报告人：吴启亮（俄亥俄大学）
邀请人：李骥
报告时间：2023年7月1日（星期六）10:00-11:30
2023年7月2日（星期日）10:00-11:30
2023年7月4日（星期二）10:00-11:30
2023年7月6日（星期四）10:00-11:30
报告地点：科技楼南楼715室
报告题目：Part I: Existence and Bifurcation Analysis of Periodic Patterns
Part II: Spectral Analysis of Periodic Patterns
Part III: Nonlinear Stability of Periodic Patterns
Part IV: Nonlinear Stability of Periodic Patterns under Spectral Degeneracies
报告摘要：The Lyapunov stability of equilibria in dynamical systems is determined by the interplay between the linearization and nonlinear terms. In this talk series, we firstly go through the existence and bifurcation analysis of periodic patterns; spectral analysis of periodic patterns, and classical results in nonlinear stability of periodic patterns. At last, we present our recent results on the case when the spectrum of the linearization is diffusively stable with high-order spectral degeneracy at the origin. Roll solutions at the zigzag boundary of the Swift-Hohenberg equation are shown to be nonlinearly stable, serving as examples that linear decays weaker than the classical diffusive decay, together with quadratic nonlinearity, still give nonlinear stability of spatially periodic patterns. The study is conducted on two physical domains: the 2D plane and the infinite 2D torus. Linear analysis reveals that, instead of the classical $t^{-1}$ diffusive decay rate, small perturbations of zigzag stable roll solutions decay with slower algebraic rates ($t^{-3/4}$ for the 2D plane; $t^{-1/4}$ for the infinite 2D torus) due to the high-order degeneracy of the translational mode at the origin in the Bloch-Fourier spaces. The nonlinear stability proofs are based on decompositions of the neutral translational mode and the faster decaying modes, and fixed-point arguments, demonstrating the irrelevancy of the nonlinear terms.
报告人简介：吴启亮，现为美国俄亥俄大学教授，研究领域为非线性动力系统微分方程，生物数学。本科毕业于中国科技大学，2013 年于美国明尼苏达大学获博士学位。后于密歇根州立大学作博士后研究。其研究获美国国家自然科学基金资助。在JDE，JMPA，JMB，PRSE等国际权威杂志发表论文数十篇。
