Yoshihisa Morita

An Entire Solution to the Lotka–Volterra Competition-Diffusion Equations

We deal with a system of Lotka–Volterra competition-diffusion equations on

Ginzburg-Landau equations and stable solutions in a rotational domain

\mathbb{R}

Destabilization of periodic solutions arising in delay-diffusion systems in several space dimensions

, which is a competing two species model with diffusion. It is known that the equations allow traveling waves with monotone profile. In this article we prove the existence of an entire solution which behaves as two monotone waves propagating from both sides of the x-axis, where an entire solution is meant by a classical solution defined for all space and time variables. The global dynamics for this entire solution exhibits the extinction of the inferior species by the superior one invading from both sides. The proof is carried out by applying the comparison principle for the competition-diffusion equations, that is, using an appropriate pair of a subsolution and a supersolution.

Ginzburg-Landau equation and stable steady state solutions in a non-trivial domain

The Ginzburg–Landau (GL) equations, with or without magnetic effect, are studied in the case of a rotational domain in

Stability and bifurcation of nonconstant solutions to a reaction–diffusion system with conservation of mass

\mathbb{R}^3

Reaction-diffusion systems in nonconvex domains: Invariant manifold and reduced form

. It can be shown that there exist rotational solutions which describe the physical state of permanent current of electrons in a ring-shaped superconductor. Moreover, if a physical parameter—called the GL parameter—is sufficiently large, then these solutions are stable, that is, they are local minimizers of an energy functional (GL energy). This is proved by the spectral analysis on the linearized equation.

Stabilization of vortices in the Ginzburg-Landau equation with a variable diffusion coefficient

We consider a diffusion equation with time delay having a stable spatially homogeneous periodic solution bifurcating from a steady state. We show that under certain circumstances the bifurcating periodic solution loses its stability very near the bifurcation point if the diffusion coefficients are sufficiently small. Such a destabilization phenomenon also occurs when in place of the diffusion coefficients, the shape of the domain is varied instead. Sufficient conditions for the occurrence of such phenomena, along with some specific examples, will be presented.

Dynamics on the global attractor of a gradient flow arising from the Ginzburg-Landau equation

The Ginzburg-Landau equation with a large parameter is studied in a bounded domain with the Neumann B.C. It is shown that many kinds of stable non-constant solutions exist in domains with some topological condition. If the space dimension is 2 or 3 and if the domain is not simply connected, this condition holds. 25 refs.

Ordinary differential equations (ODEs) on inertial manifolds for reaction-diffusion systems in a singularly perturbed domain with several thin channels

We deal with a two-component system of reaction–diffusion equations with conservation of mass in a bounded domain with the Neumann or periodic boundary conditions. This system is proposed as a conceptual model for cell polarity. Since the system has conservation of mass, the steady state problem is reduced to that of a scalar reaction–diffusion equation with a nonlocal term. That is, there is a one-to-one correspondence between an equilibrium solution of the system with a fixed mass and a solution of the scalar equation. In particular, we consider the case when the reaction term is linear in one variable. Then the equations are transformed into the same equations as the phase-field model for solidification. We thereby show that the equations allow a Lyapunov function. Moreover, by investigating the linearized stability of a nonconstant equilibrium solution, we prove that given a nondegenerate stable equilibrium solution of the nonlocal scalar equation, the corresponding equilibrium solution of the system is stable. We also exhibit global bifurcation diagrams for equilibrium solutions to specific model equations by numerics together with a normal form near a bifurcation point.

Collision and collapse of layers in a 1D scalar reaction-diffusion equation

A study is made of systems of weakly coupled, semilinear, parabolic equations, namely reaction-diffusion systems, subject to the homogeneous Neumann boundary conditions in parametrized nonconvex domains inR2. It is assumed that the domain approaches a union of two disjoint domains as the parameter varies. Under some conditions the long-time behavior of bounded solutions is discussed and the existence of a finite-dimensional invariant manifold is shown, together with its attractivity. This manifold is represented by a graph of some function defined in a possibly large bounded region of the phase space, and the original system is reduced to an ODE system on it. Since an explicit form of the reduced ODE system is given, its dynamics can be studied in detail, which in turn reveals the global dynamics of the original reaction-diffusion system. One can thereby prove, among other things, the existence of asymptotically stable equilibrium solutions of the original system having large spatial inhomogeneity. The existence and stability of a spatially inhomogeneous periodic solution of large amplitude are also discussed.