Quenching Behavior of Parabolic Problems with Localized Reaction Term

• Main Authors: Yen-Huang Hsu, 許言鍠
• Other Authors: Hon-hung Terence Liu
• Format: Others
• Language: en_US
• Published: 2008
• Online Access: http://ndltd.ncl.edu.tw/handle/32602782040633441848

碩士 === 大同大學 === 應用數學學系(所) === 96 === Let $\triangle $ be the Laplace operator in $n$ dimensional space. This paper studies the following the initial-boundary value problem with localized reaction term: \begin{align*} u_{t}(x,t)=\Delta u(x,t)+ \frac{1}{(1-u(x,t))^{p}}+\frac{1}{(1-u(x^{*},t))^{q}}, (x,t)\in B\times(0,T), \end{align*} \begin{align*} u(x,0)=u_{0}(x), \ x\in B, \end{align*} \begin{align*} u(x,t)=0, \ (x,t)\in\partial B\times(0,T), \end{align*} where $B=\{x\in \Bbb{R}^n: \|x\|<1 \}$, $\partial B=\{x\in \Bbb{R}^n: \|x\| =1\}$, $x^{*}\in B$, $0<p,\ q<\infty$, $T>0$, and $u_{0}\geq 0$. The existence and quenching behavior of the problem are studied. For the case $x^{*}=0$, the quenching rate of solution near the quenching time is investigated.