Izumi Takagi

Point-condensation for a reaction-diffusion system

We consider stationary solutions of a reaction-diffusion system for an activator and an inhibitor. Let d1 and d2 be the respective diffusion coefficients of the activator and the inhibitor. Assuming that d2 is sufficiently large, we construct stationary solutions which exhibit spiky patterns when d1 is near zero. Moreover, we study the global (in d1) structure of the solution set and show that if d2 is sufficiently large then (a) whenever bifurcation from the constant solution occurs, there exists a continuum of nonconstant solutions which connects the point-condensation solutions with the bifurcating solutions; and (b) when no bifurcation from the constant solution occurs, the point-condensation solutions are connected to another family of point-condensation solutions.

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

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.

On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem: Intermediate solutions

where A = cy=, 5 is the Laplace operator, R is a bounded smooth domain in R”, E > 0 is a constant, and the exponent p satisfies 1 < p < 2 for n 2 3 and 1 < p < 0;) for n = 2. We are especially interested in the properties of solutions of (1.1) as E tends to 0. In particular, we shall establish the existence of a “spike-layer’’ solution, and determine the location of the peak as well as the profile of the spike.

Stability of least energy patterns of the shadow system for an activator-inhibitor model

Stability of stationary solutions to the shadow system for the activator-inhibitor system proposed by Gierer and Meinhardt is considered in higher dimensional domains. It is shown that a stationary solution with minimal “energy” is stable in a weak sense if the inhibitor reacts sufficiently fast, while it is unstable whenever the reaction of the inhibitor is slow. Moreover, the loss of stability results in a Hopf bifurcation.

On the Existence and Shape of Solutions to a Semilinear Neumann Problem

In this article we shall review some recent progress in the study of the Neumann problem for a semilinear elliptic equation. Let Ω be a bounded domain in R N , N ≥ 2, with smooth boundary ∂Ω and let v denote the unit outer normal to ∂Ω. We consider the Neumann problem in which is the Laplace operator, d is a positive constant and throughout the article we assume that unless it is explicitly stated otherwise. (Most of the results below do generalize to a certain class of functions f including t p , and the reader is referred to the original papers cited.)

Representation formula for the critical points of the Tadjbakhsh-Odeh functional and its application

In order to study the buckled states of an elastic ring under uniform pressure, Tadjbakhsh and Odeh [14] introduced an energy functional which is a linear combination of the total squared curvature (elastic energy) and the area enclosed by the ring. We prove that the minimizer of the functional is not a disk when the pressure is large, and its curvature can be expressed by Jacobian elliptic cn(·) function. Moreover, the uniqueness of the minimizer is proven for certain range of the pressure.

Spike-layers in semilinear elliptic singular Perturbation Problems

The purpose of this expository paper is to describe a new method, introduced in a series of papers [LNT], [NT1,2], [NPT] and [J], in handling “spikes” (or “point-condensation” phenomena) for singularly perturbed semilinear elliptic equations of the form (1) where \( \Delta = \sum\limits_{i = 1}^n {\frac{{{\partial ^2}}}{{\partial x_i^2}}} \) is the Laplace operator in R n , and e is a small positive number.