Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations

TL;DR

This paper analyzes how two different power-type nonlinearities affect the lifespan of small solutions to semi-linear wave equations in 2D and 3D, establishing conditions for global existence and sharp lifespan bounds.

Contribution

It determines the full parameter region for global existence and provides sharp lifespan estimates when solutions blow up, advancing understanding of combined nonlinear effects.

Findings

Identified the full $(p,q)$ region for global existence in 2D and 3D.

Established sharp lower bounds for the lifespan in non-global cases.

Provided precise lifespan estimates matching upper bounds.

Abstract

This paper investigates the combined effects of two distinctive power-type nonlinear terms (with parameters $p,q>1$) in the lifespan of small solutions to semi-linear wave equations. We determine the full region of $(p,q)$ to admit global existence of small solutions, at least for spatial dimensions $n=2, 3$. Moreover, for many $(p,q)$ when there is no global existence, we obtain sharp lower bound of the lifespan, which is of the same order as the upper bound of the lifespan.