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Abstracts

XXIII conference

Perturbation of linear first-order hyperbolic systems, which have the property of superstability

Lyulko N.A.

Russia, 630090, Novosibirsk, pr.Koptyuga, 4,

1 pp. (accepted)

documentclass{mce}

\begin{document}

\title{Perturbation of linear first-order hyperbolic systems, which have property of superstability }

\author[]{Lyulko N.A.}

\contacts[]{Sobolev Institute of Mathematics, Russia, 630090, Novosibirsk, pr. Koptyuga, 4, phone: 8-923-1979374, E-mail:natlyl@mail.ru}

\maketitle

The boundary value problem in the half-strip $\Pi=[0,1]\times (0,\infty)$ for hyperbolic system

\begin{equation}

\label{eq:luleq1}

U_t= AU, \quad U(x,0)=U_0(x)

\end{equation}

has been investigated in autors work \cite{lul}. Here $U=(u_1,...,u_n)^T$ is a vector of real-valued functions; $AU=-K(x)U_x+B(x)U$, where $K(x)$ is diagonal matrix with different elements from each other $k_i(x)>0$, $(i=1,..,p)$, $k_i(x)0$ such that the solution $U(x,t)$ to this problem stabilize to zero in a finite time $T>0$ for all $U_0\in L_2(0,1)$.

In this work we consider the perturbed problem \eqref{eq:luleq1}, \eqref{eq:luleq2} with operator $AU=-K(x)U_x+(B(x)+C(x,t))U$, where $C(x,t)$ is an arbitrary matrix with diagonal elements equal to zero. We have

\textit{Тheorem.} Let the unperturbed problem \eqref{eq:luleq1}, \eqref{eq:luleq2} has the property of superstability. Then

- for any matrix $C(x,t)$ the perturbed problem has the property to increase the smoothness of solutions, that is, the solution $U(x,t)$ to this problem will be a continuous function by $t>T$ for all $U_0\in L_2(0,1)$;

- for any $\gamma>0$ exists $\varepsilon>0$ such that for any matrix $C(x,t): ||C(x,t)||_{C(\Pi)} T,$$

\begin{thebibliography}{100}

\bibitem{lul} \textit{Ёltysheva N.А.} About qualitative properties of solutions of some hyperbolic systems on the plane// \textit{Math.sb.} \textbf{135}, 2, 1988. С.186-209.

\bibitem{lul1} \textit{D.Creutz, M.Mazo,Jr. and C.Preda.} Superstability and finite time extriction for $C_0$-semigroups // arXiv:0907/4812v4[math.FA] 24 sep 2013, p.12.

\end{thebibliography}

\end{document}



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