Estimates of Lifespan and Blow-up Rates for the Wave Equation with a Time-dependent Damping and a Power-type Nonlinearity

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.