臺灣博碩士論文加值系統

centerline{k16 摘要} vspace{24pt}k14 large 設 $T$, $p$ 為
正常數且 $pgeqslant 1$, $Omega$ 為 $Bbb{R}^n$ 中平滑有界區域,
$partial Omega $ 為 $Omega$ 的邊界, 又 $Delta$ 為 Laplacian
算子。 本文探討半線性拋物非局部之邊界條件積微分方程式:
egin{align*}
u_t(t,x)-Delta u(t,x) &= left(int^{t}_{0}mid u(s,x)mid ^{p}ds
ight) u(t,x) in (0,T) imes Omega,
otag
Bu(t,x) &= int_{Omega}K(x,y)u(t,y)dy in (0,T) imes partial Omega, label{equ:main}
u(0,x) &= u_{0}(x), xin Omega,
otag
&
end{align*}
其中 $K(x,y)$ 與 $u_{0}(x)$ 為 $Omegacup partial Omega$
上的非負連續函數, $B$ 為邊界算子
egin{equation*}
Buequiv alpha_{0} rac{partial u}{partial
u}+u,
end{equation*}
$alpha_0geqslant 0$, 且 $D rac{partial u}{partial
u }$
代表 $u$ 在 $partialOmega $ 上的外法向量導數。
本文證明了解的局部存在性與唯一性，並證明爆炸的產生。

centerline{Large Abstract} aselineskip=1.5 aselineskip
vspace{24pt} large Let $T$, $p$ be positive constants with
$pgeqslant 1$, $Omega$ be a smooth bounded domain in
$Bbb{R}^n$, $partial Omega $ be the boundary of $Omega$, and
$Delta$ be the Laplacian. This paper studies the semilinear
parabolic integro-differential problems with nonlocal boundary
condition:
egin{align*}
u_t(t,x)-Delta u(t,x) &= left(int^{t}_{0}mid u(s,x)mid ^{p}ds
ight) u(t,x) in (0,T) imes Omega,
otag
Bu(t,x) &= int_{Omega}K(x,y)u(t,y)dy in (0,T) imes partial Omega,
u(0,x) &= u_{0}(x), xin Omega,
otag
&
end{align*}
where $K(x,y)$ and $u_{0}(x)$ are nonnegative continuous functions
on $Omegacup partial Omega$, and $B$ is the boundary operator
egin{equation*}
Buequiv alpha_{0} rac{partial u}{partial
u}+u,
end{equation*}
with $alpha_0geqslant 0$, and $D rac{partial u}{partial

u }$
denotes the outward normal derivative of $u$ on $partialOmega $.
The local existence and uniqueness of the solution are
investigated. Blow-up criteria for the problem is given.

1.Introduction
2.Comparison results
3.Local existence of the problem
4.Non-existence of the solution
5.References

egin{thebibliography}{99}
setcounter{page}{31}
chapter{References}
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