报 告 题 目：Aronszajn-type Topological Regularity for Nonlinear Delay Evolutions in Fr\'{e}chet Spaces

主 讲 人：王 荣 年

单 位：上海师范大学

时 间：5月12日9:00

腾讯 ID：705-143-702

摘 要：

The Aronszajn-type regularity, also called $R_\delta$\text{-}structure, was proposed for characterizing the solution set of differential equation having no uniqueness. The solution set with such regularity is homeomorphic to the intersection of a decreasing sequence of compact absolute retracts. In the scale of infinite-dimensionality, we consider, in the present paper, a nonlinear delay evolution equation involving multivalued nonlinearity and possibly unbounded operator in the principal part. Both a compact interval and a noncompact one are considered.  The nonlinearity, having nonempty, convex and closed values, is upper hemicontinuous with respect to the solution variable and the operator is nonautonomous linear, multivalued and/or nonlinear or autonomous semilinear. One considering the Cauchy problem, a basic question about whether there exists solution set carrying Aronszajn-type regularity remains unsolved when the evolution family (or nonlinear semigroup) generated by the operator lacks the compactness. One of our main attention is paid to settling this question in the affirmative and providing a feasible roadmap for how such a result could be obtained. Moreover, we prove that  the solution map, having nonempty and compact values, is an $R_{\delta}$-map, which sends any connected set into a connected set. These geometric features of the solution map are then used to deal with the existence in the large for the corresponding nonlocal Cauchy problem. Finally, several examples are worked out in detail, illustrating the applicability of our abstract results.

简 介：

王荣年, 博士, 上海师范大学教授, 博士生导师（应用数学）。目前主要从事非线性发展方程适定性、多值扰动及解集的拓扑正则性、不变流形理论等问题的研究, 完成的研究结果已被"Mathematische Annalen"、“Int. Math. Res. Notices、"Journal of Functional Analysis"、"Journal of Differential Equations""J. Phys. A: Math. Theo."等学术期刊发表. 主持承担了2项国家自然科学基金面上项目、国家自然科学基金青年项目、4项省自然科学基金项目和2项省教育厅基金项目。曾获聘广东省高等学校“千百十人才工程”省级培养对象、江西省高校中青年骨干教师等。