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PresentationsPerturbations of the wave equation which has the finite time extinctionNovosibirsk State University, Sobolev Institute of Mathematics, Russia, 630090, Novosibirsk, prosp. Koptyuga, 4, Phone: 8-9231979374, E-mail: natlyl@mail.ru
In the semi-strip $\Pi=(0,1)\times (0,\infty)$ we consider the initial-boundary value problem for the wave equation in the one-dimensional case \begin{equation} \label{eq:lyl1} u_{tt}-a^2u_{xx}=0, \qquad (x,t)\in \Pi. \end{equation} The solution $u$ to this problem fulfills the boundary conditions \begin{equation} \label{eq:lyl2} u(0,t)= p(u_t+au_x)(0,t), \qquad(u_t+au_x)(1,t)=0 \qquad t>0, \end{equation} or \begin{equation} \label{eq:lyl3} u(1,t)= q(u_t-au_x)(1,t), \qquad(u_t-au_x)(0,t)=0 \qquad t>0, \end{equation} and the initial data at $t=0$, namely \begin{equation} \label{eq:lyl4} u(x,0)= u_0(x), \qquad u_t(x,0)=u_1(x), \qquad x\in[0,1]. \end{equation} Here is $a>0$ and $p,q$ are arbitrary constants. In \cite{Lyl} it is proved that the finite time extinction for problems \eqref{eq:lyl1}, \eqref{eq:lyl2}, \eqref{eq:lyl4} and \eqref{eq:lyl1}, \eqref{eq:lyl3}, \eqref{eq:lyl4} equals $T=\dfrac{2}{a}$. A similar example in the case $p=0$ is given in \cite{Bal}.
In \cite{Lyl} along with the equation \eqref{eq:lyl1} we consider its perturbed version, namely \begin{equation} \label{eq:lyl5} u_{tt}-a^2u_{xx}+c(x,t)u=0, \qquad (x,t)\in \Pi, \end{equation} where $c(x,t)$ is a two times continuously differentiable function such that $c(x,t)$ itself and its first order and second order derivatives are bounded in $\overline{\Pi}$. In \cite{Lyl} it is proved that for any initial functions $u_0\in L_2(0,1)$, $u_1\in W_2^1(0,1)$ the solutions to problems \eqref{eq:lyl5}, \eqref{eq:lyl2}, \eqref{eq:lyl4} and \eqref{eq:lyl5}, \eqref{eq:lyl3}, \eqref{eq:lyl4} become continuously differentiable in a finite time. Moreover, if $sup_{x,t\in \overline{\Pi}}(\sum_{0\le \alpha+\beta\le 2}|D^{\alpha,\beta}_{x,t}c(x,t)|)$ is small, then the problems \eqref{eq:lyl5}, \eqref{eq:lyl2}, \eqref{eq:lyl4} and \eqref{eq:lyl5}, \eqref{eq:lyl3}, \eqref{eq:lyl4} are exponentially stable in $L_2(0,1)$
\begin{thebibliography}{100} \bibitem{Lyl} \textit{Kmit I.Y., Lyulko N.A.} Asymptotic behavior of solutions to perturbed superstable wave equations// \textit{J. Phys.: Conf. Series} \textbf{894}, 012056, 2017. \bibitem{Bal} \textit{Balakrishnan A.V.} Superstability of systems// \textit{J. Appl. Math. and Comput.} \textbf{164}, \textbf{2}, 2005, p. 321-326.
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