Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations

Abstract

We consider a class of semi-linear dissipative hyperbolic equations in which the operator associated to the linear part has a nontrivial kernel. Under appropriate assumptions on the nonlinear term, we prove that all solutions decay to 0, as $t\to \infty$, at least as fast as a suitable negative power of $t$. Moreover, we prove that this decay rate is optimal in the sense that there exists a nonempty open set of initial data for which the corresponding solutions decay exactly as that negative power of $t$.

Our results are stated and proved in an abstract Hilbert space setting, and then applied to partial differential equations.