arXiv:2401.10552v3 Announce Type: replace Abstract: In this paper, we investigate 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 the damping coefficient behaves as $b(t)\approx (1+t)^{-\beta}$. We first establish local existence and uniqueness of mild and strong solutions in weighted Sobolev spaces by deriving weighted Sobolev estimates for the linear propagators and overcoming the spatial-weight coupling in the nonlinearity through the Bessel potential estimates. Then, in the subcritical and critical cases $1<p\leq p_c:=1+\frac \sigma N$, we derive the upper estimates of the lifespan for the fractional Laplacian with $0<\sigma<2$ and time-dependent damping $\beta \in [-1, 1)$ via the framework of ordinary differential inequalities. The blow-up results, together with the global existence in the supercritical case obtained in \cite{LI}, show that the critical exponent for the fractional wave equation is $p_c=1+\frac{\sigma}{N}$ for $0<\sigma<2$. Moreover, combining with the lower estimate of lifespan derived in \cite{LI}, we conclude that the estimate in this paper is sharp. Note that our result for 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.
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