QUALITATIVE ANALYSIS OF A NONLINEAR WAVE EQUATION

We study the qualitative behavior of solutions of a wave equation with nonlinear damping and a source term. We give a characterization of blow-up of solutions, improving a previous result. When the dissipation dominates the source term, we show existence of unbounded global solutions. We use the stable (potential well) and unstable sets, introduced by Sattinger and Payne. We study all bounded global solutions, and we charaterize their convergence as $t \rightarrow \infty$. In particular, we prove that every solution, with energy larger or equal than the depth of the potential well, is global, bounded and converges to the set of nonzero equilibria.