带记忆项反应扩散方程的吸引子  

Attractors for Reaction-diffusion Equations with Memory

作　　者：谢永钦 张江卫 黄创霞 Yong Qin;XIE Jiang;Wei ZHANG;Chuang Xia HUANG(School of Mathematics and Statistics,Changsha University of Science and Technology,Changsha 410114,P.R.China)

机构地区：[1]长沙理工大学数学与统计学院,长沙410114

出　　处：《数学学报（中文版）》2021年第6期979-990,共12页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11101053,71471020);湖南省研究生科研创新项目(CX20200891)。

摘　　要：本文提出了一类双积空间压缩半群方法(见定理2.10),可作为另一种研究非线性发展方程渐近行为的途径.作为应用,我们考虑了带有衰退记忆项的反应扩散方程,证明了当初值属于L^(2)(Ω)×L_(μ)^(2)(R;H_(0)^(1)(Ω))时所对应解半群在H_^(0)^(1)(Ω)×L_(μ)^(2)(R;H_(0)^(1)(Ω))中的渐近紧性.因而得到了双空间全局吸引子A的存在性.此外,通过使用新的算子分解方法,我们证明了解的渐近正则性,得到了压缩函数.值得注意的是,非线性项f满足任意阶指数增长且全局吸引子A■D(A)×L_(μ)^(2)(R;D(A)).In this paper,the method(or technique)of bi-spaces contractive semigroup method(see Theorem 2.10)is presented in a general setting as an alternative approach to the study the asymptotic behavior of nonlinear evolutionary equation.As an application,we consider reaction-diffusion equations with fading memory,and prove the asymptotic compactness of the semigroup on H_^(0)^(1)(Ω)×L_(μ)^(2)(R;H_(0)^(1)(Ω)) with initial data in L^(2)(Ω)×L_(μ)^(2)(R;H_(0)^(1)(Ω)).Thus the bi-spaces global attractor A is confirmed.Furthermore,by using the new decomposition technique,we demonstrate the asymptotic regularity of the solution to obtain the contractive function.It is noteworthy that the nonlinearity f satisfies the polynomial growth of arbitrary order and bi-spaces global attractor A■D(A)×L_(μ)^(2)(R;D(A)).

关 键 词：反应扩散方程 双空间压缩半群 记忆项 双积空间全局吸引子 

分 类 号：O193[理学—数学]