Semi-linear Wave Equations with Effective Damping

The authors study the Cauchy problem for the semi-linear damped wave equation $$u_{tt} - Delta u + bleft( t right)u_t = fleft( u right), uleft( {0,x} right) = u_0 left( x right), u_t left( {0,x} right) = u_1 left( x right)$$ 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 |f(u)| ≈ |u| ^p in the supercritical case of $p > tfrac{2} {n}$ and $p leqslant tfrac{n} {{n - 2}}$ for n ≥ 3 is proved.