Sharp lifespan estimates of blowup solutions to semilinear wave equations with time-dependent effective damping

TL;DR

This paper provides precise lifespan estimates for blowup solutions of semilinear wave equations with time-dependent damping, analyzing how the damping's asymptotic behavior influences solution blowup timing.

Contribution

It offers the first sharp lifespan estimates for blowup solutions in semilinear wave equations with general time-dependent damping, including critical threshold cases.

Findings

Sharp lifespan estimates derived for various damping behaviors

Identification of threshold cases between effective and overdamping

Asymptotic profile of damping significantly affects blowup timing

Abstract

We consider the initial value problem for the semilinear wave equation with time-dependent effective damping. The interest is the behavior of lifespan of solutions in view of the asymptotic profile of the damping as $t\to \infty$. The result of this paper is the sharp lifespan estimates of blowup solutions for general time-dependent damping including threshold cases between effective and overdamping.