm-Liouville problem: \begin{gather} v_{ww} - \lambda^2 v =0, 0 \pi, then 0<\pi/w_0<1. On the other hand 0<\omega_0<2\pi so (k \pi/w_0) >1 for all k \geq 2. Therefore, in the trip 0 \leq {\rm Im} \lambda \leq 1 there exists only one simple eigenvalue \lambda(t)=i\pi/w_0 of the problem \eqref{e5.4}-\eqref{e5.5}. From Theorem 4.2 we obtain the following result. \begin{theorem} \label{thm5.1} Let u(x, t ) be a generalized solution of \eqref{e5.1}-\eqref{e5.3} in the space W^{1,1} ( e^{-\gamma t}, \Omega _\infty ). In addition, suppose that f_{t^k} \in L^\infty(0, \infty; L_2 (\Omega)) for k\leq h+1, f_{t^k}(x,0) = 0 for k \leq h. Then the following representation holds$$ u(x,t) = c(t)r^{\pi/w_0} P(\ln r) + u_1(x,t),$where$P$is a polynomial having order less than$1$and its coefficients are infinitely differentiable functions of$(\omega,t)$,$c_{t^k} \in L_{2, \rm loc}(0,\infty)$,$(u_1)_{t^k} \in W^{2,0}_0 (e^{-(2k+1)\gamma t}, \Omega_\infty)$for$k \leq h\$. \end{theorem} \begin{thebibliography}{00} \bibitem{a1} M. S. Agranovich and M. I. Vishik. \emph{Elliptic problems with a parameter and parabolic problems of general type.} Usp. Mat. Nauk,19, No3, 53-161 (1964). \bibitem{d1} R. Dautray and J. L. 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