具局部反應項的拋物問題之漸滅現象

Quenching Behavior of Parabolic Problems with Localized Reaction Term

許言鍠(Yen-Huang Hsu)

指導教授 ： 廖漢雄

大同大學/電機資訊學院/應用數學學系所/碩士(2008年)

摘要

令 $ riangle $
是 $n$ 維空間中的 Laplace 運算元.
本論文討論以下具局部反應項的起始邊界值問題:
egin{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 imes(0,T),
end{align*}
egin{align*}
u(x,0)=u_{0}(x), xin B,
end{align*}
egin{align*}
u(x,t)=0, (x,t)inpartial B imes(0,T),
end{align*}
其中 $B={xin Bbb{R}^n: |x|<1 }$, $partial B={xin
Bbb{R}^n: |x| =1}$, $x^{*}in B$, $00$,
以及 $u_{0}geq 0$.
本文討論解的存在性及漸滅現象. 進一步, 當 $x^{*}=0$ 時, 可找出靠近漸滅時間時的漸滅速率.

關鍵字

局部反應項 ； 拋物方程 ； 漸滅現象

並列摘要

Let $ riangle $ be the Laplace operator in
$n$ dimensional space. This paper studies the following the
initial-boundary value problem with localized reaction term:
egin{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 imes(0,T),
end{align*}
egin{align*}
u(x,0)=u_{0}(x), xin B,
end{align*}
egin{align*}
u(x,t)=0, (x,t)inpartial B imes(0,T),
end{align*}
where $B={xin Bbb{R}^n: |x|<1 }$, $partial B={xin
Bbb{R}^n: |x| =1}$, $x^{*}in B$, $00$,
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.

並列關鍵字

Localized Reaction Term ； Parabolic Equations ； Quenching Behavior

參考文獻

[1] A. Acker, W. Walter, The quenching problem for nonlinear parabolic differential

equations, {it Lecture Notes in Mathematics 564},

of solutions of parabolic equations with a singular nonlinear

term, {it Nonlinear Analysis} { f 2} (1978) 499-504.

for degenerate semilinear parabolic problems, {it Nonlinear