Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent
In this paper, we study the Cauchy problem of the fractional wave equation with time-dependent damping and the source nonlinearity $f(u)\approx |u|^p$: $$ \begin{cases} \partial_t^2u(t,x)+(-\Delta)^{\sigma/2} u(t,x)+b(t) \partial_t u(t,x) =f(u(t,x)),\ &(t,x)\ \in [0,T)\times \mathbb{R}^N,\\ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x),\ &x\ \in\ \mathbb{R}^N, \end{cases} $$ where $b(t)\approx (1+t)^{-\beta}$. In the subcritical and critical cases $1<p\leq p_c:=1+\frac \sigma N$, 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.