Wei Ming Ni

Symmetry and related properties via the maximum principle

We prove symmetry, and some related properties, of positive solutions of second order elliptic equations. Our methods employ various forms of the maximum principle, and a device of moving parallel planes to a critical position, and then showing that the solution is symmetric about the limiting plane. We treat solutions in bounded domains and in the entire space.

Semilinear elliptic equations of Matukuma-type and related topics

We investigate the structure of solutions of some semilinear elliptic equations, which include Matukuma’s equation as a special case. It is a mathematical model proposed by Matukuma, an astrophysicist, in 1930 to describe the dynamics of a globular cluster of stars. Equations of this kind have come up both in geometry and in physics, and have been a subject of extensive studies for some time. However, almost all the methods previously developed do not seem to apply to the original Matukuma’s equation. Our results cover most of the cases left open by previous works.

Turing patterns in the Lengyel-Epstein system for the CIMA reaction

The first experimental evidence of the Turing pattern was observed by De Kepper and her associates (1990) on the CIMA reaction in an open unstirred gel reactor, almost 40 years after Turings prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. In this paper we report some fundamental analytic properties of the Lengyel-Epstein system. Our result also indicates that if either of the initial concentrations of the reactants, the size of the reactor, or the effective diffusion rate, are not large enough, then the system does not admit nonconstant steady states. A priori estimates are fundamental to our approach for this nonexistence result. The degree theory was combined with the a priori estimates to derive existence of nonconstant steady states.

ALGORITHMS AND VISUALIZATION FOR SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS

In this paper, we compute and visualize solutions of several major types of semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in 2D. We present the mountain–pass algorithm (MPA), the scaling iterative algorithm (SIA), the monotone iteration and the direct iteration algorithms (MIA and DIA). Semilinear elliptic equations are well known to be rich in their multiplicity of solutions. Many such physically significant solutions are also known to lack stability and, thus, are elusive to capture numerically. We will compute and visualize the profiles of such multiple solutions, thereby exhibiting the geometrical effects of the domains on the multiplicity. Special emphasis is placed on SIA and MPA, by which multiple unstable solutions are computed. The domains include the disk, symmetric or nonsymmetric annuli, dumbbells, and dumbbells with cavities. The nonlinear partial differential equations include the Lane–Emden equation, Chandrasekhars equation, Henons equation, a singularly perturbed equation, and equations with sublinear growth. Relevant numerical data of solutions are listed as possible benchmarks for other researchers. Commentaries from the existing literature concerning solution behavior will be made, wherever appropriate. Some further theoretical properties of the solutions obtained from visualization will also be presented.

Uniqueness of solutions of nonlinear Dirichlet problems

Abstract In this paper, we consider the uniqueness of radial solutions of the nonlinear Dirichlet problem Δu + ƒ(u) = 0 in Ω with u = 0 on ∂Ω, where Δ = ∑ i = 1 n ∂ 2 ∂x i 2 ,ƒ satisfies some appropriate conditions and Ω is a bounded smooth domain in R n which possesses radial symmetry. Our uniqueness results apply to, for instance, ƒ(u) = u p , p > 1 , or more generally λu + ∑i = 1k aiupi, λ ⩾ 0, ai > 0 and pi > 1 with appropriate upper bounds, and Ω a ball or an annulus.

On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type

On deduit des estimations a priori pour des solutions positives du probleme de Neumann pour des systemes elliptiques semilineaires ainsi que pour des equations isolees semilineaires reliees a ces systemes

Singular behavior in nonlinear parabolic equations

In this paper, we study the well-posedness of the initial-boundary value problems of some quasilinear parabolic equations, namely, nonlinear heat equations and the porous medium equation in the fast-diffusion case. We establish nonunique- ness (local in time) and/or nonregularizing effect of these equations in some critical cases. The key which leads to the resolution of these problems is to study some singular solutions of the elliptic counterparts of these parabolic problems (the so-called M-solutions of the Lane-Emden equations in astrophysics). Introduction. In this article we are concerned first of all with the existence or nonexistence of singular solutions of certain semilinear elliptic boundary value problems, and secondly with the application of these results to the study of some related parabolic problems. To begin with, consider the problem

Chapter 3 Qualitative properties of solutions to elliptic problems

Qualitative properties of solutions to elliptic equations can be interpreted in an extremely broad sense to include virtually every property of solutions. This chapter discusses concrete and/or geometric properties of solutions. In particular, it emphasizes the two properties of solutions: the shape of solutions and the stability of solutions. Boundary conditions clearly play important roles in the qualitative behavior of solutions. One feature of this survey is the inclusion of comparisons of the different, sometimes opposite effects of Dirichlet and Neumann boundary conditions, whenever possible. Qualitative properties of solutions are closely related to the existence of solution. In fact, existence of solutions provides the basis for the study of qualitative properties. On the other hand, searching for solutions with particular properties in mind could provide clues for existence.

Point condensation generated by a reaction-diffusion system in axially symmetric domains

In this paper we consider the stationary problem for a reaction-diffusion system of activator-inhibitor type, which models biological pattern formation, in an axially symmetric domain. It is shown that the system has multi-peak stationary solutions such that the activator is localized around some boundary points if the activator diffuses very slowly and the inhibitor diffuses rapidly enough.

The Mathematics of Diffusion

Though it incorporates much new material, this new edition preserves the general character of the book in providing a collection of solutions of the equations of diffusion and describing how these solutions may be obtained.