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.

Appendix

In this section we include the following generalized version of Jensen inequality [11].

Lemma 8

Let ${\varPhi }$ be a convex function on ${{\mathbb {R}}}$. Let $\alpha =\alpha (x)$ be defined and non-negative almost everywhere on ${\varOmega }$, such that $\alpha$ is positive in a set of positive measure. Then, it holds

$$\begin{aligned} {\varPhi }\left( \frac{\int _{\varOmega }u(x)\alpha (x)\,dx}{\int _{\varOmega }\alpha (x)\,dx}\right) \le \frac{\int _{\varOmega }{\varPhi }(u(x))\alpha (x)\,dx}{\int _{\varOmega }\alpha (x)\,dx} \end{aligned}$$

for all non-negative functions u provided that all the integral terms are meaningful.

Proof

Let $\gamma >0$ be fixed. From the convexity of ${\varPhi }$ it follows that there exists $k\in {{\mathbb {R}}}^1$, such that

$$\begin{aligned} {\varPhi }(t)-{\varPhi }(\gamma )\ge k(t-\gamma ) \quad \text{ for } \text{ all } \quad t\ge 0. \end{aligned}$$

Putting $t=u(x)$ and multiplying the last inequality by $\alpha (x)$, we get after integration over ${\varOmega }$ that

$$\begin{aligned} \int _{\varOmega }{\varPhi }(u(x))\alpha (x)\,dx- {\varPhi }(\gamma )\int _{{\varOmega }}\alpha (x)\,dx\ge k\left( \int _{\varOmega }u(x)\alpha (x)\,dx-\gamma \int _{\varOmega }\alpha (x)\,dx \right) . \end{aligned}$$

The statement follows by putting

$$\begin{aligned} \gamma =\frac{\int _{\varOmega }u(x)\alpha (x)\,dx}{\int _{\varOmega }\alpha (x)\,dx}. \end{aligned}$$