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

Abstract

We consider the following semilinear wave equation with time-dependent damping. \begin{align} \tag{NLDW} \left\{ \begin{array}{ll} \partial_t^2 u - Δu + 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 $p_F:=1+2/n$ is the Fujita exponent. The sharp estimate of the lifespan was well studied when $1<p<p_F$. In the critical case $p=p_F$, 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 $b(t)\approx (t+1{)}^{-\beta }$ with $\beta \in [-1,1)$ by the Lai--Zhou method.