Semi-linear wave equations with effective damping

Abstract

The 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.

Keywords

Semi-linear equations / Damped wave equations / Critical exponent / Global existence

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