华东师范大学学报（自然科学版） >

2019 , Vol. 2019 >Issue 3: 13 - 23

DOI: https://doi.org/10.3969/j.issn.1000-5641.2019.03.003

数学

衰退记忆型经典反应扩散方程在非线性边界条件下解的渐近性

  • 汪璇 ,
  • 赵涛 ,
  • 张玉宝
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  • 西北师范大学 数学与统计学院, 兰州 730070
汪璇,女,博士,教授,研究方向为非线性微分方程和无穷维动力系统理论应用.E-mail:wangxuan@nwnu.edu.cn.

收稿日期: 2018-04-02

  网络出版日期: 2019-05-30

基金资助

国家自然科学基金（11761062，11561064，11661071）；西北师范大学青年教师科研能力提升计划（NWNU-LKQN-14-6）

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

  • WANG Xuan ,
  • ZHAO Tao ,
  • ZHANG Yu-bao
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  • College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received date: 2018-04-02

  Online published: 2019-05-30

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摘要

本文研究了记忆型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超临界指数增长并满足一定的平衡条件时，运用抽象函数理论和半群理论，证明了该方程的全局吸引子在L2（Ω）×Lμ2（R+；H1（Ω））中的存在性，此结果改进和推广了一些已有的结果.

关键词： 经典反应扩散方程; 非线性边界; 衰退记忆; 任意阶多项式增长

本文引用格式

汪璇 , 赵涛 , 张玉宝 . 衰退记忆型经典反应扩散方程在非线性边界条件下解的渐近性[J]. 华东师范大学学报（自然科学版）, 2019 , 2019(3) : 13 -23 . DOI: 10.3969/j.issn.1000-5641.2019.03.003

Abstract

In this paper, we study the asymptotic behavior of solutions for the classical reaction-diffusion equation with memory. Through the use of abstract function theory and semigroup theory, the existence of a global attractor in L2(Ω)×Lμ2(R+; H1(Ω)) is proven when the internal nonlinearity and boundary nonlinearity adhere to polynomial growth of arbitrary order as well as the balance condition. This result extends and improves some known results.

Key words： classical reaction-diffusion equation; nonlinear boundary; fading memory; polynomial growth of arbitrary order

参考文献

[1] DAFERMOS C M. Asymptotic stability in viscoelasticity[J]. Archive Rational Mechanics & Analysis, 1970, 37(4):297-308.
[2] GIORGI C, PATA V. Asymptotic behavior of a nonlinear hyperbolic heat equation with memory[J]. Nonlinear Differential Equations and Applications NoDEA, 2001, 8(2):157-171.
[3] CHEPYZHOV V V, MIRANVILLE A. On trajectory and global attractors for semilinear heat equations with fading memory[J]. Indiana University Mathematics Journal, 2006, 55(1):119-168.
[4] 汪璇, 朱宗伟, 马巧珍. 带衰退记忆的经典反应扩散方程的全局吸引子[J]. 数学年刊(中文版), 2014, 35(4):423-434.
[5] CARVALHO A N, OLIVA S M, PEREIRA A L, et al. Attractors for parabolic problems with nonlinear boundary conditions[J]. Journal of Mathematical Analysis and Applications, 1997, 207(1/2):409-461.
[6] YANG L, YANG M H. Attractors of the non-autonomous reaction-diffusion equation with nonlinear boundary condition[J]. Nonlinear Analysis:Real World Applications, 2010, 11(5):3946-3954.
[7] YANG L. Asymptotic regularity and attractors of the reaction-diffusion equation with nonlinear boundary condition[J]. Nonlinear Analysis:Real World Applications, 2012, 13(3):1069-1079.
[8] RODRÍGUEZ-BERNAL A, TAJDINE A. Nonlinear balance for reaction diffusion equations under nonlinear boundary conditions:Dissipativity and blow-up[J]. Journal of Differential Equations, 2001, 169(2):332-372.
[9] PATA V, SQUASSINA M. On the strongly damped wave equation[J]. Communications in Mathematical Physics, 2005, 253(3):511-533.
[10] PATA V, ZUCCHI A. Attractors for a damped hyperbolic equation with linear memory[J]. Advances in Mathematical Sciences and Applications, 2001, 11(2):505-529.
[11] SUN C Y, CAO D M, DUAN J Q. Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor[J]. Discrete and continuous dynamical systems (Series B), 2008, 9:743-761.
[12] SUN C Y, CAO D M, DUAN J Q. Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity[J]. Nonlinearity, 2006, 19(11):2645-2665.
[13] ROBINSON J C. Infinite-Dimensional Dynamical Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors[M]. Cambridge:Cambridge University Press, 2001.
[14] TEMAM R. Infinite Dimensional Dynamical System in Mechanics and Physics[M]. 2nd ed. Berlin:SpringVerlag, 1997.
[15] HALE J K. Asymptotic behavior of dissipative systems[M]. Providence, RI:American Mathematical Society, 1988.
[16] 张玉宝, 汪璇. 无阻尼弱耗散抽象发展方程的强全局吸引子[J]. 华东师范大学学报(自然科学版), 2017, 2:8-19.
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