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Steklov Institute of Mathematics at St.Petersburg

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PREPRINT 03/2006

M. I. Belishev##
DYNAMICAL INVERSE PROBLEM FOR A LAM\'E
TYPE SYSTEM

This preprint was accepted February 9, 2006

ABSTRACT:
The system under consideration is governed
by the equation $u_{tt}=\nkd u - \rmr u$ in $\Omega \times (0,T)$;
its response operator ("input $\mapsto$ output" map) $R^T$ plays
the role of the inverse data. It is shown that $R^{2T}$ determines
$\kappa \mid_{\OTp}$ and $\mu \mid_{\OTs}$, where $\OTp$ and
$\OTs$ are the subdomains of $\Omega$ filled (at the moment $T$)
with $p-$ and $s-$waves propagating from $\partial \Omega$ with
velocities $c_p = \sqrt \kappa$ and $c_s = \sqrt \mu$
correspondingly. Due to the wave splitting $u=\n p + \r s$ the
problem is reduced to the inverse problems for the acoustical and
Maxwell subsystems governed by the equations $p_{tt}=\kappa \Delta
p$ and $s_{tt}= - \mrr s$ with the response operators $R^{2T}_p$
and $R^{2T}_s$ determined by $R^{2T}$. The first problem can be
solved by the BC method (Belishev, 1986), the second one is solved
by a
version of the method based on the blow up effect. This version
is the main subject of the paper.

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