The move from Fujita type exponent to a shift of it for a class of semilinear evolution equations with time-dependent damping

Abstract

In this paper, we derive suitable optimal \(L^p-L^q\) decay estimates, \(1\le p\le 2\le q\le \infty \) , for the solutions to the \(\sigma \) -evolution equation, \(\sigma >1\) , with scale-invariant time-dependent damping and power nonlinearity  \(|u|^p\) , $$\begin{aligned} u_{tt}+(-\Delta )^\sigma u + \frac{\mu }{1+t} u_t= |u|^{p}, \quad t\ge 0, \quad x\in {{\mathbb {R}}}^n, \end{aligned}$$ where  \(\mu >0\) , \(p>1\) . The critical exponent \(p=p_c\) for the global (in time) existence of small data solutions to the Cauchy problem is related to the long time behavior of solutions, which changes accordingly \(\mu \in (0, 1)\) or \(\mu >1\) . Under the assumption of small initial data in \(L^m({{\mathbb {R}}}^n)\cap L^2({{\mathbb {R}}}^n), m=1,2\) , we find the critical exponent at low space dimension n with respect to \(\sigma \) , namely, $$\begin{aligned} p_c= \max \left\{ {{\bar{p}}}(\gamma _{m}), {{\bar{p}}} (\gamma _{m}+\mu -1) \right\} , \quad \gamma _{m}{\mathrm {\,:=\,}}\frac{n}{m\sigma }, \quad \mu >1-\gamma _m, \end{aligned}$$ where \( {{\bar{p}}}(\gamma ){\mathrm {\,:=\,}}1+ \frac{2}{\gamma }\) is the well known Fujita exponent. Hence, \(p_c={{\bar{p}}}(\gamma _{m})\) if \(\mu >1\) , whereas \(p_c={{\bar{p}}} (\gamma _{m}+\mu -1)\) is a shift of Fujita type exponent if \(\mu \in (0, 1)\) .