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

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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.