\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 125, pp. 1--21.\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-2009/125\hfil Asymptotic formulas for solutions] {Asymptotic formulas for solutions of parameter-depending elliptic boundary-value problems in domains with conical points} \author[N. T. Anh, N. M. Hung \hfil EJDE-2009/125\hfilneg] {Nguyen Thanh Anh, Nguyen Manh Hung} % in alphabetical order \address{Nguyen Thanh Anh \hfill\break Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam} \email{thanhanh@hnue.edu.vn} \address{Nguyen Manh Hung \hfill\break Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam} \email{hungnmmath@gmail.com} \thanks{Submitted March 5, 2009. Published September 4, 2009.} \subjclass[2000]{35J40, 35B40, 35P99, 47A55, 47A56} \keywords{Elliptic boundary problem; nonsmooth domains; conical point; \hfill\break\indent asymptotic behaviour} \begin{abstract} In this article, we study elliptic boundary-value problems, depending on a real parameter, in domains with conical points. We present asymptotic formulas for solutions near singular points, as linear combinations of special singular functions and regular functions. These functions and the coefficients of the linear combination are regular with respect to the parameter. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction}\label{sec1} Elliptic boundary-value problems in domains with point singularities were thoroughly investigated (see, e.g, \cite{KMR} and the extensive bibliography in this book). We are concerned with elliptic boundary-value problems depending on a real parameter in domains with conical points. These problems arise in considering initial-boundary-value problems for non-stationary equations with coefficients depending on time (see, e.g, \cite{H}, where the initial-boundary-value problem for strongly hyperbolic systems with Dirichlet boundary conditions was considered). We give here as an example the initial-boundary-value problem for the parabolic equation \begin{gather} u_t+L(x,t,\partial_x)u=f \quad\text{in } G_T, \label{eq101}\\ B_j(x,t,\partial_x)u = 0, \quad\text{on } S_T,\; j=1,\dots,m, \label{eq102}\\ u|_{t=0}=\varphi \quad\text{on } G, \label{eq103} \end{gather} where the sets $G,G_T,S_T$, and the operators $L,B_j$ are introduced in Section \ref{sec2}. For this problem we have first dealt with the unique solvability and the regularity of the generalized solution with respect to the time variable $t$ (see \cite{HA1}). After that, to investigate the regularity and the asymptotic of the solution, (\ref{eq101}) and (\ref{eq102}) are rewritten in the form \begin{gather} L(x,t,\partial_x)u= f-u_t \quad\text{in } G_T, \label{eq104}\\ B_j(x,t,\partial_x)u= 0, \quad\text{on } S_T,\; j=1,\dots,m. \label{eq105} \end{gather} Then (\ref{eq104}), (\ref{eq105}) can be regarded as a elliptic boundary-value problem depending on the parameter $t$. This approach was suggested in \cite{Kondratiev}. In the present paper we are concerned with asymptotic behaviour of the solutions near the singular points. Firstly, applying the results of the analytic perturbation theory of linear operators (\cite{Kato}) and the method of linearization of polynomial operator pencils (\cite{Markus}), we establish the smoothness with respect to the parameter of the eigenvalues, the eigenvectors of the operator pencils generated by the problems. After that, applying the well-known results for elliptic boundary-value problems (without parameter) in the considered domains, we receive the asymptotic formulas of the solutions as a sum of a linear combination of special singular functions and a regular function in which this functions and the coefficients of the linear combinations are regular with respect to the parameter. The present results will be applied to deal with the asymptotic behaviour of the solutions of initial-boundary-value problems for parabolic equations in cylinders with bases containing conical points in a forthcoming work. Our paper is organized as follows. In Section \ref{sec2}, we introduce some needed notation and definitions. We study the spectral properties of the operator pencil generated by the problem in Section \ref{sec2a}. Section \ref{sec3} is devoted to establishing the asymptotic behaviour of the solutions in a neighborhood of the conical point. \section{Preliminaries}\label{sec2} Let $G$ be a bounded domain in ${\mathbb R}^n (n \geq 2)$ with the boundary $\partial G$. We suppose that $S = \partial G\setminus\{0\}$ is a smooth manifold and $G$ in a neighborhood of the origin $0$ coincides with the cone $K=\{x: x/|x| \in \Omega\}$, where $\Omega$ is a smooth domain on the unit sphere $S^{n-1}$ in ${\mathbb R}^n$. Let $T$ be a positive real number or $T=+\infty$. If $A$ is a subset of $\mathbb{R}^n$, we set $A_T=A\times (0,T)$. For each multi-index $\alpha=(\alpha_1, \dots, \alpha_n)\in{\mathbb N}^n$, set $|\alpha|=\alpha_1+ \dots+ \alpha_n$, and $\partial^\alpha=\partial^\alpha_x=\partial^{\alpha_1}_{x_1}\dots \partial^{\alpha_n}_{x_n}$. Let us introduce some functional space used in this paper. Let $l$ be a nonnegative integer. We denote by $W_2^l(G)$ the usual Sobolev space of functions defined in $G$ with the norm $$\|u\|_{W_2^{l}(G)}=\Big(\int_G\sum_{|\alpha|\leq m} |\partial^\alpha_x u|^2 dx\Big)^{1/2},$$ and by $W_2^{l-\frac{1}{2}}(S)$ the space of traces of functions from $W_2^l(G)$ on $S$ with the norm $$\|u\|_{W_2^{l-\frac{1}{2}}(S)}=\inf\big\{\|v\|_{W_2^l(G)}: v \in {W_2^l(G)}, v|_{S}=u\big\}.$$ We define the weighted Sobolev space $V^l_{2,\gamma}(K)$ ($\gamma \in \mathbb{R}$) as the closure of $C_0^\infty(\overline{K}\setminus\{0\})$ with respect to the norm $$\|u\|_{V^l_{2,\gamma}(K)}=\Big(\sum_{|\alpha| \leq l}\int_Kr^{2(\gamma+|\alpha|-l)}|\partial^\alpha_x u|^2dx \Big)^{1/2}, \label{eq205}$$ where $r=|x|=\big(\sum_{k=1}^nx^2_k\big)^{1/2}$. If $l\geq 1$, then $V^{l-\frac{1}{2}}_{\gamma}(\partial K)$ denote the space consisting of traces of functions from $V^l_{2,\gamma}(K)$ on the boundary $\partial K$ with the norm $$\|u\|_{V^{l-\frac{1}{2}}_{\gamma}(\partial K)} =\inf\big\{\|v\|_{V^l_{2,\gamma}(K)}: v \in {V^l_{2,\gamma}(K)}, v|_{\partial K}=u\big\}. \label{eq206}$$ It is obvious from the definition that the space $V^{l+k}_{2,\gamma+k}(K)$ is continuously imbedded into the space $V^{l}_{2,\gamma}(K)$ for an arbitrary nonnegative integer $k$. An analogous assertion holds for the space $V^{l-\frac{1}{2}}_{2,\gamma}(\partial K)$. The weighted spaces $V^l_{2,\gamma}(G)$, $V^{l-\frac{1}{2}}_{\gamma}(S)$ are defined similarly as in (\ref{eq205}), (\ref{eq206}) with $K, \partial K$ replaced by $G, S$, respectively. Let $h$ be a nonnegative integer and $X$ be a Banach space. Denote by $\mathcal{B}(X)$ the set of all continuous linear operators from $X$ into itself. By $W_2^h((0,T);X)$ we denote the Sobolev space of $X$-valued functions defined on $(0,T)$ with $$\|f\|_{W_2^h((0,T);X)}=\Big(\sum_{k=0}^{h}\int_O^T \big\|\frac{d^kf(t)}{dt^k}\big\|^2_{X}dt\Big)^{1/2}<+\infty.$$ For short, we set \begin{gather*} W_2^h((0,T))=W_2^h((0,T);\mathbb{C}), \quad W_2^{l,h}(\Omega_T)=W_2^h((0,T);W_2^l(\Omega)),\\ V^{l,h}_{2,\gamma}(G_T)=W_2^h((0,T);V^l_{2,\gamma}(G)), \quad V^{l-\frac{1}{2},h}_{2,\gamma}(S_T)=W_2^h((0,T); V^{l-\frac{1}{2}}_{2,\gamma}(S)). \end{gather*} Recall that a $X$-valued function $f(t)$ defined on $[0,+\infty)$ is said to be continuous or analytic at $t=+\infty$ if the function $g(t)$ $=$ $f(\frac{1}{t})$ is continuous or analytic, respectively, at $t=0$ with a suitable definition of $g(0) \in X$. In these cases we can regard $f(t)$ as a function defined on $[0,+\infty]$ with $f(+\infty)=g(0)$. Denote by $C^\mathfrak{a}([0,T];X)$ the set of all $X$-valued functions defined and analytic on $[0,T]$ (recall that $T$ is a positive real number or $T=+\infty$). It is clear that if $f \in$ $C^\mathfrak{a}([0,T];X)$, then $f$ together with all its derivatives are bounded on $[0,T]$. Let $A$ be a subset of $\mathbb{R}^n$ and $f(x,t)$ be a complex-valued function defined on $A_T=A\times[0,T]$. We will say that $f$ belongs to the class $C^{\infty,\mathfrak{a}}(A_T)$ if and only if $f\in$ $C^a([0,T];C^l(A) )$ for all nonnegative integer $l$. Let $$L=L(x,t,\partial_x) = \sum_{|\alpha|\leq 2m} a_{\alpha}(x,t)\,\partial^\alpha_x$$ be a differential operator of order $2m$ defined in $Q$ with coefficients belonging to $C^{\infty,\mathfrak{a}}(\overline{G}_T)$ ($\overline{G}_T=\overline{G}\times [0,T]$). Suppose that $L(x,t,\partial_x)$ is elliptic on $\overline{G}$ uniformly with respect to $t$ on $[0,T]$, i.e, there is a positive constant $c_0$ such that $$|L^\circ(x,t,\xi)| \geq c_0|\xi|^{2m} \label{eq201}$$ for all $\xi \in \mathbb{R}^n$ and for all $(x,t) \in \overline{G}_T$. Here $L^\circ(x,t,\partial_x)$ is principal part of the operator $L(x,t,\partial_x)$; i.e, $L^\circ(x,t,\partial_x) = \sum_{|\alpha| = 2m} a_{\alpha}(x,t)\partial^\alpha_x.$ Let $$B_j=B_j(x,t,\partial_x)=\sum_{|\alpha|\leq \mu_j}b_{j,\alpha}(x,t) \partial^\alpha_x, \quad j =1,\dots, m,$$ be a system of boundary operators on $S$ with coefficients belonging to $C^{\infty,\mathfrak{a}}(\partial G\times [0,T])$, $\mathop{\rm ord}B_j = \mu_j \leq 2m-1$, $j=1,\dots,m$. Suppose that $\{B_j(x,t,\partial_x)\}_{j=1}^m$ is a normal system on $S$ uniformly with respect to $t$ on $[0,T]$; i.e, the two following conditions are satisfied: \begin{itemize} \item[(i)] $\mu_j \not= \mu_k$ for $j \ne k$, \item[(ii)] there are positive constants $c_j$ such that $$|B_j^\circ(x,t,\nu(x))| \geq c_j, j =1, \dots, m, \label{eq202}$$ for all $(x,t) \in S_T$. Here $B_j^\circ(x,t,\partial_x)$ is the principal part of $B_j(x,t,\partial_x)$ and $\nu(x)$ is the unit outer normal to $S$ at point $x$. \end{itemize} In this paper, we consider asymptotic behaviour near the conical point of solutions of the elliptic boundary-value problem depending on the parameter $t$: \begin{gather} L(x,t,\partial_x)u =f \quad\text{in } G_T, \label{eq203}\\ B_{j}(x,t,\partial_x)u =g_j\quad \text{on } S_T, j=1,\dots,m. \label{eq204} \end{gather} \section{Spectral properties of the pencil operator generated}\label{sec2a} Let $\mathfrak{L}=\mathfrak{L}(t,\partial_x), \mathfrak{B}_j =\mathfrak{B}_j(t,\partial_x)$ be the principal homogenous parts of $L(x,t,\partial_x)$, $B_j(x,t,\partial_x)$ at $x=0$; i.e, \begin{gather*} \mathfrak{L}=\mathfrak{L}(t,\partial_x)=\sum_{|\alpha| = 2m}a_{\alpha}(0,t)\,\partial^\alpha_x,\\ \mathfrak{B}_j=\mathfrak{B}_j(t,\partial_x)=\sum_{|\alpha|= \mu_j}b_{j\alpha}(0,t)\,\partial^\alpha_x, j =1,\dots, m. \end{gather*} It can be directly verified that the derivative $\partial^\alpha_x$ has the form $$\partial^\alpha_x = r^{-|\alpha|}\sum_{p=0}^{|\alpha|} P_{\alpha,p}(\omega,\partial_\omega)(r\partial r)^{p}, \label{eq212}$$ where $P_{\alpha,p}(\omega,\partial_\omega)$ are differential operators of order $\leq |\alpha|-p$ with smooth coefficients on $\overline{\Omega}$, $r=|x|$, $\omega$ is an arbitrary local coordinate system on $S^{n-1}$. Thus we can write $\mathfrak{L}(t,\partial_x), \mathfrak{B}_j(t,\partial_x)$ in the form \begin{gather*} \mathfrak{L}(t,\partial_x)=r^{-2m}\mathscr{L}(\omega,t,\partial_\omega,r\partial_r),\\ \mathfrak{B}_j(t,\partial_x)=r^{-\mu_j}\mathscr{B}_j(\omega,t,\partial_\omega,r\partial_r). \end{gather*} We introduce the operator $$\mathscr{U}(\lambda,t)=(\mathscr{L}(\omega,t,\partial_\omega,\lambda), \mathscr{B}_j(\omega,t,\partial_\omega,\lambda)),\lambda \in {\mathbb C}, t \in [0,T]$$ of the parameter-depending elliptic boundary-value problem \begin{gather*} \mathscr{L}(\omega,t,\partial_\omega,\lambda)u=f \quad\text{in } \Omega, \\ \mathscr{B}_j(\omega,t,\partial_\omega,\lambda)u =g_j \quad\text {on } \partial\Omega, \quad j=1,\dots, m \end{gather*} (Here the parameters are $\lambda$ and $t$). For every fixed $\lambda \in {{\mathbb C}}, t \in [0,T]$ this operator continuously maps $$\mathcal{X}\equiv W_2^l(\Omega) \text{ into } \mathcal{Y} \equiv W_2^{l-2m}(\Omega)\times \prod_{j=1}^m W_2^{l-\mu_j-\frac{1}{2}}(\partial \Omega)\;(l\geq 2m).$$ We can write $\mathscr{U}(\lambda,t)$ in the form $\mathscr{U}(\lambda,t)=A_{2m}(t)\lambda^{2m}+A_{2m-1}(t) \lambda^{2m-1}+\dots+A_{0}(t),$ where $A_{k}(t), k=2m, 2m-1, \dots, 0$ are differential operators in $\Omega$ of order $2m-k$ with coefficients belonging to $C^{\infty,\mathfrak{a}}(\Omega_T)$, especially $$A_{2m}(t)=(\sum_{|\alpha|=2m}a_{\alpha}(0,t)\omega^{\alpha},0,\dots,0). \label{eq3}$$ We mention now some well-known definitions (\cite{KMR}). Let $t_0 \in [0,T]$ fixed. If $\lambda_0 \in \mathbb{C}$, $\varphi_0 \in \mathcal{X}$ such that $\varphi_0 \not= 0, \mathscr{U}(\lambda_0,t_0)\varphi_0=0$, then $\lambda_0$ is called an eigenvalue of $\mathscr{U}(\lambda,t_0)$ and $\varphi_0 \in \mathcal{X}$ is called an eigenvector corresponding to $\lambda_0$. $\Lambda= \dim\ker\mathscr{U}(\lambda_0,t_0)$ is called the geometric multiplicity of the eigenvalue $\lambda_0$. If the elements $\varphi_1, \dots,\varphi_s$ of $\mathcal{X}$ satisfy the equations $\sum_{q=0}^{\sigma}\frac{1}{q!}\frac{d^q}{d\lambda^q} \mathscr{U}(\lambda,t_0)|_{\lambda=\lambda_0}\varphi_{\sigma-q}=0 \quad\text{for } \sigma=1,\dots,s,$ then the ordered collection $\varphi_0, \varphi_1, \dots,\varphi_s$ is said to be a Jordan chain corresponding to the eigenvalue $\lambda_0$ of the length $s+1$. The rank of the eigenvector $\varphi_0$ ($\mathop{\rm rank} \varphi_0$) is the maximal length of the Jordan chains corresponding to the eigenvector $\varphi_0$. A canonical system of eigenvectors of $\mathscr{U}(\lambda_0,t_0)$ corresponding to the eigenvalue $\lambda_0$ is a system of eigenvectors $\varphi_{1,0}, \dots,\varphi_{\Lambda,0}$ such that $\mathop{\rm rank} \varphi_{1,0}$ is maximal among the $\mathop{\rm rank}$ of all eigenvectors corresponding to $\lambda_0$ and $\mathop{\rm rank} \varphi_{j,0}$ is maximal among the $\mathop{\rm rank}$ of all eigenvectors in any direct complement in $\ker \mathscr{U}(\lambda_0,t_0)$ to the linear span of the vectors $\varphi_{1,0}, \dots,\varphi_{j-1,0}\;(j=2,\dots,\Lambda)$. The number $\kappa_{j}=\mathop{\rm rank} \varphi_{j,0}\;(j=1,\dots,\Lambda)$ are called the partial multiplicities and the sum $\kappa=\kappa_1+\dots+\kappa_\Lambda$ is called the algebraic multiplicity of the eigenvalue $\lambda_{0}$. The eigenvalue of $\lambda_0$ is called simple if both its geometric multiplicity and the rank of the corresponding eigenvector equal to one. For each $t \in [0,T]$ fixed the set of all complex number $\lambda$ such that $\mathscr{U}(\lambda,t)$ is not invertible is called the spectrum of $\mathscr{U}(\lambda,t)$. It is known that the spectrum of $\mathscr{U}(\lambda,t)$ is an enumerable set of its eigenvalues (see \cite[Th. 5.2.1]{KMR}). Moreover, there are constants $\delta, R$ such that $\mathscr{U}(\lambda,t)$ is invertible for all $t \in [0,T]$ and all $\lambda$ in the set $$\mathcal{D}:= \{\lambda \in \mathbb{C}: |\mathop{\rm Re}\lambda|\leq \delta|\mathop{\rm Im}\lambda|, |\lambda|\geq R \} \label{eq2}$$ (see \cite[Thm. 3.6.1]{KMR}). Now we use method of linearization to investigate the smoothness of the eigenvalues and the eigenvectors of $\mathscr{U}(\lambda,t)$ with respect to $t$. Without loss of generality we can assume that the operator $A_0(t)$ is invertible. Indeed, if $\lambda_0$ is an eigenvalue of $\mathscr{U}(\lambda,t)$ for all $t \in [0,T]$, then $\mathscr{U}(\lambda_0+\lambda,t) = \sum_{k=0}^{2m} \widetilde{A}_k(t)\lambda^k,$ where $\widetilde{A}_0(t)=\mathscr{U}(\lambda_0,t)$ is invertible for all $t \in [0,T]$. Setting $$\mathscr{V}(\lambda,t)=A_0^{-1}(t)\mathscr{U}(\lambda,t), D_k(t)=A_0^{-1}(t)A_k(t), k=1,\dots,2m,$$ we have the pencils of continuous operators $\mathscr{V}(\lambda,t)$, $D_k(t)$, $k=1,\dots,2m$, from $\mathcal{X}$ into itself, and $$\mathscr{V}(\lambda,t)=D_{2m}(t)\lambda^{2m}+D_{2m-1}(t) \lambda^{2m-1}+\dots+D_{1}(t)\lambda+I,$$ where $I$ is the identical operator in $\mathcal{X}$. The eigenvalues and the eigenvectors of $\mathscr{V}(\lambda,t)$ are defined analogously as of $\mathscr{U}(\lambda,t)$. We can verify directly (or see \cite[Le. 12.1]{Markus}) that $$\mathcal{I}-\lambda\mathscr{A}(t)= \mathscr{C}(t)\mathscr{E}(\lambda,t) \begin{bmatrix} \mathscr{V}(\lambda,t)&&&\\ &I&&\\ &&\ddots&\\ &&&I\\ \end{bmatrix}\mathscr{F}(\lambda,t), \label{eq1}$$ where $\mathscr{A}= \begin{bmatrix} -D_1&\dots&-D_{2m-1}&-D_{2m}\\ I&&&\\ &\ddots&&\\ &&I&\\ \end{bmatrix},\quad \mathscr{C}= \begin{bmatrix} I&-D_1&\dots&-D_{2m-1}\\ &I&&\\ &&\ddots&\\ &&&I\\ \end{bmatrix},$ $\mathscr{E}= \begin{bmatrix} I&\sum_{k=1}^{2m}D_{k}\lambda^{k-1}&\sum_{k=2}^{2m}D_{k}\lambda^{k-2}&\dots&D_{2m-1}+D_{2m}\lambda\\ &I&&&\\ &&\ddots&&\\ &&&&I\\ \end{bmatrix},$ $\mathscr{F}= \begin{bmatrix} I&&&\\ -\lambda I&I&&\\ &\ddots&\ddots&\\ &&-\lambda I&I\\ \end{bmatrix},$ (in operator matrices all the elements not indicated are assumed to be zero, and the argument $t$ has been omitted for the sake of brevity) and $\mathcal{I}$ is the identical operator in $\mathcal{X}^{2m}$. Verifying directly we see that $\mathscr{C}(t)$, $\mathscr{E}(\lambda,t)$, $\mathscr{F}(\lambda,t)$ are invertible elements of $\mathcal{B}(\mathcal{X}^{2m})$ with $\mathscr{C}^{-1}= \begin{bmatrix} I&D_1&\dots&D_{2m-1}\\ &I&&\\ &&\ddots&\\ &&&I\\ \end{bmatrix},\quad \mathscr{F}^{-1}= \begin{bmatrix} I&&&\\ -\lambda I&I&&\\ &\ddots&\ddots&\\ &&-\lambda I&I\\ \end{bmatrix},$ $\mathscr{E}^{-1}= \begin{bmatrix} I&-\sum_{k=1}^{2m}D_{k}\lambda^{k-1}&-\sum_{k=2}^{2m}D_{k}\lambda^{k-2}&\dots&-(D_{2m-1}+D_{2m}\lambda)\\ &I&&&\\ &&\ddots&&\\ &&&&I \end{bmatrix}.$ It follows from the assumption on the analyticity of coefficients of differential operators $L(x,t,\partial_x)$ and $B_j(x,t,\partial_x)$, $j=1,\dots,m$, that $\mathscr{V}(\lambda,t)$ is of the class $C^\mathfrak{a}([0,T];\mathcal{B}(\mathcal{X}))$ and $\mathscr{A}(t)$ is of the class $C^\mathfrak{a}([0,T];\mathcal{B}(\mathcal{\mathcal{X}}^{2m}))$. It is obvious that $\mathscr{U}(\lambda,t)$ and $\mathscr{V}(\lambda,t)$ have the same eigenvalues with the same multiplicities and the same corresponding eigenvectors. It follows from (\ref{eq1}) that the spectra except the zero of the pencil $\mathscr{V}(\lambda,t)$ and the operator $\mathscr{A}(t)$ coincide for all $t\in [0,T]$. We now show that for each $t\in [0,T]$ all eigenvalues of $\mathscr{V}(\lambda,t)$ and $\mathscr{A}(t)$ are nonzero. It is obvious for these of $\mathscr{V}(\lambda,t)$. Suppose $\varphi =(\varphi^{(1)},\dots,\varphi^{(2m)} ) \in \mathcal{\mathcal{X}}^{2m}, \varphi \ne 0$ such that $\mathscr{A}(t)\varphi=0$ for some $t\in [0,T]$. Then $\varphi^{(1)}=\dots=\varphi^{(2m-1)}=0$ and $D_{2m}(t)\varphi^{(2m)}=0$. This implies $A_{2m}(t)\varphi^{(2m)} = 0$, but this do not occur since $\ker A_{2m}(t)=\{0\}$ which follows from (\ref{eq3}). Now we can apply \cite[Le. 12.5, 12.8]{Markus} to conclude that the complex number $\lambda_0$ is an eigenvalue of the pencil $\mathscr{V}(\lambda,t)$ (for some $t\in [0,T]$) if and only if $\sigma_0=(\lambda_0)^{-1}$ is an eigenvalue of the operator $\mathscr{A}(t)$ with the same multiplicities. Hence for each $t\in [0,T]$ the spectrum of the operator $\mathscr{A}(t)$ is a bounded set consisting nonzero eigenvalues with finite multiplicities. \begin{lemma} \label{th21} Let $\gamma_1, \gamma_2$ be real numbers, $\gamma_1< \gamma_2$ such that the lines $\mathop{\rm Re}\lambda=\gamma_j, j=1,2$, do not contain any eigenvalue of $\mathscr{U}(\lambda,t)$ and all eigenvalues of this pencil in the strip $$\mathcal{D}_1:=\{\lambda \in \mathbb{C}: \gamma_1<\mathop{\rm Re}\lambda< \gamma_2\} \label{3eq002}$$ are simple for all $t\in [0,T]$. Then there are complex-valued functions $\lambda_k(t)$ and $\mathcal{X}$-valued functions $\varphi_{k}(t)$, $k=1,\dots,N$, which are analytic on $[0,T]$ such that, for each $t\in [0,T]$, $\{\lambda_1(t), \dots, \lambda_N(t)\}$ is the set of all eigenvalues of $\mathscr{U}(\lambda,t)$ in $\mathcal{D}_1$ and $\varphi_k(t)$ are eigenvectors corresponding to the eigenvalues $\lambda_k(t)$, $k=1,\dots,N$, respectively. \end{lemma} \begin{proof} Since the set $\mathcal{D}$ defined in (\ref{eq2}) does not contains eigenvalues of $\mathscr{U}(\lambda,t)$ for all $t\in [0,T]$, the eigenvalues of this pencil in the strip $\mathcal{D_1}$ actually are located in the bounded domain $\mathcal{D}_2=\left(\mathbb{C}\setminus \mathcal{D}\right)\cap \mathcal{D}_1$. Moreover, the boundary $\partial \mathcal{D}_2$ of $\mathcal{D}_2$ contains no eigenvalues of $\mathscr{U}(\lambda,t)$ for all $t\in [0,T]$. Let $M$ be a positive number such that $\|\mathscr{A}(t)\|0$(j=1,\dots,m)$, for all$x \in S$sufficiently near the origin and for all$t \in [0,T]$.$\mathfrak{B}_j(t,\nu(x))$can be regarded as defined on$K_T$, and$\mathfrak{B}_j(t,\nu(x)) \ne 0$for all$x \in \partial K_T$and for all$t \in [0,T]$since$\nu(x)$are the same on each axis of the cone$K$. Thus, the system$\{\mathfrak{B}_j(t,\partial_x)\}_{j=1}^m$is normal on$\partial K$for each$t \in [0,T]$. Therefore, there are boundary operators$\mathfrak{B}_{j}(t,\partial_x), \mathop{\rm ord}\mathfrak{B}_j(t,\partial_x)=\mu_j<2m, j=m+1, \dots 2m$, such that the system$\{\mathfrak{B}_j(x,t,\partial_x)\}_{j=1}^{2m}$is a Dirichlet system of order$2m$(for definition see \cite{KMR}, p. 63) on$\partial K$for each$t \in [0,T]$, and the following classical Green formula $$\int_{K}\mathfrak{L}u\overline{v}dx +\sum_{j=1}^m\int_{\partial K}\mathfrak{B}_ju \overline{\mathfrak{B}'_{j+m}v}ds=\int_{K}u\overline{\mathfrak{L}^+v} dx+\sum_{j=1}^m\int_{\partial K}\mathfrak{B}_{j+m}u\overline{\mathfrak{B}'_{j}v}ds \label{eq34a}$$ holds for$u, v \in C_0^\infty(\overline{K}\setminus\{0\} )$and for each$t \in [0,T]$. Here$\mathfrak{L}^+=\mathfrak{L}^+(t,\partial_x)$is the formal adjoint operator of$\mathfrak{L}$, i.e, $$\mathfrak{L}^+u=(-1)^{2m} \sum_{|\alpha|= 2m} \overline{a}_{\alpha}(0,t)\partial^\alpha_x u,$$ and$\mathfrak{B}'_j=\mathfrak{B}'_j(t,\partial_x)$are boundary operators of order$\mu'_j=2m-1-\mu_{j+m}$if$j\leq m$, and of order$\mu'_j=2m-1-\mu_{j-m}$if$j\geq m+1$. The coefficients of$\mathfrak{B}'_j=\mathfrak{B}'_j(t,\partial_x), j=1,\dots,2m$, are independent of the variable$x$and dependent on$t$analytically on$[0,T]$. The operators$\mathfrak{L}^{+}(t,\partial_x), \mathfrak{B}_j'(t,\partial_x)can be written in the form \begin{gather} \mathfrak{L}^{+}(t,\partial_x)=r^{-2m}\mathscr{L}^{+} (\omega,t,\partial_\omega,r\partial_r),\\ \mathfrak{B}_j'(t,\partial_x)=r^{-\mu_j'}\mathscr{B}_j' (\omega,t,\partial_\omega,r\partial_r). \end{gather} From the Green formula (\ref{eq34a}) we get the following Green formula \begin{align*} &\int_{\Omega}\mathscr{L}(t,\lambda)\widetilde{u}\overline {\widetilde{v}}dx+\sum_{j=1}^m\int_{\partial\Omega}\mathscr{B}_j (t,\lambda)\widetilde{u}\overline{\mathscr{B}'_{j+m} (t,-\overline{\lambda}+2m-n)\widetilde{v}}ds \\ &= \int_{\Omega}\widetilde{u}\overline{\mathscr{L}^+ (t,-\overline{\lambda}+2m-n)\widetilde{v}}dx+\sum_{j=1}^m \int_{\partial\Omega}\mathscr{B}_{j+m}(t,\lambda)\widetilde{u} \overline{\mathscr{B}'_{j}(t,-\overline{\lambda}+2m-n)\widetilde{v}}ds \end{align*} for \widetilde{u}, \widetilde{v} \in C^{\infty}(\overline{\Omega})$and for all$t \in [0,T]$(see \cite[p. 206]{KMR}). Here for the sake of brevity, we have omitted the arguments$\omega$and$\partial_\omega$in the operators of this formula. We denote by$\mathcal{U}^{+}(\lambda,t)$the operator of the boundary-value problem \begin{gather} \mathscr{L}^{+}(\omega,t,\partial_\omega,-\lambda+2m-n)v=f \quad\text{ in } \Omega,\\ \mathscr{B}_j^{+}(\omega,t,\partial_\omega,-\lambda+2m-n)v=g_j \quad\text{on } \partial\Omega, \quad j=1,\dots, m. \end{gather} Let$\lambda_0(t)$be an analytic function on$[0,T]$such that$\lambda_0(t)$be a simple eigenvalue of$\mathscr{U}(\lambda,t)$for each$t \in [0,T]$and let$\varphi \in C^{\infty,\mathfrak{a}}(\Omega_T)$such that$\varphi(t)$be an eigenvector of$\mathscr{U}(\lambda,t)$corresponding to the eigenvalue$\lambda_0(t)$for each$t \in [0,T]$. Then$\overline{\lambda}_0(t)$are simple eigenvalues of the pencil$\mathcal{U}^{+}(\lambda,t)$for all$t\in [0,T]$(see \cite[6.1.6]{KMR}). Moreover, there exists a function$\psi \in C^{\infty,\mathfrak{a}}(\Omega_T)$such that$\psi(t)$is an eigenvector of$\mathscr{U}^{+}(\lambda,t)$corresponding to the eigenvalues$\overline{\lambda}_0(t)$for each$t \in [0,T]$which is analogous to the case of the pencil$\mathscr{U}(\lambda,t). We claim that \begin{aligned} &(\mathscr{L}^{(1)}(\lambda_0(t),t)\varphi(t), \psi(t))_{\Omega}\\ &+\sum_{j=1}^m\big( \mathscr{B}_j^{(1)}(\lambda_0(t),t)\varphi(t),\mathscr{B}'_{j+m}(-\overline{\lambda}_0(t)+2m-n,t)\psi(t)\big)_{\partial \Omega} \ne 0, \end{aligned}\label{eq004} for allt\in [0,T]$, where $$\mathscr{L}^{(1)}(\lambda,t)=\frac{d }{d\lambda} \mathscr{L}(\lambda,t), \mathscr{B}_j^{(1)}(\lambda,t) =\frac{d }{d\lambda}\mathscr{B}_j(\lambda,t), j=1,\dots,m.$$ We prove this by contradiction. If (\ref{eq004}) is not true for some$t_0 \in [0,T]$, then one can solve with respect to$uthe following elliptic boundary-value problem \begin{align*} \mathscr{U}(\lambda,t_0)u&=\mathscr{U}^{(1)}(\lambda,t_0)\varphi(t_0)\\ &\equiv (\mathscr{L}^{(1)}(\lambda,t_0)\varphi(t_0), \mathscr{B}_1^{(1)}(\lambda,t_0)\varphi(t_0), \dots, \mathscr{B}_m^{(1)}(\lambda,t_0)\varphi(t_0)). \end{align*} This implies that the eigenvalue\lambda_0(t_0)$is not simple which is not possible. It follows from (\ref{eq004}) that we can choose$\psi(t) \in C^{\infty,\mathfrak{a}}(\Omega_T)such that \begin{aligned} &(\mathscr{L}^{(1)}(\lambda_0(t),t)\varphi(t), \psi(t))_{\Omega}\\ &+\sum_{j=1}^m\big( \mathscr{B}_j^{(1)}(\lambda_0(t),t)\varphi(t),\mathscr{B}'_{j+m}(-\overline{\lambda}_0(t)+2m-n,t)\psi(t)\big)_{\partial \Omega} = 1 \end{aligned}\label{eq007} for allt\in [0,T]$. Moreover, applying~\cite[Th. 5.1.1]{KMR}, we assert that there exists a neighborhood$U$of the origin$O$such that in$U_T=U\times [0,T] $the inverse$\mathscr{U}^{-1}(\lambda,t)$has the following representation $$\mathscr{U}^{-1}(\lambda,t)=\frac{P_{-1}(t)}{\lambda-\lambda_0(t)} +\mathscr{P}(\lambda,t), \label{eq005}$$ where$P_{-1}(t)$is a 1-dimensional operator from from$\mathcal{Y}$into$\mathcal{X}$depending analytically on$t\in [0,T]$defined by $$P_{-1}(t)v=\left\langle\left\langle v, \psi(t)\right\rangle\right\rangle \varphi(t), \quad v \in \mathcal{Y}, \label{eq006}$$ and$\mathscr{P}(\lambda,t)$is a pencil of continuous operators from$\mathcal{Y}$into$\mathcal{X}$depending analytically on both$\lambda \in \mathbb{C}$and$t \in [0,T]$. Here $\left\langle\left\langle v, \psi(t)\right\rangle\right\rangle := (v_0,\psi(t))_{\Omega}+\sum_{j=1}^{m}\big(v_j, B'_{j+m}(-\lambda_0(t)+2m-n,t)\psi(t)\big)_{\partial\Omega}$ for$v=(v_0,v_1,\dots,v_m) \in \mathcal{Y}$. \section{ Asymptotic behaviour of the solutions}\label{sec3} In this section we investigate the behaviour of the solutions of the problem \eqref{eq203}, \eqref{eq204} in a neighborhood of the conical point. First, let us introduce some needed lemmas. \begin{lemma}\label{th22} Let$u \in V^{l_1,h}_{2,\beta_1}(G_T)$be a solution of the problem \begin{gather} \mathfrak{L}(t,\partial_x)u =f\quad \text{ in } G_T, \label{eq31}\\ \mathfrak{B}_j(t,\partial_x)u =g_j \text{ on } S_T, \; j=1,\dots, m, \label{eq32} \end{gather} where$f \in V^{l_2-2m,h}_{2,\beta_2}(G_T)$,$g_j \in V^{l_2-\mu_j-\frac{1}{2},h}_{2,\beta_2}(S_T)$,$l_1, l_2 \geq 2m$,$\beta_1-l_1> \beta_2-l_2$. Suppose that the lines$\mathop{\rm Re}\lambda =-\beta_i+l_i-\frac{n}{2}\, (i=1,2)$do not contain eigenvalues of the pencil$\mathscr{U}(\lambda,t)$, and all eigenvalues of this pencil in the strip$-\beta_1+l_1-\frac{n}{2}< \mathop{\rm Re}\lambda < -\beta_2+l_2-\frac{n}{2}$are simple for all$t\in [0,T] $which are chosen to be analytic functions$\lambda_1(t), \lambda_2(t), \dots, \lambda_N(t)$defined on$[0,T]$as the result of Lemma \ref{th21}. Then there exists a neighborhood$V$of the origin of$\mathbb{R}^n$such that in$V_T$the solution$u$has representation $$u(x,t)=\sum_{k =1}^N c_k(t)r^{\lambda_k(t)} \varphi_{k}(\omega,t)+w(x,t), \label{eq33}$$ where$w\in V^{l_2,h}_{2,\beta_2}(K_T)$,$c_k(t) \in W_2^h((0,T))$and$\varphi_{k}\in C^{\infty,\mathfrak{a}}(\Omega_T)$are eigenvectors of$\mathscr{U}(\lambda,t)$corresponding to the eigenvalues$\lambda_k(t)$,$k=1,\dots,N$. \end{lemma} \begin{proof} For each$k=1,\dots,N$, let$\psi_{k}(t)$be eigenvectors the pencil$\mathscr{U}^{+}(\lambda,t)$corresponding to the eigenvalues$\overline{\lambda}_k(t)(k=1,\dots,N)$having the properties as in (\ref{eq007}). Set$v_{k}=r^{-\overline{\lambda}_k(t)+2m-n}\psi_{k }$for$k=1, \dots, N$. (i) First, we assume that the function$u$has the support contained in$U_T$, where$U$is a certain neighborhood of$0 \in \mathbb{R}^n$in which the domain$G$coincides with the cone$K$. By extension by zero to$K_T$(respectively,$\partial K_T$) we can regard$u$,$f$(respectively,$g_j$) as functions defined in$K_T$(respectively,$\partial K_T$). For each$t \in [0,T]$fixed, according to results for elliptic boundary problem in a cone (see, e.g, \cite[Th. 6.1.4, Th. 6.1.7]{KMR}), the solution$u(x,t)$admits the representation (\ref{eq33}) in$Kwith $$w(x,t)=\frac{1}{2\pi i}\int_{\mathop{\rm Re}\lambda =-\beta_2+l_2-\frac{n}{2}}r^{\lambda}\mathscr{U}^{-1} (\lambda,t)\widetilde{\mathcal{F}}(\omega,\lambda,t)d\lambda \label{eq34}$$ and \begin{align*} c_k(t)&=\big(f(.,t),v_k(.,t)\big)_{K}+\sum_{j=1}^{m}\big(g_j(.,t), B'_{j+m}v_k(.,t)\big)_{\partial K}\\ &=\big(f(.,t),v_k(.,t)\big)_{G}+\sum_{j=1}^{m}\big(g_j(.,t), B'_{j+m}v_k(.,t)\big)_{S} \end{align*} fork = 1, \dots, N $, where$\widetilde{\mathcal{F}}=(\widetilde{r^{2m}f}, \widetilde{r^{\mu_1}g_1}$,$ \dots$,$ \widetilde{r^{\mu_m}g_m)}.$Here$\widetilde{g}(\omega,\lambda,t)$denotes the Mellin transformation with respect to the variable$r$of$g(\omega,r,t)$; i.e, $\widetilde{g}(\omega,\lambda,t) = \int_0^{+\infty}r^{-\lambda-1}g(\omega, r,t)dr.$ We will prove below that$w\in V^{l_2,h}_{2,\beta_2}(K_T)$,$c_k(t) \in W_2^h((0,T)) $. Now we make clear the first one. Since there are no eigenvalues of the operator pencil$\mathscr{U}(\lambda,t)$on the line$\mathop{\rm Re}\lambda = -\beta_2+l_2-\frac{n}{2}$, from the proof of \cite[Th. 3.6.1]{KMR} we have the estimate $$\|\mathscr{U}^{-1}(\lambda,t)\widetilde{\Psi}\|^2_{W^{l}_{2} (\Omega,\lambda)} \leq C\Big(\|\widetilde{\eta}\|^2_{W^{l-2m}_2(\Omega,\lambda)} +\sum_{j=1}^{m}\|\widetilde{\eta}_j\|^2_{W^{l-\mu_j -\frac{1}{2}}_{2}(\partial\Omega,\lambda)}\Big) \label{eq35}$$ for all$\lambda$on the line$\mathop{\rm Re}\lambda = -\beta_2+l_2-\frac{n}{2}$,$t \in [0,T]$, and all$\widetilde{\Psi} = (\widetilde{\eta}, \widetilde{\eta}_1,\dots , \widetilde{\eta}_m) \in W^{l-2m}_2(\Omega) \times\prod_{k=1}^mW^{l-\mu_k-\frac{1}{2}}_2(\partial\Omega)$, where the constant$C$is independent of$\lambda$,$t$and$\widetilde{\Psi}$. Here \begin{gather*} \|u\|_{W^{l}_{2}(\Omega,\lambda)}=\|u\|_{W^{l}_{2}(\Omega)} +|\lambda|^l\|u\|_{L_{2}(\Omega)},\\ \|u\|_{W^{l}_{2}(\partial\Omega,\lambda)} =\|u\|_{W^{l-\frac{1}{2}}_{2}(\partial\Omega)} +|\lambda|^{l-\frac{1}{2}}\|u\|_{L_{2}(\partial\Omega)}, \end{gather*} which are equivalent to the norms in$W^l_2(\Omega)$,$W^{l-\frac{1}{2}}_2(\partial\Omega)$, respectively, for arbitrary fixed complex number$\lambda$. We will prove by induction on$h$that $$\|(\mathscr{U}^{-1})_{t^h}(\lambda,t) \widetilde{\Psi}\|^2_{W^{l}_{2}(\Omega,\lambda)} \leq C(h)\Big(\|\widetilde{\eta}\|^2_{W^{l-2m}_2(\Omega,\lambda)} +\sum_{j=1}^{m}\|\widetilde{\eta}_j\|^2_{W^{l-\mu_j-\frac{1}{2}}_{2} (\partial\Omega,\lambda)}\Big). \label{eq36}$$ It holds for$h=0$by (\ref{eq35}). Assume that it holds for$h-1$. From the equality $\mathscr{U}(\lambda,t)\mathscr{U}^{-1}(\lambda,t)=I,$ differentiating both sides of it$h~(h\ge1)$times with respect to$twe obtain $\sum_{k=0}^{h-1}\tbinom{h-1}{k}\mathscr{U}_{t^{h-k}} (\lambda,t)(\mathscr{U}^{-1})_{t^{k}}(\lambda,t) +\mathscr{U}(\lambda,t)(\mathscr{U}^{-1})_{t^{h}}(\lambda,t)=0.$ Rewrite this equality in the form $(\mathscr{U}^{-1})_{t^{h}}(\lambda,t)=-\mathscr{U}^{-1}(\lambda,t) \sum_{k=0}^{h-1}\tbinom{h-1}{k}\mathscr{U}_{t^{h-k}} (\lambda,t)(\mathscr{U}^{-1})_{t^{k}}(\lambda,t).$ Then (\ref{eq36}) follows from this equality and the inductive assumption. It is well-known (see \cite[Le. 6.1.4]{KMR}) that the norm (\ref{eq205}) is equivalent to $||| u |||_{V^l_{2,\beta}(K)} =\Big(\frac{1}{2\pi i}\int_{\mathop{\rm Re}\lambda=-\beta+l -\frac{n}{2}}\|\widetilde{u}(.,\lambda)\|^2_{W_2^{l}(\Omega,\lambda)} d\lambda\Big)^{1/2},$ and the norm (\ref{eq206}) is equivalent to $||| u |||_{V^{l-\frac{1}{2}}_{2,\beta}(\partial K)}=\Big(\frac{1}{2\pi i}\int_{\mathop{\rm Re}\lambda=-\beta+l-\frac{n}{2}}\|\widetilde{u}(.,\lambda)\| ^2_{W_2^{l-\frac{1}{2}}(\partial\Omega,\lambda)}d\lambda\Big)^{1/2}.$ Using these with noting $\widetilde{w}(\omega,\lambda,t) = \mathscr{U}^{-1}(\lambda,t) \widetilde{\mathcal{F}}(.,\lambda,t)$ (see \cite[Le. 6.1.3]{KMR}) and (\ref{eq35}), we get from (\ref{eq34}) that \begin{align} &\|w(.,t)\|^2_{V^{l_2}_{2,\beta_2}(K)} \nonumber \\ &\leq \frac{C}{2\pi i}\int_{\mathop{\rm Re}\lambda=-\beta_2+l_2-\frac{n}{2}} \|\widetilde{w}(.,\lambda,t)\|^2_{W_2^{l_2}(\Omega,\lambda)}d\lambda \nonumber\\ &= \frac{C}{2\pi i} \int_{\mathop{\rm Re}\lambda=-\beta_2+l_2-\frac{n}{2}} \| \mathscr{U}^{-1}(\lambda,t)\widetilde{\mathcal{F}} (.,\lambda,t)\|_{W^{l_2}_2(\Omega,\lambda)}d\lambda \nonumber\\ &\leq \frac{C}{2\pi i} \int_{\mathop{\rm Re}\lambda=-\beta_2+l_2-\frac{n}{2}} \Big(\|\widetilde{r^{2m}f(.,t)}\|^2_{W^{l-2m}_2(\Omega,\lambda)} \nonumber\\ &\quad +\sum_{j=1}^{m}\|\widetilde{r^{\mu_j}g_j(.,t)} \|^2_{W^{l-\mu_j-\frac{1}{2}}_{2}(\partial\Omega,\lambda)}\Big)d\lambda \label{eq37} \\ &\leq C\Big(\|r^{2m}f(.,t)\|^2_{V^{l_2-2m}_{2,\beta_2-2m}(K)} +\sum_{j=1}^{m}\|r^{\mu_j}g_j(.,t)\|^2_{V^{l_2-\mu_j-\frac{1}{2}} _{2,\beta_2-\mu_j}(\partial K)}\Big) \nonumber\\ &\leq C\Big(\|f(.,t)\|^2_{V^{l_2-2m}_{2,\beta_2}(K)} +\sum_{j=1}^{m}\|g_j(.,t)\|^2_{V^{l_2-\mu_j-\frac{1}{2}}_{2,\beta_2} (\partial K)}\Big) \nonumber\\ &= C\Big(\|f(.,t)\|^2_{V^{l_2-2m}_{2,\beta_2}(G)} +\sum_{j=1}^{m}\|g_j(.,t)\|^2_{V^{l_2-\mu_j-\frac{1}{2}}_{2,\beta_2}(S)} \Big) \nonumber \end{align} for allt \in [0,T]$. Here, and sometimes later, for convenience, we denote different constants by the same symbol$C$. Integrating the last inequality with respect to t from$0$to$+\infty$, we obtain$w \in {V^{l_2}_{2,\beta_2}(K_T)}$and $\|w\|^2_{V^{l_2,0}_{2,\beta_2}(K_T)}\leq C\Big(\|f\|^2_{V^{l_2-2m,0}_{2,\beta_2}(G_T)} +\sum_{j=1}^{m}\|g_j\|^2_{V^{l_2-\mu_j-\frac{1}{2},0} _{2,\beta_2}(S_T)}\Big).$ Differentiating (\ref{eq34})$h$times with respect to$t$we have $w_{t^h}(x,t)=\frac{1}{2\pi i}\int_{\mathop{\rm Re}\lambda =-\beta_2+l_2-\frac{n}{2}}r^{\lambda}\sum_{k=0}^{h} \tbinom{h}{k}(\mathscr{U}^{-1})_{t^k}(\lambda,t) \widetilde{\mathcal{F}}_{t^{h-k}}(\omega,\lambda,t)d\lambda.$ Now using (\ref{eq36}) and arguments the same as in (\ref{eq37}) we arrive at $$\|w_{t^h}\|^2_{V^{l_2,0}_{2,\beta_2}(K_T)}\leq C\sum_{k=0}^{h}\Big(\|f_{t^k}\|^2_{V^{l_2-2m,0}_{2,\beta_2}(G_T)}+\sum_{k=1}^{m}\|(g_j)_{t^k}\|^2_{V^{l_2-\mu_j-\frac{1}{2},0}_{2,\beta_2}(S_T)}\Big).$$ Therefore,$w \in V^{l_2,h}_{2,\beta_2}(K_T)$and $$\|w\|^2_{V^{l_2,h}_{2,\beta_2}(K_T)}\leq C\Big(\|f\|^2_{V^{l_2-2m,h}_{2,\beta_2}(G_T)} +\sum_{j=1}^{m}\|g_j\|^2_{V^{l_2-\mu_j-\frac{1}{2},h}_{2,\beta_2}(S_T)} \Big).$$ Now we verify that$c_k(t) \in W_2^h((0,T))$for$k = 1, \dots, N $. For some such$kput $$v(x,t)=r^{-\overline{\lambda}_k(t)+2m-n}\psi_{k}(\omega,t).$$ Using formula (\ref{eq212}), we have \begin{aligned} \partial^{\alpha}v &= r^{-|\alpha|}\sum_{p=0}^{|\alpha|}(r\partial r)^{p} r^{-\overline{\lambda}_k(t)+2m-n}P_{\alpha,p}\psi_{k}\\ &= r^{-|\alpha|-\overline{\lambda}_k(t)+2m-n} \sum_{p=0}^{|\alpha|}(-\overline{\lambda}_k(t)+2m-n)^{p} P_{\alpha,p}\psi_{k}. \end{aligned}\label{eq38} Since\mathop{\rm Re}\lambda_k(t) < -\beta_2+l_2-\frac{n}{2}$for all$t \in [0,T]$and$\lambda_k(t)$is analytic on$[0,T]$, then there is a real number$\epsilon >0$such that$\mathop{\rm Re}\lambda_k(t) \leq -\beta_2+l_2-\frac{n}{2}-2\epsilon$for all$t \in [0,T]$. Thus, it follows from (\ref{eq38}) that $|r^{-\gamma_2+l_2-2m+|\alpha|}\partial^{\alpha}v(x,t)| \leq C r^{-\frac{n}{2}+\epsilon}\sum_{p=0}^{|\alpha|} |P_{\alpha,p}\psi_{k}(\omega,t)|$ for all$(x,t) \in G_T$and all multi-index$\alpha$. This implies$v(.,t) \in V^{l}_{2,-\beta_2+l_2-2m+l}(G)$and $\|v(.,t)\|_{V^{l}_{2,-\gamma_2+l_2-2m+l}(G)} \leq C\| \psi_{k}(.,t)\|_{W_2^l(\Omega)}$ for an arbitrary integer$l. Using Fa\{a} Di Bruno's Formula for the higher order derivatives of composite functions (see, e.g, \cite{Johnson}), we have \begin{align*} v_{t^p}&=\sum_{q=0}^{p}\binom{p}{q}\big(r^{-\overline{\lambda}_k(t)+2m-n}\big)_{t^{p-q}}(\psi_{k})_{t^q}\\ &=r^{-\overline{\lambda}_k(t)+2m-n}\sum_{q=0}^{p}\binom{p}{q} \sum\frac{n!}{m_1!\dots m_n!}(\ln r)^{m_1+\dots+m_n}\\ &\quad\times \prod_{s=1}^n\Big(\frac{-\lambda_k^{(s)}(t)}{s!} \Big)^{m_s}(\psi_{ k})_{t^q}, \end{align*} where the second sum is over alln-$tuples ($m_1, \dots, m_n)$satisfying the condition $m_1+2m_2+\dots+nm_n=n.$ According to Lemma \ref{th21},$\lambda_k(t)$is analytic on$[0,T]$. Therefore, it together with its derivatives are bounded on$[0,T]$. Repeating the arguments as above, we get $\|v_{t^p}(.,t)\|_{V^{l}_{2,-\gamma_2+l_2-2m+l}(G)} \leq C\sum_{q=0}^{p}\|(\psi_{k})_{t^q}\|_{W_2^l(\Omega)}.$ Thus, we have $$\sup_{t \in [0,T]}\|v\|_{V^{l,p}_{2,-\gamma_2+l_2-2m+l}(G)} \leq C\sum_{q=0}^{p}\sup_{t \in [0,T]}\|(\psi_{ k})_{t^q}\|_{W_2^l(\Omega)} <+\infty \label{eq39}$$ for arbitrary nonnegative integers$l,p$. Set$c(t)= (f(.,t),v(.,t))_{G}$. For$p\leq h, using (\ref{eq39}), we have \begin{align*} |c_{t^p}(t)|^2 &=\Big|\sum_{q=0}^{p}\tbinom{p}{q}(f_{t^{p-q}}(.,t),v_{t^q}(.,t))_{G} \Big|^2\\ &\leq C\Big(\sum_{q=0}^{p}\|r^{\beta_2-l_2+2m}f_{t^q}\|^2_{L_2(G)}\Big)\Big(\sum_{q=0}^{p}\|r^{-\beta_2+l_2-2m}v_{t^q}\|^2_{L_2(G)}\Big)\\ &\leq C\sum_{q=0}^{p}\|f_{t^q}\|^2_{V^{l_2-2m}_{2,\beta_2}(G)}. \end{align*} This impliesc(t) \in W_2^{h}((0,T))$and $$\|c\|_{W_2^{h}((0,T))} \leq C\|f\|_{V^{l_2-2m,h}_{2,\beta_2}(G_T)}. \label{eq310}$$ Now set$c_j(t)= (g_j,B'_{j+m}v)_{S}, j=1, \dots, m. Then also using (\ref{eq39}), we have \begin{align*} &|(c_j)_{t^p}(t)|^2\\ &=\Big|\sum_{q=0}^{p}\tbinom{p}{q}((g_j)_{t^{p-q}}(.,t),v_{t^q} (.,t))_{S}\Big|^2\\ &\leq C\Big(\sum_{q=0}^{p}\|r^{\beta_2-l_2+\mu_j +\frac{1}{2}}(g_j)_{t^q}\|^2_{L_2(G)}\Big)\Big(\sum_{q=0}^{p}\|r^{-\beta_2+l_2-\mu_j-\frac{1}{2}}(B'_{j+m}v)_{t^q}\|^2_{L_2(G)}\Big)\\ &\leq C\Big(\sum_{q=0}^{p}\|(g_j)_{t^q}\|^2_{V^{l_2-\mu_j -\frac{1}{2}}_{2,\beta_2}(S)}\Big). \Big(\sum_{q=0}^{p}\|v_{t^q}\|^2_{V^{2m-\mu_j}_{2,-\beta_2+l_2 -\mu_j}(G)}\Big)\\ & \leq C \sum_{q=0}^{p}\|(g_j)_{t^q}\|^2_{V^{l_2-\mu_j -\frac{1}{2}}_{2,\beta_2}(S)}(p \leq h). \end{align*} This impliesc_j \in W_2^h((0,T))$and $$\|c_j\|_{W_2^h((0,T))} \leq C \|g_j\|_{V^{l_2-\mu_j-\frac{1}{2},h}_{2,\beta_2}(S_T)}. \label{eq311}$$ From (\ref{eq310}) and (\ref{eq311}), we can conclude that$c_k(t) \in W_2^h((0,T))$and $$\|c_k\|_{W_2^h((0,T))} \leq C\Big(\|f\|_{V^{l_2-2m,h}_{2,\beta_2}(G_T)}+\sum_{j=1}^{m} \|g_j\|_{V^{l_2-\mu_j-\frac{1}{2},h}_{2,\beta_2}(S_T)}\Big).$$ (ii) Now we consider the case$u \in V^{l_1,h}_{2,\beta_1}(G_T)$is arbitrary. Let$\eta$be an infinitely differential function with support in$U$, equal to one in a neighborhood$V$of the origin. Denote by$\mathfrak{G}$the set of all subdomain$G'$of$G$with the smooth boundary such that$G \cap U\setminus V \subset G'$. We will show that$u \in W^{l_2,h}_2(G'_T)$for all$G' \in \mathfrak{G}$. To this end, we will prove by induction on$h$that $$u_{t^k} \in W^{l_2,0}_2(G'_T) \quad \text{for k=0,\dots,h and G' \in \mathfrak{G}.} \label{eq311e}$$ According to the results on the regularity of solutions of elliptic boundary problems in smooth domains, we can conclude from (\ref{eq31}), (\ref{eq32}) that$u(.,t) \in W^{l_2}_2(G')$for each$t \in [0,T]and \begin{align*} \|u(.,t)\|_{W^{l_2}_2(G')} &\leq C\Big(\|u(.,t)\|_{W^{l_1}_{2}(G'')} +\|f(.,t)\|_{W^{l_2-2m}_{2}(G'')}\\ &\quad +\sum_{j=1}^{m}\|g_j(.,t)\|_{W^{l_2-\mu_j-\frac{1}{2}}_{2} (S\cap \partial G'')}\Big), \end{align*} whereG'' \in \mathfrak{G}$such that$\overline{G'}\subset S\cup G''$and$C$is a constant independent of$u$,$f$,$g_j$and$t$. Integrating this inequality with respect to$t$from$0$to$T$we get$u \in W^{l_2,0}_2(G'_T)$. Thus (\ref{eq311e}) holds for$h=0$. Assume that it holds for$h-1$. Differentiating equalities (\ref{eq31}), (\ref{eq32}) with respect to$th$times and using the inductive assumption, we have \begin{gather*} Lu_{t^h}=f_{t^h}-\sum_{k=0}^{h-1}\binom{h}{k}L_{t^{h-k}}u_{t^k} \in W^{l_2-2m,0}_{2}(G''_T), \\ B_ju_{t^h} = (g_j)_{t^h}-\sum_{k=0}^{h-1}\binom{h}{k}(B_j)_{t^{h-k}} u_{t^k} \in W^{l_2-\mu_j-\frac{1}{2},0}_{2,\beta_2}(S_T\cap \partial G''_T), \end{gather*} where$G', G'' \in \mathfrak{G}$,$\overline{G'}\subset S\cup G''$. Applying the arguments above for$u_{t^h}$, we get$u_{t^h} \in W^{l_2,0}_2(G'_T)$. From (\ref{eq31}) we have $$\mathfrak{L}(\eta u) = \eta f +[\mathfrak{L},\eta]u \text{ in } G_T, \label{eq311a}$$ where$[\mathfrak{L},\eta]=\mathfrak{L}\eta-\eta\mathfrak{L}$is the commutator of$\mathfrak{L}$and$\eta$. Noting that$u \in W^{l_2,h}_2(G'_T)$for all$G' \in \mathfrak{G}$and$[\mathfrak{L},\eta]$is a differential expression (acting on$u$) of order$\leq 2m-1$with coefficients having the supports contained in$U\setminus V$, we have$[\mathfrak{L},\eta]u $is in$ W^{l_2-2m,h}_{2,\beta_2}(G_T)$. So is the right-hand side of (\ref{eq311a}). Similarly, we have $$\mathfrak{B}_j(\eta u) = \eta g_j +[\mathfrak{B}_j,\eta]u \in W^{l_2-\mu_j-\frac{1}{2},h}_{2,\beta_2}(S_T) (j=1,\dots,m). \label{eq311b}$$ Applying the the part (i) above for the function$\eta u$, we conclude from (\ref{eq311a}) and (\ref{eq311b}) that$u$admits the decomposition (\ref{eq33}) in$V_T$. The theorem is proved. \end{proof} \begin{lemma} \label{le22} Let \begin{gather} f= r^{\lambda_0(t)-2m}\sum_{\sigma=0}^{s}\frac{1}{\sigma !} (\ln r)^{\sigma}f_{s-\sigma}, \\ g_j= r^{\lambda_0(t)-\mu_j}\sum_{\sigma=0}^{s}\frac{1}{\sigma !} (\ln r)^{\sigma}g_{j,s-\sigma}, \quad j=1,\dots,m, \end{gather} where$f_{\sigma}\in W_2^{l-2m,h}(\Omega_T)$,$ g_{j,\sigma} \in W_2^{l-\mu_j-\frac{1}{2},h}(\partial\Omega_T)$,$\sigma =0, \dots, s$,$ j=1,\dots,m$and$\lambda_0(t)$be a complex-valued function defined on$[0,T]$. Suppose that if$\lambda_0(t)$is an eigenvalue of$\mathscr{U}(\lambda,t)$for some$t$, then$\lambda_0(t)$are simple eigenvalues of$\mathscr{U}(\lambda,t)$for all$t \in [0,T]$. Then there exists a solution$u$of (\ref{eq31}), (\ref{eq32}) which has the form $$u= r^{\lambda_0(t)}\sum_{\sigma=0}^{s+\kappa} \frac{1}{\sigma !}(\ln r)^{\sigma}u_{s+\kappa-\sigma} \label{eq312}$$ where$u_{\sigma}\in W_2^{l,h}(\Omega_T), \sigma=0,\dots,s+\kappa$. Here$\kappa=1$or$\kappa=0$according as$\lambda_0(t)$are simple eigenvalues of$\mathscr{U}(\lambda,t)$or not. \end{lemma} \begin{proof} According to (\ref{eq005}), the inverse of$\mathscr{U}(\lambda,t)$admits the representation $\mathscr{U}^{-1}(\lambda,t) = \sum_{k=-\kappa}^{+\infty}P_{k}(t) \big(\lambda-\lambda_0(t)\big)^{k},$ where$P_-1(t)$is defined in (\ref{eq006}) for the case$\kappa=1$, and $$P_k(t)=\frac{1}{n!}\frac{\partial^k \mathscr{P}}{\partial \lambda^k}(\lambda_0(t),t)$$ for$k=0,1,\dots.$It is obvious that$P_k(t)$,$k=-\kappa, -\kappa+1,\dots$, are continuous operators from$\mathcal{Y}$into$\mathcal{X}$depending analytically on$t$on$[0,T]$. From the equality $\mathscr{U}(\lambda,t) \mathscr{U}^{-1}(\lambda,t) = \sum_{k=-\kappa}^{+\infty}\Big(\sum_{q=0}^{\kappa+k} \frac{1}{q!}\mathscr{U}^{(q)}(\lambda_{0}(t),t)P_{k-q}(t)\Big) (\lambda-\lambda_0(t))^{k} = I$ it follows that $$\sum_{q=0}^{\kappa+k} \frac{1}{q!}\mathscr{U}^{(q)} (\lambda_{0}(t),t)P_{k-q}(t) =\delta_{k,0}, \quad k= -\kappa, -\kappa+1, \dots , \label{eq313}$$ where$\delta_{k,l}$is Kronecker symbol. Let$ube the function given in (\ref{eq312}). Then \begin{align*} \mathscr{U}(r\partial_r,t)u& = r^{\lambda_0(t)}\mathscr{U}(\lambda_0(t) +r\partial_r,t) \sum_{\sigma=0}^{s+\kappa}\frac{1}{\sigma !}(\ln r)^{\sigma} u_{s+\kappa-\sigma}\\ &= r^{\lambda_0(t)}\sum_{q=0}^{2m}\frac{1}{q!}\mathscr{U}^{(q)} (\lambda_0(t),t)(r\partial_r)^{q} \sum_{\sigma=0}^{s+\kappa}\frac{1}{\sigma !}(\ln r)^{\sigma} u_{s+\kappa-\sigma}\\ &= r^{\lambda_0(t)}\sum_{\sigma=0}^{s+\kappa}\frac{1}{\sigma !} (\ln r)^{\sigma} \sum_{q=0}^{s+\kappa-\sigma}\frac{1}{q!}\mathscr{U}^{(q)} (\lambda_0(t),t)u_{s+\kappa-\sigma-q} \end{align*} Settingv_{\sigma}=(f_{\sigma},g_{1,\sigma},\dots,g_{m,\sigma})$,$\sigma =0, \dots, s$, and $u_k=\sum_{p=0}^{\min(k,s)}P_{-\kappa+k-p}(t)v_p, k=1,\dots,s+\kappa,$ we get$u_k \in W_2^{l,h}(\Omega_T), k=0,\dots,s+\kappa. Using the equality (\ref{eq313}) we have \begin{align*} &\mathscr{U}(r\partial_r,t)u\\ &= r^{\lambda_0(t)}\sum_{\sigma=0}^{s+\kappa}\frac{1}{\sigma !}(\ln r)^{\sigma}\sum_{q=0}^{s+\kappa-\sigma}\frac{1}{q!}\mathscr{U}^{(q)}(\lambda_0(t),t) \sum_{p=0}^{\min(s+\kappa-\sigma-q,s)}P_{s-\sigma-q-p}(t)v_p\\ &=r^{\lambda_0(t)} \sum_{\sigma=0}^{s + \kappa}\frac{1}{\sigma!}(\ln r)^\sigma \sum_{p=0}^{\min(s+\kappa-\sigma,s)} \sum_{q=0}^{s+\kappa-\sigma - p} \frac{1}{q!}\mathscr{U}^{(q)}(\lambda_{0}(t),t) P_{s-\sigma-p-q}(t)v_p\\ &=r^{\lambda_0(t)}\sum_{p=0}^{s}\sum_{\sigma=0}^{s+\kappa-p} \frac{1}{\sigma!}(\ln r)^\sigma \Big(\sum_{q=0}^{s+\kappa-\sigma - p} \frac{1}{q!}\mathscr{U}^{(q)}(\lambda_{0}(t),t) P_{s-\sigma-p-q}(t)\Big)v_p \\ &=r^{\lambda_0(t)} \sum_{p=0}^{s}\sum_{\sigma=0}^{s+\kappa-p} \frac{1}{\sigma!}(\ln r)^\sigma \delta_{s+\kappa-\sigma - p,0} v_p\\ &= r^{\lambda_0(t)} \sum_{p=0}^{s}\frac{1}{(s-p)!}(\ln r)^{s-p}v_p = r^{\lambda_0(t)} \sum_{p=0}^{s}\frac{1}{\sigma!}(\ln r)^{\sigma}v_{s-\sigma}. \end{align*} Rewrite this equality in the form \begin{gather*} \mathscr{L}(\omega,t,\partial_\omega,r\partial_r)u =r^{2m}f \quad \text{in } G_T, \\ \mathscr{B}_j(\omega,t,\partial_\omega,r\partial_r)u =r^{\mu_j}g_j \quad \text{ on } S_T, \; j=1,\dots, m. \end{gather*} This impliesu$is a solution of (\ref{eq31}), (\ref{eq32}), and the lemma is proved. \end{proof} \begin{lemma}\label{le23} Let$u \in V^{l,h}_{2,\beta}(G_T)$be a solution of Problem \eqref{eq203}, \eqref{eq204}, where$f \in V^{l-2m+s,h}_{2,\beta+s}(G_T)$,$g_j \in V^{l_1-\mu_j+s-\frac{1}{2},h}_{2,\beta+s}(S_T)$,$l, s, h$are nonnegative integers,$l \geq 2m$. Then$u \in V^{l+s,h}_{2,\beta+s}(G_T)$and $$\|u\|^2_{V^{l+s,h}_{2,\beta+s}(G_T)} \leq C(\|f\|^2_{V^{l-2m+s,h}_{2,\beta+s}(G_T)} +\sum_{j=1}^{m}\|g_j\|^2_{V^{l_1-\mu_j+s-\frac{1}{2},h}_{2,\beta+s}(S_T)})$$ with the constant$C$independent of$u, f$and$g_j$. \end{lemma} \begin{proof} It is only needed to show that$u_{t^k} \in V^{l+s,0}_{2,\beta+s}(G_T)$and $$\|u_{t^k} \|^2_{V^{l+s,0}_{2,\beta+s}(G_T)} \leq C(\|f\|^2_{V^{l-2m+s,k}_{2,\beta+s}(G_T)} +\sum_{j=1}^{m}\|g_j\|^2_{V^{l_1-\mu_j+s-\frac{1}{2},k}_{2,\beta+s}(S_T)}) \label{eq316a}$$ for$k=0,\dots,h$, where$C$is a constant independent of$u, f$and$g_j$. We will prove this by induction on$h$. First, we fix some$t \in [0,T]$and consider \eqref{eq203}, \eqref{eq204} as an elliptic boundary-value problem (without parameter). Applying Corollary 6.3.2 of \cite{KMR}, we conclude that$u(.,t) \in V^{l+s}_{2,\beta+s}(G)$and $$\|u(.,t) \|^2_{V^{l+s}_{2,\beta+s}(G)} \leq C(\|f(.,t)\|^2_{V^{l-2m+s}_{2,\beta+s}(G)} +\sum_{j=1}^{m}\|g_j(.,t)\|^2_{V^{l_1-\mu_j+s -\frac{1}{2}}_{2,\beta+s}(S)}),$$ where$C$is a constant independent of$u, f, g_j$and$t \in [0,T]$. Integrating both sides of this inequality with respect to$t$from$0$to$T$we get $$\|u\|^2_{V^{l+s,0}_{2,\beta+s}(G_T)} \leq C(\|f\|^2_{V^{l-2m+s,0}_{2,\beta+s}(G_T)} +\sum_{j=1}^{m}\|g_j\|^2_{V^{l_1-\mu_j +s-\frac{1}{2},0}_{2,\beta+s}(S_T)}). \label{eq318a}$$ Thus (\ref{eq316a}) holds for$h=0$. Now suppose that it is true for$h-1$($h\geq 1$). Differentiating both sides of \eqref{eq203} with respect to$th$times and using the inductive assumption, we get $$Lu_{t^h}=f_{t^h}-\sum_{k=0}^{h-1}\binom{h}{k}L_{t^{h-k}}u_{t^k} \in V^{l-2m+s,0}_{2,\beta+s}(G_T). \label{eq317a}$$ Similarly, we have $$B_ju_{t^k} = (g_j)_{t^k}-\sum_{p=0}^{k-1}\binom{k}{p} (B_j)_{t^{k-p}}u_{t^p} \in V^{l_1-\mu_j +s-\frac{1}{2},0}_{2,\beta+s}(S_T) \label{eq319a}$$ for$j=1, \dots, m$. It is the same as in (\ref{eq318a}), we get from (\ref{eq317a}) and (\ref{eq319a}) that$u_{t^k} \in V^{l+s,0}_{2,\beta+s}(G_T)$and the estimate (\ref{eq316a}) holds. The proof is complete. \end{proof} Now let us give the main result of the present paper. \begin{theorem}\label{th23} Let$u \in V^{l_1,h}_{2,\beta_1}(G_T)$be a solution of Problem \eqref{eq203}, \eqref{eq204}, where$f \in V^{l_2-2m,h}_{2,\beta_2}(G_T)$,$g_j \in V^{l_1-k_j-\frac{1}{2},h}_{2,\beta_2}(S_T)$,$l_1, l_2, h$are nonnegative integers,$l_1, l_2 \geq 2m$,$l_1-\beta_1< l_2-\beta_2$. Suppose that there are real numbers$\delta_0,\delta_2, \dots,\delta_M$such that $$\delta_0=\beta_1+l_2-l_1, \delta_{M}= \beta_2,\quad 0< \delta_{d-1}-\delta_{d}\leq 1, \quad d =1,\dots, M,$$ and the lines$\mathop{\rm Re}\lambda =-\delta_{d}+l_2-\frac{n}{2}$,$d =0,\dots, M$, do not contain eigenvalues of the pencil$\mathscr{U}(\lambda,t)$. Furthermore, suppose that all eigenvalues of this pencil in the strip$-\delta_0+l_2-\frac{n}{2}< \mathop{\rm Re}\lambda < -\delta_M+l_2-\frac{n}{2}$are simple for all$t\in [0,T] $which are chosen to be analytic functions$\lambda_1(t), \lambda_2(t), \dots, \lambda_N(t)$defined on$[0,T]$as the result of Lemma \ref{th21}. If$\lambda_{j}(t_0)= \lambda_k(t_0)+s$for some$j, k \in \{1, \dots, N\}$, for some integer$s$and$t_0 \in [0,T]$, then let$\lambda_{j}(t)= \lambda_k(t)+s$for all$t \in [0,T]$. Then the solution$u$admits the decomposition $$u=\sum_{k =1}^N\sum_{\tau =0}^{\ell_k}r^{\lambda_k(t)+\tau }P_{k,\tau }(\ln r)+w, \label{eq316}$$ where$w\in V^{l_2,h}_{2,\beta_2}(G_T)$,$P_{k,\tau}$are polynomials with coefficients belonging to$W_2^{l_2,h}(\overline{\Omega}_T)$,$\ell_k$is the minimal integer greater than$-\delta_M-\lambda_k(t)-1+l_2-\frac{n}{2}$for all$t \in [0,T]$. \end{theorem} \begin{proof} According to Lemma \ref{le23},$u \in V^{l_2,h}_{2,\delta_0}(G_T)$. Suppose, by renumbering if necessary,$\mathop{\rm Re}\lambda_1(t)<\mathop{\rm Re}\lambda_2(t)< \dots < \mathop{\rm Re}\lambda_N(t)$for all$t \in [0,T]$. For each$d \in \{1,\dots,M\}$denote by$N_{d}$the maximal integer in$\{0, 1, \dots, N\}$such that$\lambda_1(t), \lambda_2(t), \dots, \lambda_{N_d}(t)$belong to the strip$-\delta_0+l_2-\frac{n}{2}<\mathop{\rm Re}\lambda < -\delta_d+l_2-\frac{n}{2}$and by$\ell_{k,d}$the minimal integer greater than$-\delta_{d}-\lambda_k(t)-1+l_2-\frac{n}{2}$for all$t \in [0,T]$. We will prove by induction on$d (1\leq d \leq M)$that the function$u$can be represented in the form $$u=\sum_{k =1}^{N_d}\sum_{\tau =0}^{\ell_{k,d}}r^{\lambda_k(t)+\tau } P_{k,\tau}^{(d)}(\ln r)+u_{d}, \label{eq317}$$ where$P_{k,\tau}^{(d)}$are polynomials with coefficients belonging to$W_2^{l_2,h}(\Omega_T)$and$u_{d}\in V^{l,h}_{2,\delta_{d}}(G_T)$. Then (\ref{eq317}) for$d=M$proves the theorem. We rewrite \eqref{eq203}, \eqref{eq204} in the form \begin{gather} \mathfrak{L}u=f+(\mathfrak{L}-L)u\equiv f+L'u \quad\text{in } G_T, \label{eq324}\\ \mathfrak{B}_ju=g_j+(\mathfrak{B}_j-B_{j})u\equiv g_j+B_{j}'u \quad \text{on } S_T, j=1,\dots, m. \label{eq325} \end{gather} We write $L'u=\sum_{|\alpha|=2m}(a_\alpha(0,t)-a_\alpha(x,t))\partial^{\alpha}u+\sum_{|\alpha|\leq 2m-1}a_\alpha(x,t)\partial^{\alpha}u\equiv L_1u+L_2u.$ Since$|a_{\alpha}(x,t)-a_{\alpha}(0,t)|\leq Cr$, and$u \in V^{l_2,h}_{2,\delta_{0}}(G_T)$, we have$L_1u \in V^{l_2-2m,h}_{2,\delta_{0}-1}(G_T) $. Otherwise,$L_2u \in V^{l_2-2m+1,h}_{2,\delta_{0}}(G_T) \subset V^{l_2-2m,h}_{2,\delta_{0}-1}(G_T)$. Since$\delta_0+1\geq \delta_1$and$V^{l_2-2m,h}_{2,\delta_{0}-1}(G_T) \subset V^{l_2-2m,h}_{2,\delta_{1}}(G_T)$. Therefore,$f+L'u \in V^{l_2-2m,h}_{2,\delta_{1}}(G_T)$. Similarly,$g_j+B_{j}'u \in V^{l_2-k_j-\frac{1}{2},h}_{2,\delta_1}(S_T)$. Now we can apply Lemma \ref{th22} to conclude that $$u= \sum_{k=1}^{N_2}r^{\lambda_k(t)}c_{k}(t)\varphi_{k}+u_1,$$ where$c_{k}(t) \in W_2^h((0,T))$,$u_1 \in V^{l_2,h}_{2,\delta_{1}}(G_T)$. Thus (\ref{eq317}) holds for$d=1$with$P_{k,\tau}^{(1)}(\ln r)=c_{k}(t)\varphi_{k}$. We assume now (\ref{eq317}) is true for some$d(1\leq d \leq M-1)$. Then we can rewrite \eqref{eq203}, \eqref{eq204} in the form \begin{gather} \mathfrak{L}u_d =f +(\mathfrak{L}-L)u_d-L z\quad \text{in } G_T, \label{eq322}\\ \mathfrak{B}_ju_d =g_j+(\mathfrak{B}_j-B_{j})u_d-B_{j}z \quad \text{on } S_T, j=1, \dots, m, \label{eq323} \end{gather} where $$z=\sum_{k =1}^{N_d}\sum_{\tau =0}^{\ell_{k,d}}r^{\lambda_k(t) +\tau }P_{k,\tau}^{(d)}(\ln r).$$ Since the coefficients$a_{\alpha}, |\alpha|\leq 2m$, belong to the class$C^{\infty,\mathfrak{a}}(\overline{G}_T)$, then, for an arbitrary nonnegative integer$k$, they admit representation $$a_\alpha =a_{\alpha}(\omega,r,t)= \sum_{\delta=0}^{k} r^{\delta}a_{\alpha}^{(\delta)}(\omega,t)+r^{k+1} a_{\alpha}^{(k+1)}(\omega,r,t)$$ where$a_{\alpha}^{(\delta)} \in C^{\infty,\mathfrak{a}} (\overline{\Omega}_T) $for$\delta = 0,\dots, k$, and$ a_{\alpha}^{(k+1)} \in C^{\infty,\mathfrak{a}} (\overline{G}_T)$. Thus, we can write the operator$L$in the form $$L=\sum_{\delta=0}^{\ell_{k,d+1}}r^{-2m+\delta} \mathscr{L}^{(\delta)}(\omega,t,\partial_{\omega},r\partial_r) +r^{-2m+\ell_{k,d+1}+1}\mathscr{L}^{(\ell_{k,d+1}+1)} (\omega,r,t,\partial_{\omega},r\partial_r)$$ where$\mathscr{L}^{(\delta)}(\omega,t,\partial_{\omega},r\partial_r), \delta =0,\dots,\ell_{k,d+1}$, and$ \mathscr{L}^{(\ell_{k,d+1}+1)} (\omega,r,t,\partial_{\omega},r\partial_r)$are polynomials of$\partial_\omega$and$r\partial r$of order not greater than$2m$with coefficients in$C^{\infty,\mathfrak{a}}(\overline{\Omega}_T)$and$C^{\infty,\mathfrak{a}}(\overline{G}_T), respectively. \begin{aligned} &L\Big(\sum_{\tau =0}^{\ell_{k,d}}r^{\lambda_k(t)+\tau } P_{k,\tau}^{{(d)}}(\ln r)\Big)\\ &=\sum_{\delta=0}^{\ell_{k,d+1}}\sum_{\tau=0}^{\ell_{k,d}} r^{-2m+\lambda_k(t)+\tau +\delta}\mathscr{L}^{(\delta)} (\omega,t,\partial_{\omega},\lambda_k(t)+\tau+r\partial_r) P_{k,\tau}^{{(d)}}(\ln r)\\ &\quad + \sum_{\tau =0}^{\ell_{k,d}}r^{-2m+\lambda_k(t) +\ell_{k,d+1}+1+\tau } \mathscr{L}^{(\ell_{k,d+1}+1)} (\omega,r,t,\partial_{\omega},\lambda_k(t) +\tau+ r\partial_r)P_{k,\tau}^{{(d)}}(\ln r). \end{aligned} \label{eq318} Write the first term of the right-hand side of (\ref{eq318}) in the form $$\sum_{\tau=0}^{\ell_{k,d+1}}r^{-2m+\lambda_k(t) +\tau}\Psi_{k,\tau}^{{(d+1)}}(\ln r) +\sum_{\tau=\ell_{k,d+1}+1}^{\ell_{k,d}+\ell_{k,d+1}} r^{-2m+\lambda_k(t)+\tau}\Psi_{k,\tau}^{{(d+1)}}(\ln r), \label{eq319}$$ where\Psi_{k,\tau}^{{(d+1)}}$are polynomials with coefficients belonging to$W_2^{l_2,h}(\Omega_T)$. Since$\ell_{k,d+1}>-\delta_{d+1}-\lambda_k(t)-1+l_2-\frac{n}{2}$for all$t \in [0,T]$, then$-2m+\lambda_k(t)+\ell_{k,d+1}+1> -\delta_{d+1}+l_2-2m-\frac{n}{2}$for all$t \in [0,T]$. Thus, the final terms in (\ref{eq318}) and (\ref{eq319}) belong to$V^{l-2m,h}_{2,\delta_{d+1}}(G_T)$. Hence, (\ref{eq318}) can be rewritten in the form $$L\Big(\sum_{\tau =0}^{\ell_{k,d}}r^{\lambda_k(t)+\tau } P_{k,\tau}^{{(d)}}(\ln r)\Big)=\sum_{\tau=0}^{\ell_{k,d+1}} r^{-2m+\lambda_k(t)+\tau}\Psi_{k,\tau}^{{(d+1)}}(\ln r) + w_k$$ where$w_k \in V^{l-2m,h}_{2,\delta_{d+1}}(G_T)$. Analogously, we can write $$B_j\Big(\sum_{\tau =0}^{\ell_{k,d}}r^{\lambda_k(t)+\tau } P_{k,\tau}^{{(d)}}(\ln r)\Big)=\sum_{\tau=0}^{\ell_{k,d+1}} r^{-k_j+\lambda_k(t)+\tau}\Psi_{k,\tau,j}^{{(d+1)}}(\ln r) + w_{k,j}$$ for$j=1,\dots,m$, where$\Psi_{k,\tau,j}^{{(d+1)}}$are polynomials with coefficients belonging to$W_2^{l_2,h}(\partial\Omega_T)$and$w_{k,j} \in V^{l-k_j-\frac{1}{2},h}_{2,\delta_{d+1}}(\partial\Omega_T)$. According to Lemma \ref{le22}, there are polynomials$\Phi_{k,\tau}^{{(d+1)}}$with coefficients belonging to$W_2^{l_2,h}(\Omega_T)$such that \begin{gather} \mathfrak{L}(t,\partial_x)\Big(r^{\lambda_k(t)+\tau } \Phi_{k,\tau}^{{(d+1)}}(\ln r)\Big)=r^{-2m+\lambda_k(t) +\tau}\Psi_{k,\tau}^{{(d+1)}}(\ln r) \text{ in } G_T, \label{eq320}\\ \mathfrak{B}_j(t,\partial_x)\Big(r^{\lambda_k(t)+\tau } \Phi_{k,\tau}^{{(d+1)}}(\ln r)\Big)=r^{-k_j+\lambda_k(t) +\tau}\Psi_{k,\tau,j}^{{(d+1)}}(\ln r) \text{ on } S_T. \label{eq321} \end{gather} Set $$v=\sum_{k =1}^{N_d}\sum_{\tau=0}^{\ell_{k,d+1}} r^{\lambda_k(t)+\tau }\Phi_{k,\tau}^{{(d+1)}}(\ln r).$$ Now we rewrite (\ref{eq322}), (\ref{eq323}) in the form \begin{gather} \mathfrak{L}(t,\partial_x)(u_d+v) =f+(\mathfrak{L}-L)u_d +\sum_{k =1}^{N_d}w_{k}\quad\text{in } G_T,\label{eq326}\\ \mathfrak{B}_j(t,\partial_x)(u_d+v)=g_j+(\mathfrak{B}_j-B_j)u_d +\sum_{k =1}^{N_d}w_{k,j} \quad\text{on } S_T,\; j=1,\dots,m. \label{eq327} \end{gather} It is the same as in (\ref{eq324}), (\ref{eq325}) that$(\mathfrak{L}-L)u_d \in V^{l-2m,h}_{2,\delta_{d+1}}(G_T)$and$(\mathfrak{B}_j-B_j)u_d \in V^{l-k_j -\frac{1}{2},h}_{2,\delta_{d+1}}(S_T)$. Thus the right-hand sides of (\ref{eq326}), (\ref{eq327}) belong to$V^{l-2m,h}_{2,\delta_{d+1}} (G_T)$,$V^{l-k_j-\frac{1}{2},h}_{2,\delta_{d+1}}(S_T)$, respectively. Now we can apply Lemma \ref{th22} to conclude from (\ref{eq326}), (\ref{eq327}) that $$u_d+v=\sum_{k =1}^{N_{d+1}}\sum_{\tau =0}^{\ell_{k,d+1}} r^{\lambda_k(t)+\tau }\Upsilon_{k,\tau }^{(d+1)}(\ln r)+u_{d+1},$$ where$\Upsilon_{k,\tau }^{(d+1)}$are polynomials with coefficients belonging to$W_2^{l_2,h}(\Omega_T)$and$u_{d+1}\in V^{l_2,h}_{2,\delta_{d+1}}(G_T)$. Setting$P_{k,\tau }^{(d+1)}=P_{k,\tau }^{(d)} - \Phi_{k,\tau }^{(d+1)} +\Upsilon_{k,\tau }^{(d+1)}$for$k=1,\dots,N_d$, and$P_{k,\tau }^{(d+1)}(\ln r)=\Upsilon_{k,\tau }^{(d+1)}(\ln r)$for$k=N_{d+1}$, we have $u=\sum_{k =1}^{N_{d+1}}\sum_{\tau =0}^{\ell_{k,d+1}}P_{k,\tau }^{(d+1)}(\ln r)+u_{d+1}.$ This implies that (\ref{eq317}) holds for$d+1$. The proof is complete. \end{proof} \noindent{\bf Acknowledgment.} This work was supported by National Foundation for Science and Technology Development (NAFOSTED), Vietnam. \begin{thebibliography}{00} \bibitem{Kato} T. Kato; \emph{Perturbation theory for linear operators}, Springer-Verlag, 1966. \bibitem{Kondratiev} V. A. Kondrat'ev, O. A. Oleinik; \emph{Boundary value problems for partial differential equations in nonsmooth domains}, Usp. Math. Nauka 38 (1983), N$^\circ\$\,3 (230), 3-76 (in Russian). \bibitem{KMR} V. A. Kozlov, V. G. Maz'ya and J. Rossmann; \emph{Elliptic boundary problems in domains with point singularities}, Mathematical Surveys and Monographs 52, Amer. Math. Soc., Providence, Rhode Island, 1997. \bibitem{KMR1} Kozlov, V. A., Maz'ya, V. G., Rossmann, J.; \emph{Spectral problems associated with corner singularities of solutions to elliptic equations}, Mathematical Surveys and Monographs 85, Amer. Math. Soc., Providence, Rhode Island, 2001. \bibitem{H} N. M. Hung; \emph{Asymptotic behaviour of solutions of the first boundary-value problem for strongly hyperbolic systems near a conical point at the boundary of the domain}, Mat. Sb., 190 (1999), No. 7, 103-126. \bibitem{HA1} N. M. Hung and N. T. Anh; \emph{Regularity of solutions of initial-boundary-value problems for parabolic equations in domains with conical points}, J. Differential Equations 245 (2008), 1801-1818. \bibitem{Johnson} W. P. Johnson; \emph{The Curious History of Fa\{a} di Bruno's Formula}, American Mathematical Monthly, Vol. 109, March 2002, 217-234. \bibitem{MP} V. G. Maz'ya and B. A. Plamenevskii; \emph{Coefficients in the asymptotics of the solutions of an elliptic boundary-value problem in a cone}. J. Sov. Math. 9 (1978), 750-764. \bibitem{MP1} V. G. Maz'ya and B. A. Plamenevskii; \emph{On the coefficients in the asymptotic of solutions of the elliptic boundary problem in domains with conical points}. Amer. Math. Soc. Trans. (2) 123 (1984), 57-88. \bibitem{Markus} A. S. Markus; \emph{Introduction to the spectral theory of polynomial operator pencils}, Translations of Mathematical Monographs, vol. 71, American Mathematical Society, Providence, Rhode Island 1988. \bibitem{ReedSimon} M. Reed, B. Simon; \emph{Methods of Modern Mathematical Physics, IV: Analysis of Operators}, Academic Press, London, 1978. \end{thebibliography} \end{document}