Русский
 ! Вопросы и ответы

## Presentations

### Perturbations of the wave equation which has the finite time extinction

Lyulko N.A.

Novosibirsk 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

\label{eq:lyl1}

The solution $u$ to this problem fulfills the boundary conditions

\label{eq:lyl2}

or

\label{eq:lyl3}

and the initial data at $t=0$, namely

\label{eq:lyl4}

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

\label{eq:lyl5}

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 $L_2(0,1)$