On a framework of scattering for dissipative systems

Abstract

In this paper we study the existence of scattering solutions for some dissipative systems which contain elastic wave with dissipative boundary conditions in a half space of R3 (cf. Dermenjian-Guillot [1]). First we give a framework based on the idea of Simon [18] and apply it to elastic wave mentioned above. In applying the abstract framework, we shall use the Mellin transformation (cf. Perry [14]) as a key tool. Let H be separable Hilbert space with inner 〈· ·〉H. The norm is denoted by ‖ · ‖H. Let { ( )} ≧0 and { 0( )} ∈R be a contraction semigroup in H and a unitary group in H0, respectively. We denote the generator of ( ) and 0( ) by and 0, respectively ( ( ) = − and 0( ) = − 0 ). We make the following assumptions on and 0. (A1) σ( 0) = σ ( 0) = R or [0 ∞). (A2) ( − )−1 − ( 0 − )−1 defined as a form is extended to a compact operator in H. (A3) There exist non-zero projection operators in H, + and −, such that + + − = and