Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent

Abstract

In this paper, we study the Cauchy problem of the fractional wave equation with time-dependent damping and the source nonlinearity $f(u)\approx |u{|}^p$: $$\{\begin{array}{ll}{\partial }_t^{2}u(t,x)+(-\Delta {)}^{\sigma /2}u(t,x)+b(t){\partial }_{t}u(t,x)=f(u(t,x)),& (t,x)\in [0,T)\times {\mathbb{R}}^N,\\ u(0,x)=u_0(x),{\partial }_{t}u(0,x)=u_1(x),& x\in {\mathbb{R}}^N,\end{array}$$ where $b(t)\approx (1+t{)}^{-\beta }$. In the subcritical and critical cases $1<p\le p_c:=1+\frac{\sigma }{N}$, we derive the upper estimates of the lifespan for fractional Laplacian with $0<\sigma <2$ and time-dependent damping $\beta \in [-1,1)$ by the framework of ordinary differential inequality. The blow-up results, with the global existence in the supercritical case $p_c<p<\frac{N}{N-\sigma }$ obtained in [19], shows that the critical exponent for the fractional wave quation is $p_c=1+\frac{\sigma }{N}$ for $0<\sigma <2$. Moreover, together with the lower estimate of lifespan derived in [19], we could conclude that the estimate in this paper is sharp. Note that the our result of the critical case is completely new even in the classical case $b(t)=1$. We also consider the case of $\beta =1$, and obtain the upper estimate of the lifespan.