Existence and asymptotic stability for the wave equation on compact manifolds with nonlinearities of arbitrary growth

Abstract

We study the wellposedness, stabilization and blow up of solutions of the wave equation with nonlinearities of arbitrary growth and locally distributed nonlinear dissipation posed in a 2-dimensional compact Riemannian manifold $(M,g)$ without boundary. Differently of the previous literature we give a different proof based on the truncation of the original problem and passage to the limit in order to obtain in one shot, the energy identity as well as the Observability Inequality, which are the essential ingredients to obtain uniform decay rates of the energy. One advantage of our proof, even in the case of subcritical, critical or super critical growth, is that the decay rate is independent of the nonlinearity. We can also treat the focusing case for those solutions with energy less than $d$ of the ground state, where $d$ is the level of the Mountain Pass Theorem.