  In this paper, we study the Cauchy problem of the nonlinear wave equation
with fractional Laplacian and time-dependent damping. Firstly, we derive the
weighted Sobolev estimate of the solution operators for the linear wave
equation with the damping of constant coefficient, and prove the local
existence and uniqueness in the weighted Sobolev space for the power-type
nonlinearity and $b(t)\in L^\infty$, by the contraction mapping principle.
Secondly, we consider the case of the source nonlinearity $f(u)\approx |u|^p$.
In the subcritical and critical cases $1<p\leq p_c=1+\frac \sigma N$, based on
the blow-up result on the ordinary differential inequality, we could prove the
blow-up of the solution and obtain the upper bound of the lifespan. And the
upper bound of the lifespan in the critical case is independent on the
coefficient of the time-dependent damping and is completely new even if the
classical case $b(t)=1$.
