Existence of a Global Attractor for Semilinear Dissipative Wave Equations on N

We consider the semilinear hyperbolic problem utt+δut−φ(x) Δu+λf(u)=η(x), x N, 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( N). The energy space 0= 1, 2( N)×L2g( N) 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 0. The compactness of the embedding 1, 2( N) L2g( N) is widely applied.