Hiroshi Matano - Academia.edu (original) (raw)

Papers by Hiroshi Matano

arXiv (Cornell University), Jul 3, 2019

arXiv (Cornell University), Apr 5, 2010

arXiv (Cornell University), Jul 11, 2018

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arXiv (Cornell University), Nov 2, 2017

arXiv: Analysis of PDEs, 2018

We study an e-dependent stochastic Allen-Cahn equation with a mild random noise on a bounded doma... more We study an e-dependent stochastic Allen-Cahn equation with a mild random noise on a bounded domain in R n , n ≥ 2. Here e is a small positive parameter that represents formally the thickness of the solution interface, while the mild noise ξ e (t) is a smooth random function of t of order O(e −γ) with 0 < γ < 1/3 that converges to white noise as e → 0 +. We consider initial data that are independent of e satisfying some non-degeneracy conditions, and prove that steep transition layers-or interfaces-develop within a very short time of order e 2 | ln e|, which we call the "generation of interface". Next we study the motion of those transition layers and derive a stochastic motion law for the sharp interface limit as e → 0 +. Furthermore, we prove that the thickness of the interface for e small is indeed of order O(e) and that the solution profile near the interface remains close to that of a (squeezed) travelling wave; this means that the presence of the noise does not...

In this paper we investigate the dynamical properties of a spatially periodic reaction-diffusion ... more In this paper we investigate the dynamical properties of a spatially periodic reaction-diffusion system whose reaction terms are of hybrid nature in the sense that they are partly competitive and partly cooperative depending on the value of the solution. This class of problems includes various biologically relevant models and in particular many models focusing on the Darwinian evolution of species. We start by studying the principal eigenvalue of the associated differential operator and establishing a minimal speed formula for linear monotone systems. In particular, we show that the generalized Dirichlet principal eigenvalue and the periodic principal eigenvalue may not coincide when the reaction matrix is not symmetric, in sharp contrast with the case of scalar equations. We establish a sufficient condition under which equality holds for the two notions. We also show that the propagation speed may be different depending on the direction of propagation, even in the absence of a firs...

Journal de Mathématiques Pures et Appliquées, 2019

Calculus of Variations and Partial Differential Equations, 2019

Communications in Contemporary Mathematics, 2019

This work focuses on dynamics arising from reaction-diffusion equations, where the profile of pro... more This work focuses on dynamics arising from reaction-diffusion equations, where the profile of propagation is no longer characterized by a single front, but by a layer of several fronts which we call a propagating terrace. This means, intuitively, that transition from one equilibrium to another may occur in several steps, that is, successive phases between some intermediate stationary states. We establish a number of properties on such propagating terraces in a one-dimensional periodic environment under very wide and generic conditions. We are especially concerned with their existence, uniqueness, and their spatial structure. Our goal is to provide insight into the intricate dynamics arising from multistable nonlinearities.

Archive for Rational Mechanics and Analysis, 2015

Communications on Pure and Applied Mathematics, 2016

The bidomain model is the standard model describing electrical activity of the heart. Here we stu... more The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen‐Cahn equation) in two spatial dimensions. In the bidomain Allen‐Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen‐Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen‐Cahn equation in striking contrast to the classical or anisotropic Allen‐Cahn equations. We identify two types of instabilities, one with respect to long‐wavelength perturbations, the other...

Journal of Dynamics and Differential Equations, 2015

Journal of the European Mathematical Society, 2015

Transactions of the American Mathematical Society, 2010

Archive for Rational Mechanics and Analysis, 2014

Proceedings of the London Mathematical Society, 2014

Transactions of the American Mathematical Society, 2014

We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic ... more We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic nonlinearities – including multi-stable ones – and study the asymptotic behavior of solutions with Heaviside type initial data. Our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front, but by a layer of several fronts which we call a terrace. Existence and convergence to such a terrace is proven by using an intersection number argument, without much relying on standard linear analysis. Hence, on top of the peculiar phenomenon of propagation that our work highlights, several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before.

SIAM Journal on Mathematical Analysis, 2009

SIAM Journal on Mathematical Analysis, 2005

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2010

We consider the spatially inhomogeneous and anisotropic reaction–diffusion equation ut = m(x)−1 d... more We consider the spatially inhomogeneous and anisotropic reaction–diffusion equation ut = m(x)−1 div[m(x)ap(x,∇u)] + ε−2f(u), involving a small parameter ε > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. More precisely, we show that, within the time-scale of order ε2|ln ε|, the unique weak solution uε develops a steep transition layer that separates the regions {uε ≈ 0} and {uε | 1}. Then, on a much slower time-scale, the layer starts to propagate. Consequently, as ε → 0, the solution uε converges almost everywhere (a.e.) to 0 in Ω−t and 1 in Ω+t , where Ω−t and Ω+t are sub-domains of Ω separated by an interface Гt, whose motion is driven by its anisotropic mean curvature. We also prove that the thickness of the transition layer is of order ε.