Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case

TL;DR

This paper proves the global existence of solutions for a wave equation with time-dependent damping in the overdamping case, showing small data do not lead to blow-up in the energy space for subcritical nonlinearities.

Contribution

It establishes the global well-posedness for small initial data in the overdamping case, a result not previously known for this specific damping regime.

Findings

Small data solutions exist globally in the energy space

No blow-up occurs for small data in the overdamping case

Results differ from effective or non-effective damping scenarios

Abstract

We study global existence of solutions to the Cauchy problem for the wave equation with time-dependent damping and a power nonlinearity in the overdamping case. We prove the global well-posedness for small data in the energy space for the whole energy-subcritical case. This result implies that small data blow-up does not occur in the overdamping case, different from the other cases, i.e. effective or non-effective damping.