Existence of a Global Attractor for Semilinear Dissipative Wave Equations on [formula omitted

Abstract

We consider the semilinear hyperbolic problem utt+δut−φ(x)Δu+λf(u)=η(x), x∈RN, t>0, with the initial conditions u(x, 0)=u0(x) and ut(x, 0)=u1(x) in the case where N⩾3 and (φ(x))−1:=g(x) lies in LN/2(RN). The energy space X0=D1, 2(RN)×L2g(RN) is introduced, to overcome the difficulties related with the noncompactness of operators which arise in unbounded domains. We derive various estimates to show local existence of solutions and existence of a global attractor in X0. The compactness of the embedding D1, 2(RN)⊂L2g(RN) is widely applied.