Critical regularity of nonlinearities in semilinear classical damped wave equations

Abstract

In this paper we consider the Cauchy problem for the semilinear damped wave equation $u_{tt}-\Delta u+u_t=h(u);u(0;x)=f(x);u_t(0;x)=g(x);$ where $h(s)=|s{|}^{1+2/n}\mu (|s|)$. Here n is the space dimension and $\mu$ is a modulus of continuity. Our goal is to obtain sharp conditions on $\mu$ to obtain a threshold between global (in time) existence of small data solutions (stability of the zerosolution) and blow-up behavior even of small data solutions.

14 pages