Stabilization of solutions of an exterior boundary value problem for some class of evolution systems

Abstract

(1) utt + L(x,Dx)u = 0 , u|t=0 = f1(x) , ut|t=0 = f2(x) , Bj(x,Dx,Dt)u|S = 0 , j = 1, . . . , l , where the boundary operators Bj are such that problem (1) is well posed. Our purpose is to study asymptotics as t → ∞ of solutions of this problem for some class of operators L and Bj . Many authors have studied the behavior as t → ∞ of solutions of exterior boundary value problems for hyperbolic equations. These questions have been most completely investigated for the wave equation: qualified decay has been obtained (powerlike in the case of an even number of space variables and exponential in the case of an odd number). The work in this direction can be divided into two groups, each with its own approach. The first approach consists in application of the Laplace transform in the variable t with subsequent study of the analytic properties of the resolvent and its behaviour for the large and small values of the spectral parameter. In [14], [19], [20] this program is realized for various boundary problems for the wave equation (−L(x,Dx) = ∆n, the Laplace operator in x1, . . . , xn; n = 2, 3) in the exterior of a bounded convex domain. The second approach, based on nonstandard energy identities, was applied by C. Morawetz to the wave equation with the Dirichlet boundary condition