Asymptotic behaviors of solutions for time dependent damped wave equations

In this paper, we consider the Cauchy problem for the wave equation with time dependent damping b ( t ) u t and absorbed semilinear term | u | ρ − 1 u . Here, b ( t ) = b 0 ( 1 + t ) − β with − 1 < β < 1 and b 0 > 0 . Using the weighted energy method, we obtain the L 1 and L 2 decay rates of the solution, which coincide to those for self-similar solutions to the corresponding parabolic equation when 1 < ρ < ρ F ( N ) : = 1 + 2 N .