Critical exponent for the one-dimensional wave equation with a space-dependent scale invariant damping and time derivative nonlinearity

TL;DR

This paper determines the critical exponent for the global existence and blow-up of solutions to a one-dimensional wave equation with space-dependent damping and nonlinear time derivatives, extending previous results.

Contribution

First to identify the critical exponent range for this class of wave equations with space-dependent damping and nonlinearities.

Findings

Global existence for small data with energy estimates

Blow-up results under positive initial data

Critical exponent given by p_G(1+μ_0)=1+2/μ_0 for μ_0 in (0,1]

Abstract

We investigate in this paper the Cauchy problem of the one-dimensional wave equation with space-dependent damping of the form $\mu_0(1+x^2)^{-1/2}$, where $\mu_0>0$, and time derivative nonlinearity. We establish global existence of mild solutions for small data compactly supported by employing energy estimates within suitable Sobolev spaces of the associated homogeneous problem. Furthermore, we derive a blow-up result under some positive initial data by employing the test function method. This shows that the critical exponent is given by $p_G(1+\mu_0)=1+2/\mu_0$, when $\mu_0\in (0,1]$, where $p_G$ is the Glassey exponent. To the best of our knowledge, this constitutes the first identification of the critical exponent range for this class of equations. As by product, we extend the global existence result to a more general class of space/time nonlinearities of the form $f(\partial_tu,\partial_x u)=|\partial_x u|^{q}$ or $f(\partial_tu,\partial_x u)=|\partial_tu|^{p}|\partial_x u|^{q}$, with $p,q>1$.