The lifespan of solutions of semilinear wave equation with weighted nonlinearity

Abstract

We investigate the lifespan of solutions to a specific variant of the semilinear wave equation, which incorporates weighted nonlinearity $$ u_{tt}-u_{xx} =|x|^α|u|^p, \quad\mbox{for}\;\;\; (t,x)\in (0,\infty)\times\mathbb{R}, $$ 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.

The preprint requires further refinement, including additional examples and applications. We will continue to address these aspects