A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations

Abstract

We give necessary and sufficient conditions for existence of global and nonglobal solutions of a nonlinear wave equation in a bounded domain. We consider nonlinear dissipation and a nonlinear source term. We also analyze the qualitative behavior of solutions forwards and backwards for the wave equation without dissipation. In this case we present characterizations of blow-up and asymptotic behavior. Finally, we extend some of our results to a nonlinear Kirchhoff equation. We use the concepts of stable and unstable sets introduced by Payne and Sattinger in 1975.