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Ryunosuke Mori

Publications and source records attributed to Ryunosuke Mori.

5 recordsLinked to original sources
arXiv2024

The long-time behavior of solutions of a three-component reaction-diffusion model for the population dynamics of farmers and hunter-gatherers: the different motility case

In this paper, we investigate the spreading properties of solutions of the Aoki-Shida-Shigesada model. This model is a three-component reaction-diffusion system that delineates the geographical expansion of an initially localized population of farmers into a region occupied by hunter-gatherers. By considering the scenario where farmers and hunter-gatherers possess identical motility, Aoki et al. previously concluded, through numerical simulations and some formal linearization arguments, that there are four different types of spreading behaviors depending on the parameter values. In this paper, we concentrate on the general case for which farmers and hunter-gatherers possess different motility. By providing more sophisticated estimates, we not only theoretically justify the spreading speed of the Aoki-Shida-Shigesada model, but also establish sharp estimates for the long-time behaviors of solutions. These estimates enable us to validate all four types of spreading behaviors observed by Aoki et al..

math.AP↗
arXiv2021

Brakke's formulation of velocity and the second order regularity property

Suppose that a family of $k$-dimensional surfaces in $\mathbb R^n$ evolves by the motion law of $v=h+u^\perp$ in the sense of Brakke's formulation of velocity, where $v$ is the normal velocity vector, $h$ is the generalized mean curvature vector and $u^\perp$ is the normal projection of a given vector field $u$ in a dimensionally sharp integrability class. When the flow is locally close to a time-independent $k$-dimensional plane in a weak sense of measure in space-time, it is represented as a graph of a $C^{1,\alpha}$ function over the plane. On the other hand, it is not known if the graph satisfies the PDE of $v=h+u^\perp$ pointwise in general. For this problem, when $k=n-1$ and under the additional assumption that the distributional time derivative of the graph is a signed Radon measure, it is proved that the graph satisfies the PDE pointwise. An application to a short-time existence theorem for a surface evolution problem is given.

math.AP↗
arXiv2017

Validity of formal expansions for singularly perturbed competition-diffusion systems

We consider a two-species competition-diffusion system involving a small parameter $\varepsilon>0$ and discuss the validity of formal asymptotic expansions of solutions near the sharp interface limit $\varepsilon\approx0$. We assume that the corresponding ODE system has two stable equilibria. As in the scalar Allen--Cahn equation, it is known that the motion of the sharp interfaces of such systems is governed by the mean curvature flow with a driving force. The formal expansion also suggests that the profile of the transition layers converges to that of a traveling wave solution as $\varepsilon\rightarrow0$. In this paper, we rigorously verify this latter ansatz for a large class of initial data. The proof relies on a rescaling argument, the super--subsolution method and a Liouville type theorem for eternal solutions of parabolic systems. Roughly speaking, the Liouville type theorem states that any eternal solution that lies between two traveling waves is itself a traveling wave. The same Liouville type theorem was established for the scalar Allen--Cahn equation by Berestycki and Hamel. In view of their importance, we prove the Liouville type theorems in a rather general framework, not only for two-species competition-diffusion systems but also for $m$-species cooperation-diffusion systems possibly with time periodic or spatially periodic coefficients.

math.AP↗
arXiv2017

A variational problem associated with the minimal speed of traveling waves for spatially periodic KPP type equations

We consider a variational problem associated with the minimal speed of pulsating traveling waves of the equation $u_t=u_{xx}+b(x)(1-u)u$, $x\in{\mathbb R},\ t>0$, where the coefficient $b(x)$ is nonnegative and periodic in $x\in{\mathbb R}$ with a period $L>0$. It is known that there exists a quantity $c^*(b)>0$ such that a pulsating traveling wave with the average speed $c>0$ exists if and only if $c\geq c^*(b)$. The quantity $c^*(b)$ is the so-called minimal speed of pulsating traveling waves. In this paper, we study the problem of maximizing $c^*(b)$ by varying the coefficient $b(x)$ under some constraints. We prove the existence of the maximizer under a certain assumption of the constraint and derive the Euler--Lagrange equation which the maximizer satisfies under $L^2$ constraint $\int_0^L b(x)^2dx=β$. The limit problems of the solution of this Euler--Lagrange equation as $L\rightarrow0$ and as $β\rightarrow0$ are also considered. Moreover, we also consider the variational problem in a certain class of step functions under $L^p$ constraint $\int_0^L b(x)^pdx=β$ when $L$ or $β$ tends to infinity.

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