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

TL;DR

This paper establishes optimal decay rates for solutions to a broad class of semi-linear dissipative hyperbolic equations, showing solutions diminish at a specific rate over time, with some solutions decaying exactly at that rate.

Contribution

It introduces a general framework for decay estimates in semi-linear dissipative hyperbolic equations with nontrivial kernels, proving optimal decay rates and their attainability.

Findings

Solutions decay at least as fast as a negative power of time.

Decay rate is proven to be optimal for a nonempty set of initial data.

Results are applicable to various partial differential 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 -> +infinity, 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.