An extension of the concept of gradient semigroups which is stable under perturbation
An extension of the concept of gradient semigroups which is stable under perturbation
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DOI：
10.1016/j.jde.2009.01.007
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发表时间：
2009-04
影响因子：
2.4
通讯作者：
A. Carvalho;J. Langa
中科院分区：
文献类型：
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作者：
A. Carvalho;J. Langa
在这篇文章中，我们引入了梯度类非线性半群的概念，作为梯度非线性半群（具有李雅普诺夫函数的那些，见[J. K.黑尔，耗散系统的渐近行为，数学调查学报，第25卷，美国数学学会，1989]）和一个具有类梯度吸引子的非线性半群。证明了类梯度非线性半群的扰动保持类梯度非线性半群。此外，对于非自治动力系统，我们引入了类梯度演化过程的概念，证明了类梯度非线性半群的非自治扰动是类梯度演化过程.对于类梯度非线性半群和发展过程，我们证明了它们的吸引子在扰动下的连续性、特征和（拉回和前向）指数吸引，推广了[A.N. Carvalho，J.A.朗加，J.C.罗宾逊，A. Suárez，Characterization of non-autonomous attractor of a perturbed gradient system，J. Differential Equations 236（2007）570-603]的特征和[A.V. Babin，M.I.张文，等.演化方程中的吸引子.数学与应用，2000，24（1）：111 - 112. Vol. 25，North-Holland，Amsterdam，1992] on exponential attraction.
In this article we introduce the concept of a gradient-like nonlinear semigroup as an intermediate concept between a gradient nonlinear semigroup (those possessing a Lyapunov function, see [J.K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol. 25, Amer. Math. Soc., 1989]) and a nonlinear semigroup possessing a gradient-like attractor. We prove that a perturbation of a gradient-like nonlinear semigroup remains a gradient-like nonlinear semigroup. Moreover, for non-autonomous dynamical systems we introduce the concept of a gradient-like evolution process and prove that a non-autonomous perturbation of a gradient-like nonlinear semigroup is a gradient-like evolution process. For gradient-like nonlinear semigroups and evolution processes, we prove continuity, characterization and (pullback and forwards) exponential attraction of their attractors under perturbation extending the results of [A.N. Carvalho, J.A. Langa, J.C. Robinson, A. Suárez, Characterization of non-autonomous attractors of a perturbed gradient system, J. Differential Equations 236 (2007) 570–603] on characterization and of [A.V. Babin, M.I. Vishik, Attractors in Evolutionary Equations, Stud. Math. Appl., vol. 25, North-Holland, Amsterdam, 1992] on exponential attraction.