We consider the following semilinear wave equation with time-dependent damping. \begin{align*} \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 $n \in \mathbb N$, $p > 1$, $\varepsilon>0$, 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 [15] 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.

中文翻译：

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具有时变阻尼的半线性波动方程寿命的精确估计

我们考虑以下具有时变阻尼的半线性波动方程。\ begin {align *} \ 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 *}，其中$ n \ in \ mathbb N $，$ p> 1 $，$ \ varepsilon> 0 $和$ b（t）\ approx（t + 1）^ {-\ beta} $ $ \ beta \ in [-1,1）$。众所周知，当$ 1 <p <p_F $时，发生小数据爆炸，另一方面，当$ p> p_F $时，小数据全局存在，其中$ p_F：= 1 + 2 / n $是Fujita。指数。当$ 1 <p <p_F $时，对寿命的精确估计得到了很好的研究。在临界情况$ p = p_F $中，还对寿命的较低估计值进行了研究。最近，当$ p = p_F $和$ b（t）= 1 $时，Lai和Zhou [15]获得了寿命的最高估计值。在本文中，我们给出了$ p = p_F $和$ b（t）\ approx（t + 1）^ {-\ beta} $且$ \ beta \ in [-1， 1）通过赖周法计算。