Stability and blow-up questions for semilinear wave equations on IR^{N}. ^{∗}

Nikos M. Stavrakakis Department of Mathematics National Technical University Zografou Campus, GR157 80 Athens Greece

                nikolas@central.ntua.gr

We consider the semilinear dissipative hyperbolic problem

u_{tt}(x, t) + du_{t}(x, t) − Δu(x, t) + λg(x) f(u(x, t)) = 0, x ∈ IR^{N}, t ≥ 0,

with the initial conditions u(x, 0) = u_{0}(x) and u_{t}(x, 0) = u_{1}(x) in the case where N ≥ 3 and the indefinite weight g lies in L^{N/2}(IR^{N}) ∩ L^{∞}(IR^{N}). The existence of global and exponential attractors in X_{0} := D^{1,2}(IR^{N})×L_{|g|}^{2}(IR^{N}) and the finiteness of their Hausdorff and fractal dimensions, respectively, will be discussed. Also questions for the blow-up of solutions in finite time will be considered, when the initial data are sufficiently large. The energy space X_{0} is proved to be the natural setting for the problem and the compactness of the embedding D^{1,2} ⊂ L_{|g|}^{2} is widely applied.

∗Keywords and Phrases: dynamical systems, attractors, exponential attractors, hyperbolic equations, nonlinear problems, semigroups, unbounded domains, Hausdorff dimension, fractal dimension, generalised Sobolev spaces, blow-up.

AMS Subject Classification: 35B30, 35B40, 35B45, 35L15, 35L70, 35L80, 58F39.