论文标题
具有空间不均匀吸收的扩散方程的有限时间灭绝
Finite time extinction for a diffusion equation with spatially inhomogeneous strong absorption
论文作者
论文摘要
具有强吸收$$ $$ \ partial_t u-Δu^m+| x | x |^σu^q = 0，\ qquad（t，x）\ in（0，\ infty） $ q \ in（0,1）$和$σ> 0 $。引入关键指数$σ^*：= 2（1-q）/（M-1）$ $ m> 1 $和$σ_*= \ infty $ for $ m = 1 $，已知有限的时间在[0，σ^*）中以$σ\ in [0，σ^*）$而灭绝。当$ m> 1 $和$σ\geσ^*$时，有限时间灭绝的发生被证明是特定的初始条件类别的，从而补充了在非扩散范围内的$σ$中可用的结果，并显示出其清晰度。
The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $$\partial_t u-Δu^m+|x|^σu^q=0, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ with $m\geq1$, $q\in(0,1)$ and $σ>0$, is addressed. Introducing the critical exponent $σ^* := 2(1-q)/(m-1)$ for $m>1$ and $σ_*=\infty$ for $m=1$, extinction in finite time is known to take place for $σ\in [0,σ^*)$ and an alternative proof is provided therein. When $m>1$ and $σ\ge σ^*$, the occurrence of finite time extinction is proved for a specific class of initial conditions, thereby supplementing results on non-extinction that are available in that range of $σ$ and showing their sharpness.
