A note on the lifespan of solutions to the semilinear damped wave equation

Abstract

This paper concerns estimates of the lifespan of solutions to the semilinear damped wave equation $\square u+\Phi (t,x)u_t=|u|^p$ in $(t,x)\in [0,\infty )\times \mathbf {R}^n$, where the coefficient of the damping term is $\Phi (t,x)=\langle x\rangle ^{-\alpha }(1+t)^{-\beta }$ with $\alpha \in [0,1),\ \beta \in (-1,1)$ and $\alpha \beta =0$. Our novelty is to prove an upper bound of the lifespan of solutions in subcritical cases $1<p<2/(n-\alpha )$.