Applied dynamical systems and asymptotic analysis

本文的研究是应用动力系统和渐近分析两个领域的结合，以及它们之间的相互作用。未来五年的主要研究方向包括行波和速度选择的研究，表面上的模式形成（尖峰、激波和斑点模式）和渐近分析（如奇异摄动和边界层理论）。研究范围和目标是：1)分析反应扩散模型的波传播模式和波速选择力学。最近对空间扩散和时间演化联合效应的研究取得了一些有趣的反应扩散方程。这些模型在物理、生物和化学科学中得到了广泛的研究。对这些流行系统的非线性动力学（例如行波前，它们在有界域中的稳态稳定性）的定性和定量分析是我未来研究的主要目的。特别地，我们对行波模式的存在和波速选择力学感兴趣。2)表面反应扩散方程模拟的斑图形成和亚稳运动分析。在各种物理、生物和化学模型中，模式形成和指数慢内层运动（亚稳态运动）的表征将是我未来研究的另一个主题。我的目标是对表面上表现出非反应扩散系统的斑点层溶液的形成和运动进行定性和定量分析。3)曲面上奇异摄动偏微分方程解的分析与逼近。我将研究曲面上奇异摄动偏微分方程边值问题的近似解。给出了近似解的误差范围，为以往文献中的一些形式分析提供了严格的论证。

The research in this proposal is a combination of two fields:applied dynamical systems and asymptotic analysis, as well as theinteraction between them. Main research topics in future five years include the study of traveling waves and speed selection, pattern formation(spike, shock and spotpatterns) on surfaces and asymptotic analysis (e.g., singular perturbation and boundary layer theory). The research scope and objectives are: 1) Analysis of wave-propagation patterns and wave-speed selection mechanics forreaction diffusion models. The recent progresstowards the combined effect of spatial diffusion and temporalevolution leads to some interesting reaction diffusion equations. These models have been extensively studied in physical, biological and chemicalsciences. The qualitative andquantitative analysis of nonlinear dynamics (e.g. travelingwavefronts, stability of their steady state in a bounded domain)of these popular systems are the main purposeof my research in the future. In particular, we are interested in the existence of traveling wave-patterns and the wave-speed selection mechanics.2) Analysis of pattern formations and metastable-motion of spot patternsmodeled by reaction diffusion equations on surfaces. Thecharacterization of pattern formations and exponentially slow internal layer motion(metastable-motion) in various physical, biological and chemicalmodels will be another subject of my future research. My goal isto give qualitative and quantitative analysis of formation and movement of spot layer solutions that exhibit inreaction-diffusion systems on surfaces.3) Analysis and approximation of solutions to singularly perturbedpartial differential equations on surfaces. I will study the approximation of solutions to boundary value problems for singularly perturbedpartial differential equations on surfaces. Error bounds of the approximate solutions will be provided and this will give rigorous arguments to some formal analysis in past references.