Global existence of solutions for semilinear damped wave equations with variable coefficients

TL;DR

This paper investigates the conditions under which solutions to semilinear damped wave equations with variable coefficients exist globally, focusing on the interplay between coefficient behavior and nonlinear power p.

Contribution

It establishes criteria for global existence of solutions based on variable coefficient properties and nonlinearity power p, extending previous results to more general settings.

Findings

Global existence is proven for small initial data under certain coefficient conditions.

The relation between coefficient decay/growth and the critical exponent p is characterized.

Results generalize known cases to variable coefficient scenarios.

Abstract

We consider the Cauchy problem for the damped wave equations with variable coefficients a(x) having power type nonlinearity |u|^p. We discuss the global existence of solutions for small initial data and investigate the relation between the range of a(x) and the order p.