On The Global Attractor For A Semilinear Dissipative Wave Equations With Linear Memory On R~N

We consider the following semilinear dissipative wave equations with linearthe difficulties related with the non-compactness of operators, which arise in un-bounded domains. This paper studies the existence of global attractor on the energyspace, and the estimates on the dimension of the global attractor. In the study of theexistence of attractors, the first step is to show the existence of a attracting set on thespaceX0. Here assumputions on the decay of the memory kernel plays afundamental role, which makes sure that the image of the absorbing set isasymptotically compact. Since the Poincaré embedding is no longer compact in theunbounded domain case, we split the semigroup S （t）into the sum:S（t）=S₁（t）S+2（t）.WhereS₁（t）is consistent tight andS2（t）exponentially decays arbitrarily small inthe long time, so we come to the conclusion by virtue of the Poincaré compactnessproperty. In studying the dimension estimation of global attractors, firstly we provethe Fréchet differentiability of the semigroup S （t）on the space X₀. The estimateson the Hausdorff and fractal dimension are in terms of given parameters, due toLiouville formula and an asymptotic estimate for the eigenvalues a of the eigenvalueproblem-φ（x）△u=au,x R^N.