The lifespan of solutions of semilinear wave equation with weighted nonlinearity

TL;DR

This paper studies how long solutions to a semilinear wave equation with weighted nonlinearity exist before blowing up, focusing on the effects of the weight and initial data size.

Contribution

It provides new insights into the lifespan of solutions for wave equations with weighted nonlinearities, extending previous results to include weights and varying parameters.

Findings

Derived lifespan estimates depending on weight and initial data size

Identified critical exponents for solution blow-up

Extended existing theories to weighted nonlinear wave equations

Abstract

We investigate the lifespan of solutions to a specific variant of the semilinear wave equation, which incorporates weighted nonlinearity $$ u_{tt}-u_{xx} =|x|^\alpha |u|^p, \quad\mbox{for}\;\;\; (t,x)\in (0,\infty)\times\mathbb{R}, $$ where $p>1$, $\alpha\in\mathbb{R}$. We explore the behavior of solutions for small initial data, considering the influence of weighted nonlinearities on the lifespan.