单调区域上非线性退化抛物方程的长时间性态  

Long Time Behavior of Nonlinear Degenerate Parabolic Equations in Monotone Domains

作　　者：李心 褚锦芳 LI Xin;CHU Jinfang(School of Science,Yanshan University,Qinhuangdao 066004,China)

机构地区：[1]燕山大学理学院,河北秦皇岛066004

出　　处：《山西大学学报（自然科学版）》2024年第4期677-690,共14页Journal of Shanxi University(Natural Science Edition)

基　　金：国家自然科学基金(11801493);河北省自然科学基金(A2022203004);河北省教育厅高等学校科技计划青年基金项目(QN2020203)。

摘　　要：单调区域上的非线性退化抛物方程固有的非自治性和退化性使得对该方程的研究具有本质性的困难。本文研究了该方程满足能量等式变分解的适定性以及拉回吸引子的存在性。首先,利用惩罚法得到该方程的扰动方程满足能量等式变分解的适定性。其次,针对扰动方程,利用正则性方法得到该方程极限解的存在性,进而得到原方程满足能量等式变分解的适定性。最后,根据变分解的适定性构造非自治动力系统的双参数半群,结合一致能量耗散估计和收缩函数法证明了该半群存在渐近紧的拉回吸收集,从而建立了此类系统的拉回吸引子。The nonlinear degenerate parabolic equation in monotone domain has essential difficulties because of its inherent non-autonomy and degeneracy.This paper mainly studied the well-posedness of the variational solution of satisfying the energy equality and the existence of the pull-back attractor.First,the penalty method was used to obtain the well-posedness of variational solution of the perturbation equation which satisfies the energy equality.Second,for the perturbed equation,the existence of the limit solution of the equation was attained by the regularity method,further gained the well-posedness of the variational solution of original equation which satisfies the energy equality.Final,based on the well-posedness of variational solutions,we constructed a two-parameter semigroup of non-autonomous dynamical system,and the existence of asymptotically compact pullback absorbing sets for such system was proved by combining the uniform energy dissipation estimates and the contraction function method,thus,the pull-back attractor of this kind of system was established from the preceding analysis.

关 键 词：退化抛物方程 单调区域 惩罚法 适定性 拉回吸引子 

分 类 号：O175.2[理学—数学]