中科院数学与系统科学研究院

数学研究所

非线性分析研讨班

时  间：2021.12.07（星期二）下午

地  点：腾讯会议：281 782 379

n  下午14：00-15：00

报告人：唐春雷 教授（西南大学）

题  目：Ground state solutions of Schrödinger equations with Hardy potential

摘  要：In this talk, the aim is to present the state of the articles around the mathematical study of Schrodinger equation and its variants. Depending on an external potential, the results will be split into two main parts, namely constant potential and non-constant potential (Singular potential and non-singular potential). We also report our recent works about the existence of ground state solutions to Schrodinger equations with Hardy potential.

n  下午15：00-16：00

报告人：艾尼·吾甫尔 教授 （新疆大学）

题  目：可靠性模型的动态分析

摘  要：介绍我们在可靠性模型的动态分析方面的研究成果。首先介绍问题的来源，然后以两个同型部件和一个修理设备（修理工）构成的可修复系统为例介绍我们在可靠性模型的适定性，解的渐近行为等方面取得的研究成果。第一步介绍两个同型部件和一个修理设备构成的系统，然后用补充变量方法建立描述该系统的数学模型，该模型是带积分边界条件的、由一阶的偏微分积分方程组描述。其次，通过选择状态空间、主算子及其定义域将该模型转化成 Banach 空间中的抽象 Cauchy 问题，第三步证明该模型的主算子生成正压缩 C0-半群，由此推出该模型存在唯一的概率瞬态解。第四步，证明该模型的主算子生成的算子半群是拟紧算子，0 是该主算子及其共轭算子的几何重数为1的特征值，从而推出该C0-半群指数收敛于一个投影算子。最后运用算子形式的残数定理求出该投影算子的表达式，由此推出该模型的时间依赖解指数收敛于该模型的稳态解。

n  下午16：00-17：00

报告人：缑天祥 博士（西安交通大学）

题 目：Higher topological type semiclassical states for fractional nonlinear elliptic equations

摘  要：In this talk, we are concerned with semiclassical states to fractional nonlinear elliptic equations in the Sobolev subcritical, critical and supercritical cases. We construct positive semiclassical states and an infinite sequence of sign-changing semiclassical states with higher energies clustering near the local minimum points of the potential. The solutions are of higher topological type, which are obtained from a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem. They correspond to critical points of the underlying energy functional at energy levels where compactness condition breaks down. The proofs are mainly based on penalization methods, s-harmonic extension theories and blow-up arguments along with local type Pohozaev identities. This is a joint work with. Prof. Shaowei Chen.