Semi-linear wave equations with effective damping

AbstractThe authors study the Cauchy problem for the semi-linear damped wave equation $$u_{tt} - \Delta u + b\left( t \right)u_t = f\left( u \right), u\left( {0,x} \right) = u_0 \left( x \right), u_t \left( {0,x} \right) = u_1 \left( x \right)$$ 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.