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 $$\begin{aligned} u_{tt} - {\varDelta }u + u_t= h(u), \quad u(0,x)=\phi (x), \quad u_t(0,x)= \psi (x), \end{aligned}$$ where $$h(s)=|s|^{1+ \frac{2}{n}}\mu (|s|)$$ . 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 zero solution) and blow-up behavior even of small data solutions.