The lifespan of solutions of semilinear wave equation with weighted nonlinearity

We investigate the lifespan of solutions to a specific variant of the semilinear wave equation, which incorporates weighted nonlinearity $$u_{tt}-u_{xx}=\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }\mathrm{\mid }u{\mathrm{\mid }}^p,\text{quadfor}(t,x)\in (0,\mathrm{\infty })\times \mathbb{R},$$ 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.