Kimie Nakashima

Multi-layered stationary solutions for a spatially inhomogeneous Allen–Cahn equation

Abstract We consider stationary solutions of a spatially inhomogeneous Allen–Cahn-type nonlinear diffusion equation in one space dimension. The equation involves a small parameter e , and its nonlinearity has the form h ( x ) 2 f ( u ), where h ( x ) represents the spatial inhomogeneity and f ( u ) is derived from a double-well potential with equal well-depth. When e is very small, stationary solutions develop transition layers. We first show that those transition layers can appear only near the local minimum and local maximum points of the coefficient h ( x ) and that at most a single layer can appear near each local minimum point of h ( x ). We then discuss the stability of layered stationary solutions and prove that the Morse index of a solution coincides with the total number of its layers that appear near the local maximum points of h ( x ). We also show the existence of stationary solutions having clustering layers at the local maximum points of h ( x ).

Singular Limit of a Spatially Inhomogeneous Lotka–Volterra Competition–Diffusion System

We discuss the generation and the motion of internal layers for a Lotka-Volterra competition-diffusion system with spatially inhomogeneous coefficients. We assume that the corresponding ODE system has two stable equilibria (ū, 0) and (0, ) with equal strength of attraction in the sense to be specified later. The equation involves a small parameter ϵ, which reflects the fact that the diffusion is very small compared with the reaction terms. When the parameter ϵ is very small, the solution develops a clear transition layer between the region where the u species is dominant and the one where the v species is dominant. As ϵ tends to zero, the transition layer becomes a sharp interface, whose motion is subject to a certain law of motion, which is called the “interface equation”. A formal asymptotic analysis suggests that the interface equation is the motion by mean curvature coupled with a drift term. We will establish a rigorous mathematical theory both for the formation of internal layers at the initial stage and for the motion of those layers in the later stage. More precisely, we will show that, given virtually arbitrary smooth initial data, the solution develops an internal layer within the time scale of O(ϵ2logϵ) and that the width of the layer is roughly of O(ϵ). We will then prove that the motion of the layer converges to the formal interface equation as ϵ → 0. Our results also give an optimal convergence rate, which has not been known even for spatially homogeneous problems.

Non-local effects in an integro-PDE model from population genetics

In this paper, we study the following non-local problem: \begin{equation*} \begin{cases} \displaystyle u_t=d{1\over\rho}\nabla\cdot(\rho V\nabla u)+b(\bar{u}-u)+ g(x) u^2(1-u) &\displaystyle \quad \textrm{in} \; \Omega\times (0,\infty),\\[3pt] \displaystyle 0\leq u\leq 1 & \quad\displaystyle \textrm{in}\ \Omega\times (0,\infty),\\[3pt] \displaystyle \nu \cdot V\nabla u=0 &\displaystyle \quad \textrm{on} \; \partial\Omega\times (0,\infty).\vspace*{-2pt} \end{cases} \end{equation*} This model, proposed by T. Nagylaki, describes the evolution of two alleles under the joint action of selection, migration, and partial panmixia – a non-local term , for the complete dominance case, where g ( x ) is assumed to change sign at least once to reflect the diversity of the environment. First, properties for general non-local problems are studied. Then, existence of non-trivial steady states, in terms of the diffusion coefficient d and the partial panmixia rate b , is obtained under different signs of the integral ∫ Ω g ( x ) dx . Furthermore, stability and instability properties for non-trivial steady states, as well as the trivial steady states u ≡ 0 and u ≡ 1 are investigated. Our results illustrate how the non-local term – namely, the partial panmixia – helps the migration in this model.