Banach 空间上半线性非自治发展方程的伪轨跟踪性

数学物理学报 ›› 2025, Vol. 45 ›› Issue (4): 1144-1160.

Banach 空间上半线性非自治发展方程的伪轨跟踪性

涂坤(),丁惠生^*()

江西师范大学数学与统计学院南昌 330022

收稿日期:2024-09-11 修回日期:2025-01-26 出版日期:2025-08-26 发布日期:2025-08-01
通讯作者: *E-mail: dinghs@mail.ustc.edu.cn
作者简介:E-mail: tukun@jxnu.edu.cn
基金资助:
国家自然科学基金(12361023);江西省双千计划(jxsq2019201001);江西省自然科学基金重点项目(20242BAB26001)

Shadowing Properties of Semilinear Nonautonomous Evolution Equations on Banach Spaces

Tu Kun(),Ding Huisheng^*()

School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022

Received:2024-09-11 Revised:2025-01-26 Online:2025-08-26 Published:2025-08-01
Supported by:
NSFC(12361023);Double Thousand Plan of Jiangxi Province(jxsq2019201001);Key Project of Jiangxi Provincial NSF(20242BAB26001)

摘要/Abstract

摘要：

该文讨论了 Banach 空间 $X$ 上半线性非自治发展方程

$u'(t) = A(t)u(t)+f(t,u(t)), \ \ t \in \mathbb{R}$

的伪轨跟踪性, 其中线性算子 $A(t) : D(A(t)) \subset X \rightarrow X$ 未必有界且 $u'(t)=A(t)u(t)$ 具有指数二分性. 该文首先在 $f$ 满足经典 Lipschitz 条件和更弱的 $BS^p$ 型 Lipschitz 条件下建立了伪轨跟踪性结果, 然后进一步引入了 $L^p$ 伪轨和 $L^p$ 伪轨跟踪性的概念并建立了相应的跟踪性定理, 最后给出了一个抛物型偏微分方程的例子作为抽象结果的应用. 相比已有文献, 该文不但减弱了非线性项的 Lipschitz 条件以及引入和讨论了新的 $L^p$ 伪轨跟踪性, 而且最重要的是允许 $A(t)$ 为无界算子从而抽象结果可以应用到偏微分方程.

关键词: 抽象发展方程, 指数二分性, 伪轨跟踪性

Abstract:

This paper discusses the shadowing properties of the semilinear nonautonomous evolution equation

$u'(t) = A(t)u(t) + f(t, u(t)), \ \ t \in \mathbb{R}$

on a Banach space $X$, where the linear operator $A(t) : D(A(t)) \subset X \rightarrow X$ may not be bounded and $u'(t)=A(t)u(t)$ admits exponential dichotomy. This paper first establishes shadowing properties under the classical Lipschitz condition and a weaker $BS^p $ type Lipschitz condition for $f$. Then we further introduce the concepts of $L^p$ pseudo orbits and $L^p$ shadowing property, establishing corresponding shadowing theorem. Finally, an example of a parabolic partial differential equation is provided as an application of the abstract results. Compared to existing literature, this paper not only weakens the Lipschitz condition for the nonlinear term and introduces and discusses the new $L^p$ shadowing property, but most importantly, it allows $A(t)$ to be an unbounded operator, thereby enabling the abstract results to be applied to partial differential equations.

Key words: abstract evolution equation, exponential dichotomy, shadowing property

中图分类号:

O177.92

涂坤, 丁惠生. Banach 空间上半线性非自治发展方程的伪轨跟踪性[J]. 数学物理学报, 2025, 45(4): 1144-1160.

Tu Kun, Ding Huisheng. Shadowing Properties of Semilinear Nonautonomous Evolution Equations on Banach Spaces[J]. Acta mathematica scientia,Series A, 2025, 45(4): 1144-1160.

参考文献 26

[1]	Al-Nayef A, Diamond P, Kloeden P, et al. Bi-shadowing and delay equations. Dynam Stability Systems, 1996, 11(2): 121-134
[2]	Anosov D V. On a class of invariant sets of smooth dynamical systems. Proc 5th Int Conf on Nonlin Oscill, 1970, 2: 39-45
[3]	Arendt W, Batty C J K, Hieber M, et al. Vector-Valued Laplace Transforms and Cauchy Problems. Basel: Verlag, 2011
[4]	Backes L, Dragičević D. A general approach to nonautonomous shadowing for nonlinear dynamics. Bull Sci Math, 2021, 1.0: 102996
[5]	Backes L, Dragičević D. Hyers-Ulam stability for hyperbolic random dynamics. Fund Math, 2021, 2.5(1): 69-90
[6]	Backes L, Dragičević D. Parameterized shadowing for nonautonomous dynamics. J Math Anal Appl, 2024, 5.9(1): 127584
[7]	Backes L, Dragičević D. Shadowing for infinite dimensional dynamics and exponential trichotomies. Proc Roy Soc Edinburgh, 2021, 1.1(3): 863-884
[8]	Backes L, Dragičević D, Pituk M, Singh L. Weighted shadowing for delay differential equations. Arch Math, 2022, 1.9(5): 539-552
[9]	Bowen R. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Berlin: Springer-Verlag, 1975
[10]	Chow S N, Lin X B, Palmer K J. A shadowing lemma with applications to semilinear parabolic equations. SIAM J Math Anal, 1989, 20(3): 547-557
[11]	D'Aniello E, Darji U B, Maiuriello M. Generalized hyperbolicity and shadowing in $L^{p}$ spaces. J Differential Equations, 2021, 2.8: 68-94
[12]	Dragičević D. Generalized dichotomies and Hyers-Ulam stability. Results Math, 2024, 79(1): Article 37
[13]	Dragičević D, Pituk M. Shadowing for nonautonomous difference equations with infinite delay. Appl Math Lett, 2021, 1.0: 107284
[14]	Engel K J, Nagel R. One-parameter Semigroups for Linear Evolution Equations. New York: Springer-Verlag, 2000
[15]	Gao S. A shadowing lemma for random dynamical systems. J Appl Anal Comput, 2021, 11(6): 3014-3030
[16]	Henry D. Geometric Theory of Semilinear Parabolic Equations. Berlin: Springer-Verlag, 1981
[17]	Meyer K R, Sell G R. An analytic proof of the shadowing lemma. Funkcial Ekvac, 1987, 30: 127-133
[18]	Palmer K J. Exponential dichotomies, the shadowing lemma and transversal homoclinic points. Dynamics Reported, 1988, 1: 265-306
[19]	Palmer K J. Shadowing and Silnikov chaos. Nonlinear Anal, 1996, 27(9): 1075-1093
[20]	Palmer K J. Shadowing in Dynamical Systems:Theory and Applications. Dordrecht: Springer Science, 2000
[21]	Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983
[22]	Pilyugin S Y, Sakai K. Shadowing and Hyperbolicity. Cham: Springer, 2017
[23]	Pilyugin S Y. Shadowing in Dynamical Systems. Berlin: Springer-Verlag, 1999
[24]	Pilyugin S Y. Shadowing in structurally stable flows. J Differential Equations, 1997, 1.0: 238-265
[25]	Sell G R, You Y. Dynamics of Evolutionary Equations. New York: Springer-Verlag, 2002
[26]	Yang X F, Lu T X, Pi J M, Jiang Y X. On shadowing system generated by a uniformly convergent mappings sequence. J Dyn Control Syst, 2023, 29(3): 691-702