带有分布导数的反应扩散方程在$\Bbb R ^{n}$中全局吸引子的存在性

数学物理学报 ›› 2023, Vol. 43 ›› Issue (1): 82-92.

带有分布导数的反应扩散方程在$\Bbb R ^{n}$中全局吸引子的存在性

朱凯旋^1,^*(),孙涛¹(),谢永钦²()

¹湖南文理学院数理学院湖南常德 415000
²长沙理工大学数学与统计学院长沙 410114

收稿日期:2021-10-12 修回日期:2022-08-05 出版日期:2023-02-26 发布日期:2023-03-07
通讯作者: ^*朱凯旋, E-mail: zhukx12@163.com
作者简介:孙涛, E-mail: suntao5771@163.com|谢永钦, E-mail: xieyq@csust.edu.cn
基金资助:
湖南省自然科学基金(2022JJ30417);湖南省教育厅科学研究基金(21A0414);湖南省教育厅科学研究基金(21B0617);湖南文理学院科技创新团队项目(数值计算和随机过程及应用)

Global Attractors for a Class of Reaction-Diffusion Equations with Distribution Derivatives in $\Bbb R ^{n}$

Zhu Kaixuan^1,^*(),Sun Tao¹(),Xie Yongqin²()

¹College of Mathematics and Physics Science, Hunan University of Arts and Science, Hunan Changde 415000
²School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114

Received:2021-10-12 Revised:2022-08-05 Online:2023-02-26 Published:2023-03-07
Supported by:
Hunan Province Natural Science Foundation of China(2022JJ30417);Scientific Research Fund of Hunan Provincial Education Department(21A0414);Scientific Research Fund of Hunan Provincial Education Department(21B0617);Hunan University of Arts and Science(STIT): Numerical Calculation & Stochastic Process with Their Applications

摘要/Abstract

摘要：

该文考虑 $\Bbb R ^{n}$ 中一类带有加权项 $V(x)$ 和分布导数的反应扩散方程解的长时间行为, 证明了 $\big(L^{2}(\Bbb R ^{n}), L^{2}(\Bbb R ^{n})\cap L^{p}(\Bbb R ^{n})\big)$ -全局吸引子的存在性; 而且对任意的 $\delta_{1}\in [0,+\infty)$, 该吸引子能够在 $L^{2}\cap L^{p+\delta_{1}}$ -拓扑下吸引 $L^{2}(\Bbb R ^{n})$ 中的有界集.

关键词: 反应扩散方程, 渐近高阶可积性, 全局吸引子

Abstract:

In this paper, we consider the long-time behavior of solutions for a class of reaction-diffusion equations in $\Bbb R ^{n}$ with weighted terms $V(x)$ and some distribution derivatives in inhomogeneous terms. We prove that the $\big(L^{2}(\Bbb R ^{n}), L^{2}(\Bbb R ^{n})\cap L^{p}(\Bbb R ^{n})\big)$-global attractors indeed can attract every bounded subset of $L^{2}(\Bbb R ^{n})$ with the $L^{2}\cap L^{p+\delta_{1}}$-norm for any $\delta_{1}\in [0,+\infty)$.

Key words: Reaction-diffusion equations, Asymptotic higher-order integrability, Global attractors

中图分类号:

O193

朱凯旋, 孙涛, 谢永钦. 带有分布导数的反应扩散方程在$\Bbb R ^{n}$中全局吸引子的存在性[J]. 数学物理学报, 2023, 43(1): 82-92.

Zhu Kaixuan, Sun Tao, Xie Yongqin. Global Attractors for a Class of Reaction-Diffusion Equations with Distribution Derivatives in $\Bbb R ^{n}$[J]. Acta mathematica scientia,Series A, 2023, 43(1): 82-92.

参考文献

[1]	Alikakos A D. An application of the invariance principle to reaction-diffusion equations. J Differential Equations, 1979, 33: 201-225
[2]	Babin A V, Vishik M I. Attractors of Evolution Equations. Amsterdam: North-Holland, 1992
[3]	Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer, 2011
[4]	Cao D M, Sun C Y, Yang M H. Dynamics for a stochastic reaction-diffusion equation with additive noise. J Differential Equations, 2015, 259: 838-872
[5]	Cholewa J W, Dlotko T. Global Attractors in Abstract Parabolic Problems. Cambridge: Cambridge university Press, 2000
[6]	Hale J K. Asymptotic Behavior of Dissipative Systems. Providence: American Mathematival Society, 1988
[7]	Lions J L. Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Paris: Dunod, 1969
[8]	Marion M. Attractors for reactions-diffusion equations: existence and estimate of their dimension. Appl Anal, 1987, 25: 101-147
[9]	Marion M. Approximate inertial manifolds for reaction-diffusion equations in high space dimension. J Dynamic Differential Equations, 1989, 1: 245-267
[10]	Robinson J C. Infinite-Dimensional Dynamical Systems:An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge: Cambridge University Press, 2001
[11]	Sun C Y, Yuan L L, Shi J C. Higher-order integrability for a semilinear reaction-diffusion equation with distribution derivatives in $\Bbb R ^{N}$. Appl Math Lett, 2013, 26: 949-956
[12]	Sun C Y, Yuan Y B. $L^{p}$ -type pullback attractors for a semilinear heat equation on time-varying domains. Proc Roy Soc Edinburgh Sect A, 2015, 145: 1029-1052
[13]	Sun C Y, Zhong C K. Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains. Nonlinear Anal, 2005, 63: 49-65
[14]	Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. New York: Springer, 1997
[15]	Wang B. Attractors for reaction-diffusion equations in unbounded domains. Physica D, 1999, 128: 41-52
[16]	Xiao Y P, Sun C Y. Higher-order asymptotic attraction of pullback attractors for a reaction-diffusion equation in non-cylindrical domains. Nonlinear Anal, 2015, 113: 309-322
[17]	Xie Y Q, Li Q S, Huang C X, Jiang Y J. Aattractors for the semilinear reaction-diffusion equation with distribution derivatives. J Math Phys, 2013, 54(9): 092701
[18]	Zhang J, Zhong C K. The existence of global attractors for a class of reaction-diffusion equations with distribution derivatives terms in $\Bbb R ^{n}$. J Math Anal Appl, 2015, 427: 365-376
[19]	Zhong C K, Yang M H, Sun C Y. The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations. J Differential Equations, 2006, 223: 367-399
[20]	Zhu K X, Zhou F. Continuity and pullback attractors for a non-autonomous reaction-diffusion equation in $\Bbb R ^{N}$. Comput Math Appl, 2016, 71: 2089-2105