Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on R~N

摘要

We discuss estimates of the Hausdorff and fractal dimension of a global attractor for the semilinear wave equation u_(tt) + δu_t - φ (x)Δu + λf(u) = η(x), x ∈ R~N, t ≥ 0, with the initial conditions u(x, 0) = u_0(x) and u_t(x,0) = u_1(x), where N ≥ 3, δ > 0 and (φ(x))~(-1) := g(x) lies in L~(N/2)(R~N)∩L~∞(R~N). The energy space X_0 = D~(1,2)(R~N) * L_g~2(R~N) is introduced, to overcome the difficulties related with the non-compactness of operators, which arise in unbounded domains. The estimates on the Hausdorff dimension are in terms of given parameters, due to an asymptotic estimate for the eigenvalues μ of the eigenvalue problem -φ(x)Δu = μu, x ∈ R~N.