﻿ 具有分布导数的非经典反应扩散方程的指数吸引子 Exponential Attractor for Nonclassical Reaction-Diffusion Equation with Distributed Derivative

Vol. 08  No. 07 ( 2019 ), Article ID: 31443 , 7 pages
10.12677/AAM.2019.87150

Exponential Attractor for Nonclassical Reaction-Diffusion Equation with Distributed Derivative

Liyun Yan, Yonghua Ren*

School of Mathematics, Taiyuan University of Technology, Jinzhong Shanxi

Received: July 4th, 2019; accepted: July 19th, 2019; published: July 26th, 2019

ABSTRACT

In this paper, the existence of exponential attractor for a class of nonclassical reaction-diffusion equation with distributed derivative under homogeneous Neumann boundary conditions is studied by using the method of constructing extrusion property when the nonlinear term satisfies the growth of any polynomial.

Keywords:Reaction-Diﬀusion Equation, Distribution Derivative, Extrusion Property, Exponential Attractor

1. 引言

$\left\{\begin{array}{l}{u}_{t}-\Delta {u}_{t}-\Delta u+\eta u+f\left(u\right)={D}_{i}{h}^{i}+h\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(x,t\right)\in \Omega ×{R}^{+}\\ u=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }x\in \partial \Omega \\ u\left(x,0\right)={u}_{0}\left(x\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \Omega \end{array}$ (1)

${u}_{t}-\Delta u=f\left(u\right)+g\left(x\right)$

${u}_{t}-\Delta {u}_{t}-\Delta u=f\left(u\right)+g\left(x\right)$

$\left\{\begin{array}{l}{u}_{t}-\Delta u+g\left(u\right)={D}_{i}{f}^{i}+\text{ }f\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(x,t\right)\in \Omega ×{R}^{+}\\ u=0\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \partial \Omega \\ u\left(x,0\right)={u}_{0}\left(x\right)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }x\in \Omega \end{array}$

${f}^{\prime }\left(s\right)\ge -l,\text{\hspace{0.17em}}\forall s\in R$ (2)

${C}_{1}{|s|}^{p}-{C}_{0}\le f\left(s\right)s\le {C}_{2}{|s|}^{p}+{C}_{3},\text{\hspace{0.17em}}p\ge 2,\text{\hspace{0.17em}}\forall s\in R$ (3)

2. 预备知识

2.1. 常用空间

$H={L}^{2}\left(\Omega \right),V={H}_{0}^{1}\left(\Omega \right)$$\left(\cdot \text{ },\text{ }\text{ }\cdot \right)$$‖\text{ }\cdot \text{ }‖$ 分别表示H中的内积和范数，用 $‖\text{ }\cdot \text{ }‖$ 表示V的范数。

2.2. 基本定义及定理

1) M有有限的分形维数；

2) M为正不变集 $S\left(t\right)$ $S\left(t\right)M\subset M$$\forall t>0$

3) M为算子半群 ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ 的指数吸引集，即存在常数 $\alpha ,\beta >0$ 使得对任意的 $u\in B$

$dis{t}_{H}\left(S\left(t\right)u,M\right)\le \alpha {\text{e}}^{-\beta t},\forall t>0$

${‖\left(1-{P}_{N}\right)\left(Su-Sv\right)‖}_{E}\le {‖{P}_{N}\left(Su-Sv\right)‖}_{E}$

${‖\left(Su-Sv\right)‖}_{E}\le \frac{1}{8}{‖u-v‖}_{E}$

3. ${H}_{0}^{1}$ 中的指数吸引子

$u\in C\left(\left[0,T\right],H\right);u\in {L}^{2}\left(\left[0,T\right];V\right)\cap {L}^{p}\left(\left(0,T\right);\Omega \right)$

$S\left(t\right){u}_{0}=u\left(t\right),\forall t\ge 0$

$B=\stackrel{¯}{\underset{0\le t\le T}{\cup }S\left(t\right){B}_{0}}$

${|S\left(t\right){u}_{0}|}_{p}^{2}\le M,\forall t\ge T,{u}_{0}\in B$

$\underset{{u}_{0}\in B}{\mathrm{sup}}{‖{u}_{t}\left(t\right)‖}^{2}\le L,\forall t\ge 0$

${v}_{t}-\Delta {v}_{t}-\Delta v+{f}^{\prime }\left(u\right)v=0$ (4)

$\frac{\text{1}}{\text{2}}\frac{\text{d}}{\text{d}t}\left({‖\nabla v‖}^{2}+{‖\Delta v‖}^{2}\right)+{‖\Delta v‖}^{2}+\eta {‖\nabla v‖}^{2}\le l{‖\nabla v‖}^{2}$ (5)

$‖u\left(t\right)-v\left(t\right)‖\le {\text{e}}^{{c}_{2}t}‖u\left(0\right)-v\left(0\right)‖$ (6)

${\omega }_{t}-\Delta {\omega }_{t}-\Delta \omega +\eta \omega =f\left(v\right)-f\left(u\right)$ (7)

$-\Delta \omega$ 与式(7)在H中作内积

$\left(-\Delta \omega ,{\omega }_{t}\right)+\left(\Delta \omega ,\Delta {\omega }_{t}\right)+\left(\Delta \omega ,\Delta \omega \right)-\left(\Delta \omega ,\eta \omega \right)=\left(f\left(v\right)-f\left(u\right),-\Delta \omega \right)$ (8)

$\frac{\text{1}}{\text{2}}\frac{\text{d}}{\text{d}t}\left({‖\nabla \omega ‖}^{2}+{‖\Delta \omega ‖}^{2}\right)+{‖\Delta \omega ‖}^{2}+\eta {‖\nabla \omega ‖}^{2}=\left(f\left(v\right)-f\left(u\right),-\Delta \omega \right)$ (9)

$\begin{array}{c}|{\int }_{\Omega }\left(f\left(v\right)-f\left(u\right)\right)\left(-\Delta \omega \right)\text{d}x|\le {\int }_{\Omega }|{f}^{\prime }\left(\theta v+\left(1-\theta u\right)\right)|{|\Delta \omega |}^{2}\text{d}x\text{\hspace{0.17em}}\left(0<\theta <1\right)\\ \le c{\int }_{\Omega }l{|\Delta \omega |}^{2}\text{d}x\le {c}_{1}{|\Delta \omega |}^{2}\end{array}$ (10)

$\frac{\text{d}}{\text{d}t}\left({‖\nabla \omega ‖}^{2}+{‖\Delta \omega ‖}^{2}\right)\le {c}_{2}\left({‖\nabla \omega ‖}^{2}+{‖\Delta \omega ‖}^{2}\right)$ (11)

${‖\nabla \omega \left(t\right)‖}^{2}+{‖\Delta \omega \left(t\right)‖}^{2}\le {\text{e}}^{{c}_{2}t}\left({‖\nabla \omega \left(0\right)‖}^{2}+{‖\Delta \omega \left(0\right)‖}^{2}\right)$ (12)

${‖S\left({t}_{1}\right){u}_{1}-S\left({t}_{2}\right){u}_{2}‖}_{\text{2}}\le {‖S\left({t}_{1}\right){u}_{1}-S\left({t}_{1}\right){u}_{2}‖}_{\text{2}}+{‖S\left({t}_{1}\right){u}_{2}-S\left({t}_{2}\right){u}_{2}‖}_{\text{2}}$ (13)

${‖u\left({t}_{1}\right)-u\left({t}_{2}\right)‖}_{\text{2}}\le |{\int }_{{t}_{1}}^{{t}_{2}}{‖{u}_{t}\left(y\right)‖}_{2}\text{d}y|\le L|{t}_{1}-{t}_{2}|$ (14)

${‖S\left({t}_{1}\right){u}_{1}-S\left({t}_{2}\right){u}_{2}‖}_{\text{2}}\le L\left[|{t}_{1}-{t}_{2}|+{‖{u}_{1}-{u}_{2}‖}_{\text{2}}\right]$ (15)

${‖\left(1-P\right)\left(S\left({t}_{\ast }\right){u}_{0}-S\left({t}_{\ast }\right){v}_{0}\right)‖}_{2}>{‖P\left(S\left({t}_{\ast }\right){u}_{0}-S\left({t}_{\ast }\right){v}_{0}\right)‖}_{2}$

${‖S\left({t}_{\ast }\right){u}_{0}-S\left({t}_{\ast }\right){v}_{0}‖}_{2}\le \frac{1}{8}{‖{u}_{0}-{v}_{0}‖}_{2}$

$0<{\lambda }_{1}\le {\lambda }_{2}\le {\lambda }_{3}\le \cdots \le {\lambda }_{N}\le \cdots ,{\lambda }_{N}\to \infty$

$A{\omega }_{i}={\lambda }_{i}{\omega }_{i},i=1,2,\cdots$ (16)

${H}_{N}=Span\left\{{\omega }_{1},{\omega }_{2},\cdots ,{\omega }_{N}\right\}$${P}_{N}:H\to {H}_{N}$ 是正交投影，记 ${Q}_{N}=1-{P}_{N}$ 是在 ${H}_{N}$ 上正交完备化的正交投影， $\omega ={P}_{N}\omega +{Q}_{N}\omega \triangleq p+q$ 。假设 $|{P}_{N}\omega \left(t\right)|\le |{Q}_{N}\omega \left(t\right)|$

$-\Delta q$ 与式(7)在H中作内积，

$\left(-\Delta q,{\omega }_{t}\right)+\left(\Delta q,\Delta {\omega }_{t}\right)+\left(\Delta q,\Delta \omega \right)-\left(\Delta q,\eta \omega \right)=\left(f\left(v\right)-f\left(u\right),-\Delta q\right)$ (17)

$\frac{\text{1}}{\text{2}}\frac{\text{d}}{\text{d}t}\left({‖\nabla q‖}^{2}+{‖\Delta q‖}^{2}\right)+{‖\Delta q‖}^{2}+\eta {‖\nabla q‖}^{2}=\left(f\left(v\right)-f\left(u\right),-\Delta q\right)$ (18)

$|{\int }_{\Omega }\left(f\left(v\right)-f\left(u\right)\right)\left(-\Delta q\right)\text{d}x|\le {c}_{3}{\int }_{\Omega }|\omega ||\Delta q|\text{d}x\le \frac{{‖\Delta q‖}^{2}}{2}+\frac{{c}_{3}}{2}{‖\omega ‖}^{2}$ (19)

$\frac{\text{d}}{\text{d}t}\left({‖\nabla q‖}^{2}+{‖\Delta q‖}^{2}\right)+{‖\Delta q‖}^{2}\le {c}_{3}{‖\omega ‖}^{2}$ (20)

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({‖\nabla q‖}^{2}+{‖\Delta q‖}^{2}\right)+\frac{{‖\Delta q‖}^{2}}{\text{2}}+\frac{{\lambda }_{N+1}}{\text{2}}{‖\nabla q‖}^{2}\\ \le {c}_{3}{‖\omega ‖}^{2}\le {c}_{3}{‖p+q‖}^{2}\le 2{c}_{3}{‖q‖}^{2}\le 2{c}_{3}{\lambda }_{N+1}^{-1}{‖\nabla q‖}^{2}\\ \le {c}_{4}{\lambda }_{N+1}^{-1}{‖\Delta \omega ‖}^{2}\le {c}_{4}{\lambda }_{N+1}^{-1}{\text{e}}^{{c}_{2}t}{‖\Delta \omega \left(0\right)‖}^{2}\end{array}$ (21)

$\frac{\text{d}}{\text{d}t}\left({‖\nabla q‖}^{2}+{‖\Delta q‖}^{2}\right)+\frac{{‖\Delta q‖}^{2}}{\text{2}}+\frac{{\lambda }_{1}}{\text{2}}{‖\nabla q‖}^{2}\le {c}_{4}{\lambda }_{N+1}^{-1}{\text{e}}^{{c}_{2}t}{‖\Delta \omega \left(0\right)‖}^{2}$ (22)

${c}_{5}=\mathrm{min}\left\{\frac{1}{2},\frac{{\lambda }_{1}}{2}\right\}$，有：

$\frac{\text{d}}{\text{d}t}\left({‖\nabla q‖}^{2}+{‖\Delta q‖}^{2}\right)+{c}_{5}\left({‖\nabla q‖}^{2}+{‖\Delta q‖}^{2}\right)\le {c}_{4}{\lambda }_{N+1}^{-1}{\text{e}}^{{c}_{2}t}{‖\Delta \omega \left(0\right)‖}^{2}$ (23)

$\begin{array}{c}{‖\nabla q\left(t\right)‖}^{2}+{‖\Delta q\left(t\right)‖}^{2}\le {\text{e}}^{-{c}_{5}t}\left({‖\nabla q\left(0\right)‖}^{2}+{‖\Delta q\left(0\right)‖}^{2}\right)+{c}_{6}{\lambda }_{N+1}^{-1}{\text{e}}^{{c}_{2}t}{‖\Delta \omega \left(0\right)‖}^{2}\\ \le {c}_{7}\left({\text{e}}^{-{c}_{5}t}+{c}_{8}{\lambda }_{N+1}^{-1}{\text{e}}^{{c}_{2}t}\right){‖\Delta \omega \left(0\right)‖}^{2}\end{array}$ (24)

${‖\Delta \omega \left(t\right)‖}^{2}\le \text{2}{‖\Delta q\left(t\right)‖}^{2}\le {c}_{\text{9}}\left({\text{e}}^{-{c}_{5}t}+{c}_{\text{10}}{\lambda }_{N+1}^{-1}{\text{e}}^{{c}_{2}t}\right){‖\Delta \omega \left(0\right)‖}^{2}$ (25)

${t}_{\ast }>0$，使得 ${c}_{9}{\text{e}}^{-{c}_{5}{t}_{\ast }}\le \frac{1}{128}$，固定 ${t}_{\ast }$，且N足够大，使得 ${c}_{9}{c}_{10}{\lambda }_{N+1}^{-1}{\text{e}}^{{c}_{2}{t}^{\ast }}\le \frac{1}{128}$ 。有：

$‖\Delta \omega \left({t}_{\ast }\right)‖\le \frac{\text{1}}{\text{8}}‖\Delta \omega \left(0\right)‖$ (26)

$M=\underset{0\le t\le {t}_{\ast }}{\cup }S\left(t\right){M}_{\ast }$

Exponential Attractor for Nonclassical Reaction-Diffusion Equation with Distributed Derivative[J]. 应用数学进展, 2019, 08(07): 1284-1290. https://doi.org/10.12677/AAM.2019.87150

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18. NOTES

*通讯作者。