Estimates of lifespan and blow-up rates for the wave equation with a   time-dependent damping and a power-type nonlinearity
  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. 
Kazumasa Fujiwara, Masahiro Ikeda and Yuta Wakasugi
math.AP