| 非柱形区域上的非自治反应扩散方程解的长时间行为研究 |
| Alternative Title | The long-time behaviors of solutions for non-autonomous reaction-diffusion equations on non-cylindrical domains
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| 肖艳萍 |
| Thesis Advisor | 孙春友
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| 2014-12-07
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| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
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| Degree Name | 博士
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| Keyword | 非柱形区域
反应扩散方程
弱解
变分解
拉回 ~$\mathscr{D}$-吸引子
高阶可积性
连续性
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| Abstract | 本文主要研究如下定义在非柱形区域上的非自治反应扩散方程解的长时间行为.我们的主要工作是建立新的方法(框架)和先验估计, 对非线性项及外力项不增加任何额外假设,
特别地, 对外力项不做任何光滑性假设, 证明已知的 ~$(L^2,L^2)$ 型拉回~$\mathscr{D}$-吸
引子事实上可以按 ~$L^{2+\delta}$ ($\forall~\delta\in [0,\infty)$) 范数吸引相同的
集族 ~$\mathscr{D}$, 并进一步证明其 ~$H_0^1$ 拉回~$\mathscr{D}$-吸引性.
需要指出的是即使对定义在柱形区域上的非自治系统来说, 当 ~$f\in
L^2_{loc}(\mathbb{R};$ $L^2(\x))$ 时,
目前已知最好的结果是~$(L^2,L^p)$ (这里 ~$p$ 满足 ~$\eqref{g1}$) 拉回
~$\mathscr{D}$-吸引性, 对于 ~$s>0$, 关于 ~$(L^2,L^{p+s})$
拉回吸引的相关结论还未见到; 而当 ~$f\in
L^2_{loc}(\mathbb{R};H^{-1}(\x))$ 时, 最好的结果是 ~$(L^2,L^2)$ 拉回
~$\mathscr{D}$-吸引性, 对于~$(L^2,L^{q})$$(q>2)$ 拉回
~$\mathscr{D}$-吸引是否成立仍是开问题. 我们建立在变区域上的定理
\ref{main11} 和定理 \ref{main} 不但解决了上述问题,
而且也解决了变区域系统上类似的开问题. 另一方面, 关于解对初值按
~$H^1$ 范数的连续性, 就柱形区域上的非自治系统而言没有任何相关结论.
本文对任意空间维数 ~$N$ 和非线性项的任意增长阶 ~$p\geqslant 2$,
给出了变区域上解在~$H^1$ 空间中关于初值的连续依赖性定理 \ref{continuity}, 该定理解决了包含柱形区域非自治系统在内存在的相应的开问题. 同时, 我们给出的抽象结论和先验估计, 对其它耗散型方程解的长时间行为研究也有一定的借鉴作用. |
| Other Abstract | In this dissertation, we consider the long-time
behaviors of solutions for the following reaction-diffusion
equation defined on non-cylindrical domains.
The main works of this paper are to establish new methods
(framework) and a priori estimates to prove, without any additional
assumptions, especially, no any smoothness assumptions on the
forcing term, that the known $(L^2,L^2)$ pullback
$\mathscr{D}$-attractor indeed can attract the same
class $\mathscr{D}$
in $L^{2+\delta}$-norm ($\delta\in
[0,\infty)$ is arbitrary) and $H^1$-norm.
We point out that even for the cylindrical domains case, up to now,
the best known results about the attraction associated to the
corresponding non-autonomous system above is $(L^2,L^p)$ (the power
$p$ comes from ~$\eqref{g1}$) pullback $\mathscr{D}$-attraction for
$f\in L^2_{loc}(\mathbb{R};L^2(\x))$, and there is no any result
about ~$(L^2,L^{p+s})$ pullback ~$\mathscr{D}$-attraction for $s>0$;
as $f\in L^2_{loc}(\mathbb{R};H^{-1}(\x))$, the best attraction is
$(L^2,L^2)$ pullback $\mathscr{D}$-attraction, and the $(L^2,L^{q})$
pullback attraction remains open for any $q>2$. Our main results
($Theorems$ \ref{main11} and \ref{main}) established for
non-cylindrical domains case solve (even for the non-autonomous
system defined on cylindrical domains) the problems mentioned above.
On the other hand, for non-autonomous system defined on cylindrical
domains, there is no any result about the continuity of solutions
w.r.t. initial data in $H^1$. Here, for any space dimension and any
$p\in [2,\infty)$, we obtain the continuity of solutions w.r.t.
initial data in $H^1$, see $Theorem$ \ref{continuity}, which makes
up the gap even for cylindrical domains case. Moreover, we emphasize
that our method, results and proof scheme are applicable to other dissipative equations. |
| URL | 查看原文
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| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/225445
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
肖艳萍. 非柱形区域上的非自治反应扩散方程解的长时间行为研究[D]. 兰州. 兰州大学,2014.
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