The sharp estimate of the lifespan for the semilinear wave equation with time-dependent damping

We consider the following semilinear wave equation with time-dependent damping. \begin{align} \tag{NLDW} \left{ \begin{array}{ll} \partial_t^2 u - \Delta u + b(t)\partial_t u = |u|^{p}, & (t,x) \in [0,T) \times $\mathbb{R}$^n, \ u(0,x)=\varepsilon u_0(x), u_t(0,x)=\varepsilon u_1(x), & x \in $\mathbb{R}$^n, \end{array} \right. \end{align} where $n\in \mathbb{N}$, $p>1$, $\epsilon >0$, and $b(t)\approx (t+1{)}^{-\beta }$ with $\beta \in [-1,1)$. It is known that small data blow-up occurs when $1<p<p_F$ and, on the other hand, small data global existence holds when $p>p_F$, where $p_F:=1+2\mathrm{/}n$ is the Fujita exponent. The sharp estimate of the lifespan was well studied when $1<p< p_F$\mathrm{.}Inthecriticalcase$p=p_F$, the lower estimate of the lifespan was also investigated. Recently, Lai and Zhou obtained the sharp upper estimate of the lifespan when $p=p_F$ and $b(t)=1$. In the present paper, we give the sharp upper estimate of the lifespan when $p=p_F$ and $b(t)\thickapprox (t+1)^{-\beta}$with$\beta \in [-1,1)$ by the Lai–Zhou method.