Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables

TL;DR

This paper proves that solutions to a one-dimensional semilinear wave equation with certain non-effective damping terms blow up in finite time, confirming the conjecture that the critical exponent matches that of the undamped wave equation.

Contribution

It establishes finite-time blow-up results for the wave equation with time and space-dependent damping, confirming the critical exponent conjecture in one dimension.

Findings

Solutions blow up in finite time under non-effective damping

Critical exponent matches that of the undamped wave equation

Damping regarded as a perturbation does not prevent blow-up

Abstract

In this paper, we give a small data blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We show that if the damping term can be regarded as perturbation, that is, non-effective damping in a certain sense, then the solution blows up in finite time for any power of nonlinearity. This gives an affirmative answer for the conjecture that the critical exponent agrees with that of the wave equation when the damping is non-effective in one space dimension.