应用数学系学术报告

--- 分析、偏微分方程与动力系统讨论班(2024秋季第10讲)

On dynamics of the semilinear wave equations outside the unit ball (II)

杨建伟 (北京理工大学)

时间：11月22日(周五)下午 15:00-16:00

地点：沙河主楼E502

摘要: We study the longtime dynamics of the solutions of semilinear wave equations with an energy supercritical growth. The theory of nonlinear wave equations has attacted intense study in the latest three decades.  The equation in question is a typical model of infinite dimensional dynamical system so that plenty of wave phenomena arising from strong physical background are predictable by studying this model. The ongoing research in the academic community focus on the energy critical case, culminating the celebrated soliton resolution theory  of Duyckaerts Kenig Merle (DKM). Concerning the energy supercritical case, very few results are known and many things to be explored. In this talk, we shall investigate its natural extension on the exterior domain, with a complerely new structure compared with the whole space. We give a rather precise description on the dynamics in a neighborhood of stationary solutions. We clarify the use of DKM's  channel of energy method, and present some recent results on the global centerstable manifold theory, highlighting the one-pass theorem, as well as the instability  of blowup solutions. New blowup dynamics are first obtained for excited states in this model whereas it’s out of reach by the current method for the standard model on the whole space. This work is joint with Thomas Duyckaerts (LAGA, Paris Sorbonne Nord).  

报告人简介: 杨建伟，北京理工大学数学科学学院特别副研究员，研究方向为调和分析与偏微分方程。2015年博士毕业于中国工程物理研究院。后在北京大学数学中心从事博士后研究，导师为田刚院士。2016-2017 年于法国巴黎北部大学伽利略研究所从事博士后研究，导师为Thomas Duyckaerts 教授，2019 年获得cergy-pontoise 大学Labex 项目，于LAGA 从事学术访问工作。2018年入职北京理工大学。杨建伟与合作者在Kakeya问题，拉普拉斯的特征函数估计问题等调和分析热点问题中已做出多项重要成果，在非线性波动方程的动力学行为等国际前沿课题做出重要工作，相关成果发表在Analysis & PDE，Annales de l’institut Fourier, Journal of Functional Analysis, Communications in Contemporary Mathematics 等国际期刊上。

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