Small data global existence for the semilinear wave equation with space-time dependent damping

TL;DR

This paper investigates the global existence of solutions for a semilinear wave equation with space-time dependent damping, establishing conditions under which small data lead to global solutions without requiring compact support.

Contribution

It proves the existence of unique global solutions for small data when the nonlinearity exceeds the critical exponent, extending results to non-compactly supported data.

Findings

Global solutions exist for nonlinearity above the critical exponent

No assumption of compact support on initial data

Open problem remains for sub-critical exponent cases

Abstract

In this paper we consider the critical exponent problem for the semilinear wave equation with space-time dependent damping. When the damping is effective, it is expected that the critical exponent agrees with that of only space dependent coefficient case. We shall prove that there exists a unique global solution for small data if the power of nonlinearity is larger than the expected exponent. Moreover, we do not assume that the data are compactly supported. However, it is still open whether there exists a blow-up solution if the power of nonlinearity is smaller than the expected exponent.