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Scattering by periodically moving obstacles
B. R. Vainberg
Abstract:
Suppose
$x\in\mathbf R^n$,
$L_0(\partial_t,\partial_x)$ is a homogeneous hyperbolic matrix,
$U_0(t)$ is the operator taking the Cauchy data for the system
$L_0u=0$ for
$t=0$ into the corresponding data at time
$t$, and
$U(t)$ is the analogous operator constructed from the exterior mixed problem for the hyperbolic system
$Lu=0$. It is assumed that the boundary of the domain and the coefficients of the operator
$L$ are periodic in
$t$ with period
$T$,
$L=L_0$ for
$|x|\gg1$, the noncapturing condition is satisfied, the matrix
$L_0(0,\partial_x)$ is elliptic, and the energy of solutions of the exterior problem is uniformly bounded for
$t\geqslant 0$.
Under these conditions it is proved that the space
$H$ generated by the eigenfunctions of the monodromy operator
$V=U(T)$ with eigenvalues on the unit circle is finite dimensional; for initial data
$f$ with compact support the asymptotics of the solution
$U(t)f$ of the exterior problem as
$t\to\infty$ is obtained; in particular, it is shown that
$U(t)f\sim U(t)Pf$,
$t\to\infty$, where
$P$ is the operator of projection onto
$H$; and existence of the wave operators constructed on the basis of
$U_0(t)$ and
$U(t)$ and of the scattering operator is proved.
UDC:
517.9
MSC: Primary
35P25,
35L30; Secondary
35B40 Received: 18.05.1990
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