Nonexistence of global solutions of wave equations with weak time-dependent damping and combined nonlinearity

TL;DR

This paper investigates the nonexistence of global solutions for wave equations with weak, time-dependent damping and combined nonlinearities, showing that scattering damping does not influence blow-up behavior.

Contribution

It extends previous work by analyzing combined nonlinearities in damped wave equations and demonstrates the non-effect of scattering damping on solution blow-up.

Findings

Scattering damping does not affect blow-up behavior.

Wave equations with combined nonlinearities lack global solutions under certain conditions.

The results generalize previous findings for equations with single nonlinearities.

Abstract

In our previous two works, we studied the blow-up and lifespan estimates for damped wave equations with a power nonlinearity of the solution or its derivative, with scattering damping independently. In this work, we are devoted to establishing a similar result for a combined nonlinearity. Comparing to the result of wave equation without damping, one can say that the scattering damping has no influence.