Life-Span of Classical Solutions to a Semilinear Wave Equation with Time-Dependent Damping

Abstract

This paper is concerned with the Cauchy problem for a semilinear wave equation with a time-dependent damping. In case that the space dimension $n=1$ and the nonlinear power is bigger than 2, the life-span $\widetilde T(\varepsilon)$ and global existence of the classical solution to the problem has been investigated in a unified way. More precisely, with respect to different values of an index $K$, which depends on the time-dependent damping and the nonlinear term, the life-span $\widetilde T(\varepsilon)$  can be estimated below by $\varepsilon^{-\frac{p}{1-K}}$, $e^{\varepsilon^{-p}}$ or $+\infty$, where $\varepsilon$ is the scale of the compact support of the initial data.