Mathematical Sciences: Qualitative Behavior of Solutions of Elliptic and Parabolic Partial Differential Equations.

这项工作追求两个基本问题， 椭圆和抛物偏微分方程的数学分析 非线性微分方程 一是关于 退化抛物反应扩散方程的动力学 这些方程出现在种群动力学模型中， 扩散取决于密度。 这里的主要目标是 来构造和分类平衡点之间的连接轨道。 DaPrato和Grisvard的抽象抛物方程理论 将用于这项工作。 第二个目标是确定 当应用线性化稳定性原理时， 失败时的替换原则。 的应用 所提出的方法已成功地在过去的工作有关 退化抛物方程 另一个主题， 源自微分几何，涉及一个 曲面上的曲线由曲率向量（ 曲线）。 抛物型偏微分方程表示 将沿着曲线的流动与以下乘积相连接的运动 法向和测地曲率向量。 鉴别 在几何学中，这个方程形式上作为梯度流出现， 长度功能。 微分方程的平衡点 是曲面上的测地线 关于nodal的早期结果 线性抛物方程解的性质将是 用于试图阐明交叉点的性质， 曲率缩短方程的解。 与此相关 将研究测地线性质的交点 在相同的同伦类中。 这个问题的变化 出现在冶金学的枝晶形成理论中。

This work pursues two fundamental problems in the mathematical analysis of elliptic and parabolic partial differential equations of nonlinear type. The first concerns the dynamics of degenerate parabolic reaction diffusion equations. These equations occur in models of population dynamics with diffusion depending on the density. The main objective here is to construct and classify connecting orbits between equilibria. A theory of abstract parabolic equations of DaPrato and Grisvard will be used in this effort. A second objective is to determine when the principle of linearized stability applies and to seek replacement principles when it fails. Application of the proposed method has worked successfully in past work on related degenerate parabolic equations. The other main theme, which derives from differential geometry, concerns the motion of a curve on a surface governed by the curvature vector (of the curve). A parabolic partial differential equation represents the motion connecting the flow along the curve with the product of the normal and the geodesic curvature vectors. In differential geometry, this equation arises formally as the gradient flow of the length function. The equilibria of the differential equation are the geodesics on the surface. Earlier results on nodal properties of solutions of linear parabolic equations will be used in attempts to shed light on the intersection properties of solutions of the curvature shortening equation. Related to this will be the study of intersections of properties of geodesics within the same homotopy classes. Variations of this problem occur in the theory of dendrite formation in metallurgy.