Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity

TL;DR

This paper investigates the blow-up behavior and lifespan estimates of solutions to a semilinear wave equation with time-dependent damping, providing sharp upper and lower bounds based on different analytical methods.

Contribution

It offers new sharp lifespan estimates and blow-up rates for the wave equation with time-dependent damping and subcritical nonlinearity, using ODE and scaling variable techniques.

Findings

Blow-up rates are characterized for the wave equation with effective damping.

Sharp lifespan estimates are derived for subcritical nonlinearities.

Upper and lower bounds are established using different analytical approaches.

Abstract

We study blow-up behavior of solutions for the Cauchy problem of the semilinear wave equation with time-dependent damping. When the damping is effective, and the nonlinearity is subcritical, we show the blow-up rates and the sharp lifespan estimates of solutions. Upper estimates are proved by an ODE argument, and lower estimates are given by a method of scaling variables.