Finite time extinction for a diffusion equation with spatially inhomogeneous strong absorption

IF 1.8 4区 数学 Q1 MATHEMATICS Differential and Integral Equations Pub Date : 2023-11-01 DOI:10.57262/die036-1112-1005
Razvan Gabriel Iagar, Philippe Laurençot
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引用次数: 0

Abstract

The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $$ \partial_t u-\Delta u^m+|x|^{\sigma}u^q=0, \qquad (t,x)\in(0,\infty)\times \mathbb R ^N, $$ with $m\geq1$, $q\in(0,1)$ and $\sigma > 0$, is addressed. Introducing the critical exponent $\sigma^* := 2(1-q)/(m-1)$ for $m > 1$ and $\sigma^*=\infty$ for $m=1$, extinction in finite time is known to take place for $\sigma\in [0,\sigma^*)$ and an alternative proof is provided therein. When $m > 1$ and $\sigma\ge \sigma^*$, 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 $\sigma$ and showing their sharpness.
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空间非均匀强吸收扩散方程的有限时间消光
讨论了具有强吸收的扩散方程$$ \partial_t u-\Delta u^m+|x|^{\sigma}u^q=0, \qquad (t,x)\in(0,\infty)\times \mathbb R ^N, $$ ($m\geq1$, $q\in(0,1)$和$\sigma > 0$)的有界非负解的有限时间消光现象。引入$m > 1$的临界指数$\sigma^* := 2(1-q)/(m-1)$和$m=1$的临界指数$\sigma^*=\infty$,已知$\sigma\in [0,\sigma^*)$在有限时间内发生消光,并给出了另一种证明。当$m > 1$和$\sigma\ge \sigma^*$时,证明了特定的一类初始条件存在有限时间消光,从而补充了在$\sigma$范围内的非消光结果,显示了它们的明晰性。
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来源期刊
Differential and Integral Equations
Differential and Integral Equations MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
2.40
自引率
0.00%
发文量
0
审稿时长
6-12 weeks
期刊介绍: Differential and Integral Equations will publish carefully selected research papers on mathematical aspects of differential and integral equations and on applications of the mathematical theory to issues arising in the sciences and in engineering. Papers submitted to this journal should be correct, new, and of interest to a substantial number of mathematicians working in these areas.
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