Blow-up and lifespan estimate for wave equations with critical damping term of space-dependent type related to Glassey conjecture

TL;DR

This paper investigates the blow-up behavior of wave equations with space-dependent damping and nonlinearities, establishing conditions under which solutions blow up in finite time and identifying the critical exponent regions.

Contribution

It provides new blow-up criteria for wave equations with space-dependent damping, extending the understanding of critical exponents in higher dimensions and for different initial data supports.

Findings

Blow-up region in higher dimensions is p in (1, p_G(N+μ)]

Blow-up region independent of μ is p in (1, 1+2/N)

Results apply to small initial data with compact and noncompact support

Abstract

The main purpose of the present paper is to study the blow-up problem of the wave equation with space-dependent damping in the \textit{scale-invariant case} and time derivative nonlinearity with small initial data. Under appropriate initial data which are compactly supported, by using a test function method and taking into account the effect of the damping term ($\frac{\mu}{\sqrt{1+|x|^2}}u_t$), we provide that in higher dimensions the blow-up region is given by $p \in (1, p_G(N+\mu)]$ where $p_G(N)$ is the Glassey exponent. Furthermore, we shall establish a blow-up region, independent of $\mu$ given by $p\in (1, 1+\frac{2}{N}),$ for appropriate initial data in the energy space with noncompact support.