Critical exponent for nonlinear wave equations with damping and potential terms

TL;DR

This paper determines the critical exponent for nonlinear wave equations with damping and potential terms, revealing how lower order terms and initial data decay influence solution lifespan and thresholds.

Contribution

It introduces a new critical exponent for equations with combined damping and potential terms, considering their specific relation and initial data decay effects.

Findings

Critical exponent differs from classical cases without combined terms.

Initial data decay rate affects the solution's lifespan.

Sharp bounds for maximal existence time are established.

Abstract

The aim of this paper is to determine the critical exponent for the nonlinear wave equations with damping and potential terms of the scale invariant order, by assuming that these terms satisfy a special relation. We underline that our critical exponent is different from the one for related equations such as the nonlinear wave equation without lower order terms, only with a damping term, and only with a potential term. Moreover, we study the effect of the decaying order of initial data at spatial infinity. In fact, we prove that not only the lower order terms but also the order of the initial data affects the critical exponent, as well as the sharp upper and lower bounds of the maximal existence time of the solution.