带有衰退记忆的非自治经典反应扩散方程在非线性边界下解的渐近性

doi:10.3969/j.issn.1000-5641.201911046

华东师范大学学报（自然科学版） ›› 2021, Vol. 2021 ›› Issue (1): 16-27.doi: 10.3969/j.issn.1000-5641.201911046

带有衰退记忆的非自治经典反应扩散方程在非线性边界下解的渐近性

梁玉婷, 汪璇*()

西北师范大学数学与统计学院, 兰州　730070

收稿日期:2019-11-25 出版日期:2021-01-25 发布日期:2021-01-28
通讯作者: 汪璇 E-mail:wangxuan@nwnu.edu.cn
基金资助:
国家自然科学基金(11761062, 11561064, 11661071)

Asymptotic behavior of solutions for the non-autonomous classical reaction-diffusion equation with nonlinear boundary conditions and fading memory

Yuting LIANG, Xuan WANG*()

College of Mathematics and Statistics, Northwest Normal University, Lanzhou　730070, China

Received:2019-11-25 Online:2021-01-25 Published:2021-01-28
Contact: Xuan WANG E-mail:wangxuan@nwnu.edu.cn

摘要/Abstract

摘要：

本文研究了带有衰退记忆的非自治经典反应扩散方程解的长时间动力学行为, 当内部非线性项和边界非线性项均以任意阶多项式增长并满足一定的平衡条件, 且外力项仅为平移有界而非平移紧时, 运用收缩函数方法和过程理论, 证明了一致吸引子在 $L^{2}(\Omega)\times L_\mu^2(\mathbb R^+; H_{0}^1(\Omega))$ 中的存在性及其吸引子的拓扑结构. 该结果改进和推广了一些已有的结果.

关键词: 非自治经典反应扩散方程, 一致吸引子, 非线性边界, 衰退记忆, 任意阶多项式增长

Abstract:

In this paper, we study the long-time dynamic behavior of solutions for the non-autonomous classical reaction-diffusion equation with nonlinear boundary conditions and fading memory, where the internal nonlinearity and boundary nonlinearity adheres to polynomial growth of arbitrary order as well as the balance condition. In addition, the forcing term is translation bounded, rather than translation compact, by use of contractive function method and process theory. The existence and the topological structure of uniform attractors in $L^{2}(\Omega)\times L_\mu^2(\mathbb R^+; H_{0}^1(\Omega))$ are proven. This result extends and improves existing research in the literature.

Key words: non-autonomous classical reaction-diffusion equation, uniform attractors, nonlinear boundary, fading memory, polynomial growth of arbitrary order

中图分类号:

O175.29

梁玉婷, 汪璇. 带有衰退记忆的非自治经典反应扩散方程在非线性边界下解的渐近性[J]. 华东师范大学学报（自然科学版）, 2021, 2021(1): 16-27.

Yuting LIANG, Xuan WANG. Asymptotic behavior of solutions for the non-autonomous classical reaction-diffusion equation with nonlinear boundary conditions and fading memory[J]. Journal of East China Normal University(Natural Science), 2021, 2021(1): 16-27.

参考文献 18

1	DAFERMOS C M. Archive Rational Mechanics Analysis, Asymptotic stability in viscoelasticity. 1970, 37 (4): 297- 308. doi: 10.1007/BF00251609
2	YANG L. Nonlinear Anal, Uniform attractors for the closed process and application to the reaction-diffusion equation with dynamical boundary condition. 2009, 71, 4012- 4025. doi: 10.1016/j.na.2009.02.083
3	YANG L. Nonlinear Analysis: Real World Applications, Asymptotic regularity and attractors of the reaction-diffusion equation with nonlinear boundary condition. 2012, 13 (3): 1069- 1079. doi: 10.1016/j.nonrwa.2011.02.024
4	YANG L, YANG M H. Nonlinear Analysis: Real World Applications, Attractors of the non-autonomous reaction-diffusion equation with nonlinear boundary condition. 2010, 11 (5): 3946- 3954. doi: 10.1016/j.nonrwa.2010.03.002
5	ZHONG C K, YANG M H, SUN C Y. Journal of Differential Equations, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations. 2006, 223, 367- 399. doi: 10.1016/j.jde.2005.06.008
6	SUN C Y, WANG S H, ZHONG C K. Acta Math Sin(Engl Ser), Global attractors for a nonclassical diffusion equation. 2007, 23, 1271- 1280. doi: 10.1007/s10114-005-0909-6
7	SUN C Y, YANG M H. Asymptot Anal, Dynamics of the nonclassical diffusion equation. 2008, 59, 51- 81. doi: 10.3233/ASY-2008-0886
8	SONG H T, MA S, ZHONG C K. Nonlinearity, Attractors of non-autonomous reaction-diffusion equations. 2009, 22, 667- 681. doi: 10.1088/0951-7715/22/3/008
9	CHEPYZHOV V V, MIRANVILLE A. Indiana University Mathematics Journal, On trajectory and global attractors for semilinear heat equations with fading memory. 2006, 55 (1): 119- 168. doi: 10.1512/iumj.2006.55.2597
10	GIORGI C, PATA V. Nonlinear Differential Equations Applications Nodea, Asymptotic behavior of a nonlinear hyperbolic heat equation with memory. 2001, 8 (2): 157- 171. doi: 10.1007/PL00001443
11	汪璇, 朱宗伟, 马巧珍. 数学年刊(中文版), 带衰退记忆的经典反应扩散方程的全局吸引子. 2014, 35 (4): 423- 434.
12	PATA V, SQUASSINA M. Communications in Mathematical Physics, On the strongly damped wave equation. 2005, 253 (3): 511- 533. doi: 10.1007/s00220-004-1233-1
13	PATA V, ZUCCHI A. Advances in Mathematical Sciences and Applications, Attractors for a damped hyperbolic equation with linear memory. 2001, 11 (2): 505- 529.
14	RODRÍGUEZ-BERNAL A, TAJDINE A. Journal of Differential Equations, Nonlinear balance for reaction diffusion equations under nonlinear boundary conditions: Dissipativity and blow-up. 2001, 169 (2): 332- 372. doi: 10.1006/jdeq.2000.3903
15	SUN C Y, CAO D M, DUAN J Q. Nonlinearity, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity. 2006, 19 (11): 2645- 2665. doi: 10.1088/0951-7715/19/11/008
16	XIE Y Q, LI Y A, ZENG Y. Uniform attractors for nonclassical diffusion equations with memory[J]. Journal of Function Spaces, 2016 (3/4): 1-11.
17	CHEPYZHOV V V, VISHIK M I. Attractors for Equations of Mathematical Physics [M]. Providence, RI: Amer Math Soc, 2001.
18	SUN C, CAO D, DUAN J. SIAM Journal on Applied Dynamical Systems, Uniform attractors for non-autonomous wave equations with nonlinear damping. 2007, 6 (2): 293- 318. doi: 10.1137/060663805

带有衰退记忆的非自治经典反应扩散方程在非线性边界下解的渐近性

Asymptotic behavior of solutions for the non-autonomous classical reaction-diffusion equation with nonlinear boundary conditions and fading memory

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