Asymptotic behaviors of solutions for time dependent damped wave equations (Q1034562)

The authors discuss the Cauchy problem to the equation \(u_{tt}-\Delta u+b_0(1+t)^{-\beta }u_t+u\left|u\right|^{\rho -1}=0\) in \(\mathbb R^{+}\times\mathbb R^N,\) where \(\beta \in (-1,1),\) the initial data have compact supports and \(\rho >1\) for \(N=1,2\) else \(1<\rho <\frac{N+2}{N-2}.\) They prove \(L^1\) and \(L^2\) decay estimates for a weak solution for \(t\to \infty .\)