Semi-linear wave equations with effective damping

The authors study the Cauchy problem for the semi-linear damped wave equation in any space dimension. It is assumed that the time-dependent damping term is effective. The global existence of small energy data solutions for polynomial nonlinear term in the supercritical case is proved.The authors study the Cauchy problem for the semi-linear damped wave equation utt - Δu + b(t)ut = f(u), u(0,x) =u0(x), ut(0,x) = u1(x) in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for pipef(u)pipe ≈ pipeupipep in the supercritical case of p > 1 + 2/n and p ≤ n/n-2 for n ≥ 3 is proved. © 2013 Fudan University and Springer-Verlag Berlin Heidelberg