Hirokazu Ninomiya

Entire solutions of reaction—diffusion equations with balanced bistable nonlinearities

This paper deals with entire solutions of a bistable reaction—diffusion equation for which the speed of the travelling wave connecting two constant stable equilibria is zero. Entire solutions which behave as two travelling fronts approaching, with super-slow speeds, from opposite directions and annihilating in a finite time are constructed by using a quasi-invariant manifold approach. Such solutions are shown to be unique up to space and time translations.

A competition-diffusion system approximation to the classical two-phase Stefan problem

A new type of competition-diffusion system with a small parameter is proposed. By singular limit analysis, it is shown that any solution of this system converges to the weak solution of the two-phase Stefan problem with reaction terms. This result exhibits the relation between an ecological population model and water-ice solidification problems.

Diffusion-Induced Blowup in a Nonlinear Parabolic System

A two-component semilinear parabolic system on a bounded domain with Neumann boundary conditions is studied. It is shown that for a certain kind of nonlinearity, the blowup of solutions may occur when the diffusion coefficients are not equal, though the corresponding ODE possesses a globally stable equilibrium.

Diffusion-induced extinction of a superior species in a competition system

This paper discusses the dynamics of bi-stable competition systems. It is assumed that in the initial state one species is numerically superior to the other species at every point of their habitat. If the two species do not migrate, then the superior species will wipe out the inferior one. It is shown, however, that the superior species may become extinct if the diffusion is taken into consideration.

Convergence and blow-up of solutions for a complex-valued heat equation with a quadratic nonlinearity

This paper is concerned with the Cauchy problem for a system of parabolic equations which is derived from a complex-valued equation with a quadratic nonlinearity. First we show that if the convex hull of the image of initial data does not intersect the positive real axis, then the solution exists globally in time and converges to the trivial steady state. Next, on the onedimensional space, we provide some solutions with nontrivial imaginary parts that blow up simultaneously. Finally, we consider the case of asymptotically constant initial data and show that, depending on the limit, the solution blows up nonsimultaneously at space infinity or exists globally in time and converges to the trivial steady state.

Stacked Fronts for Cooperative Systems with Equal Diffusion Coefficients

In this paper m-component cooperative systems with equal diffusion coefficients are considered. The kinetic system without diffusion possesses

A REACTION-DIFFUSION APPROXIMATION TO A CROSS-DIFFUSION SYSTEM

m+1

Reaction, diffusion and non-local interaction

equilibria where one is stable and the others, including the origin, are unstable. For the diffusion-cooperative system, one might think that the transition from the origin to the stable equilibrium causes the formation of traveling fronts as observed in the Fisher-KPP equation. However, this is not always true. It is shown that under certain conditions all components of the solutions of the cooperative system propagate at the same speed as seen in the Fisher-KPP equation, while under certain other conditions some components of the solutions do not develop into single traveling fronts. In these latter cases those components may develop into stacked fronts where each front propagates at a different speed than the others.

Chapter 2 – Fast Reaction Limit of Competition-Diffusion Systems

In this paper it is discussed whether reaction and linear diffusion bring about a effect of nonlinear diffusion or not. It is proved that a cross–diffusion system for two competitive species is realized in a singular limit of a reaction–diffusion system with a small parameter under some assumptions.

Traveling curved fronts of anisotropic curvature flows

Recent years have seen the introduction of non-local interactions in various fields. A typical example of a non-local interaction is where the convolution kernel incorporates short-range activation and long-range inhibition. This paper presents the relationship between non-local interactions and reaction–diffusion systems in the following sense: (a) the relationship between the instability induced by non-local interaction and diffusion-driven instability; (b) the realization of non-local interactions by reaction–diffusion systems. More precisely, it is shown that the non-local interaction of a Mexican-hat kernel destabilizes the stable homogeneous state and that this instability is related to diffusion-driven instability. Furthermore, a reaction–diffusion system that approximates the non-local interaction system with any even convolution kernel is shown to exist.