Blow-up for Strauss type wave equation with damping and potential

TL;DR

This paper investigates blow-up phenomena in nonlinear wave equations with spatially dependent damping and potential, establishing lifespan estimates and supporting a conjecture about critical exponents.

Contribution

It provides new blow-up results and lifespan estimates for Strauss type wave equations with critical damping and potential, advancing understanding of blow-up laws.

Findings

Established blow-up results in critical and sub-critical cases

Derived upper bounds for the lifespan of solutions

Supported conjectures on the nature of critical exponents

Abstract

We study a kind of nonlinear wave equations with damping and potential, whose coefficients are both critical in the sense of the scaling and depend only on the spatial variables. Based on the earlier works, one may think there are two kinds of blow-up phenomenons when the exponent of the nonlinear term is small. It also means there are two kinds of law to determine the critical exponent. In this paper, we obtain a blow-up result and get the estimate of the upper bound of the lifespan in critical and sub-critical cases. All of the results support such a conjecture, although for now, the existence part is still open.