A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations

TL;DR

This paper introduces a new method using a Lyapunov functional to analyze blow-up behavior in perturbed semilinear wave equations with subcritical nonlinearities, providing insights into their blow-up rates.

Contribution

It develops a novel approach with a Lyapunov functional in similarity variables to study blow-up in perturbed semilinear wave equations, differing from previous methods.

Findings

Derived a Lyapunov functional for perturbed equations

Established blow-up rate estimates using the functional

Demonstrated the method's novelty compared to unperturbed cases

Abstract

We consider in this paper some class of perturbation for the semilinear wave equation with subcritical (in the conformal transform sense) power nonlinearity. We first derive a Lyapunov functional in similarity variables and then use it to derive the blow-up rate. Though the result is similar to the unperturbed case in its statements, this is not the case of our method, which is new up to our knowledge.