arXiv subjects

Hiroshi Matano

Publications and source records attributed to Hiroshi Matano.

At least 19 recordsLinked to original sources

arXiv2026

Front propagation in hybrid reaction-diffusion epidemic models with spatial heterogeneity. Part II: Pulsating traveling waves

We consider a two-species reaction-diffusion system in one space dimension that is derived from an epidemiological model in a spatially periodic environment with two types of pathogens: the wild type and the mutant. The system is of a hybrid nature, partly cooperative and partly competitive, but neither of these entirely. As a result, the comparison principle does not hold. In the previous work, we studied the propagation properties of the solutions to the Cauchy problem for this system and showed, among other things, that the spreading speeds of the fronts to the right and to the left directions, denoted by $ c^*_R$ and $ c^*_L$, can be characterized by using certain principal eigenvalues, and studied the homogenization limit as the spatial period $L$ tends to $0$, and also discussed the long-time behavior of solutions behind the fronts. In the present paper we prove the existence of pulsating traveling waves in the right direction (resp. left direction) with speed $c$ for any $c\geq c^*_R$ (resp. $c\geq c^*_L$), where $c^*_R$ and $c^*_L$ denote the aforementioned spreading speeds in the right and left directions. We also prove that the leading edge of any traveling wave has the exponential decay rate that is anticipated from formal linear analysis, thus extending part of the results of Hamel 2008 to systems of equations. Finally, we present an example in which the two speeds $c^*_R$ and $c^*_L$ are different. This is done by considering a multi-scale singular limit problem. This result highlights a marked difference between our system and scalar KPP type equations.

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arXiv2025

Front propagation on a general metric graph

We consider a bistable reaction-diffusion equation on a metric graph that is a generalization of the so-called star graphs. More precisely, our graph $\Omega$ consists of a bounded finite metric graph $D$ of arbitrary configuration and a finite number of branches $\Omega_1,\ldots,\Omega_N\,(N\geq 2)$ of infinite length emanating from some of the vertices of $D$. Each $\Omega_i\,(i=1,\ldots,N)$ is called an ``outer path''. Our goal is to investigate the behavior of the front coming from infinity along a given outer path $\Omega_i$ and to discuss whether or not the front propagates into other outer paths $\Omega_j\,(j\ne i)$. Unlike the case of star graphs, where $D$ is a single vertex, the dynamics of solutions can be far more complex and may depend sensitively on the configuration of the center graph $D$. We first focus on general principles that hold regardless of the structure of the center graph $D$. Among other things, we introduce the notion ``limit profile'', which allows us to define ``propagation'' and ``blocking'' without ambiguity, then we prove transient properties, that is, propagation $\Omega_i\to \Omega_j$ and $\Omega_j\to \Omega_k$ imply propagation $\Omega_i\to \Omega_k$. Next we consider perturbations of the graph $D$ while fixing the outer paths $\Omega_1,\ldots,\Omega_N$ and prove that if, for a given choice of $i,j$, propagation $\Omega_i\to \Omega_j$ occurs for a graph $D$, then the same holds for any graph $D'$ that is sufficiently close to $D$ (robustness under perturbation). We also consider several specific classes of graphs, such as those with a ``reservoir'' type subgraph, and study their intriguing properties.

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arXiv2024

Front propagation in hybrid reaction-diffusion epidemic models with spatial heterogeneity. Part I: Spreading speed and asymptotic behavior

We consider a two-species reaction-diffusion system in one space dimension that is derived from an epidemiological model in a spatially periodic environment with two types of pathogens: the wild type and the mutant. The system is of a hybrid nature, partly cooperative and partly competitive, but neither of these entirely. As a result, the comparison principle does not hold for the whole system. We study spreading properties of solution fronts when the infection is localized initially. We show that there is a well-defined spreading speed both in the right and left directions and that it can be computed from the linearized equation at the leading edge of the propagation front. Next we study the case where the coefficients are spatially homogeneous and show that, when spreading occurs, every solution to the Cauchy problem converges to the unique positive stationary solution as $t\to\infty$. Finally we consider the case of rapidly oscillating coefficients, that is, when the spatial period of the coefficients, denoted by $\varepsilon$, is very small. We show that there exists a unique positive stationary solution, and that every positive solution to the Cauchy problem converges to this stationary solution as $t\to\infty$. We then discuss the homogenization limit as $\varepsilon\to 0$.

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arXiv2024

Spreading properties of the Fisher--KPP equation when the intrinsic growth rate is maximal in a moving patch of bounded size

This paper is concerned with spreading properties of space-time heterogeneous Fisher--KPP equations in one space dimension. We focus on the case of everywhere favorable environment with three different zones, a left half-line with slow or intermediate growth, a central patch with fast growth and a right half-line with slow or intermediate growth. The central patch moves at various speeds. The behavior of the front changes drastically depending on the speed of the central patch. Among other things, intriguing phenomena such as nonlocal pulling and locking may occur, which would make the behavior of the front further complicated. The problem we discuss here is closely related to questions in biomathematical modelling. By considering several special cases, we illustrate the remarkable diversity of possible behaviors. In particular, when the central patch has constant size and constant speed, we provide a complete set of explicit formulas for the spreading speed.

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arXiv2024

Front propagation through a perforated wall

We consider a bistable reaction-diffusion equation $u_t=Δu +f(u)$ on $\mathbb{R}^N$ in the presence of an obstacle $K$, which is a wall of infinite span with many holes. More precisely, $K$ is a closed subset of $\mathbb{R}^N$ with smooth boundary such that its projection onto the $x_1$-axis is bounded and that $\mathbb{R}^N \setminus K$ is connected. Our goal is to study what happens when a planar traveling front coming from $x_1 = -\infty$ meets the wall $K$.We first show that there is clear dichotomy between "propagation" and "blocking". In other words, the traveling front either passes through the wall and propagates toward $x_1=+\infty$ (propagation) or is trapped around the wall (blocking), and that there is no intermediate behavior. This dichotomy holds for any type of walls of finite thickness. Next we discuss sufficient conditions for blocking and propagation. For blocking, assuming either that $K$ is periodic in $y:=(x_2,\ldots, x_N)$ or that the holes are localized within a bounded area, we show that blocking occurs if the holes are sufficiently narrow. For propagation, three different types of sufficient conditions for propagation will be presented, namely "walls with large holes", "small-capacity walls", and "parallel-blade walls". We also discuss complete and incomplete invasions.

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arXiv2022

Radial terrace solutions and propagation profile of multistable reaction-diffusion equations over $\mathbb R^N$

We study the propagation profile of the solution $u(x,t)$ to the nonlinear diffusion problem $u_t-Δu=f(u)\; (x\in \mathbb R^N,\;t>0)$, $u(x,0)=u_0(x) \; (x\in\mathbb R^N)$, where $f(u)$ is of multistable type: $f(0)=f(p)=0$, $f'(0)<0$, $f'(p)<0$, where $p$ is a positive constant, and $f$ may have finitely many nondegenerate zeros in the interval $(0, p)$. The class of initial functions $u_0$ includes in particular those which are nonnegative and decay to 0 at infinity. We show that, if $u(\cdot, t)$ converges to $p$ as $t\to\infty$ in $L^\infty_{loc}(\mathbb R^N)$, then the long-time dynamical behavior of $u$ is determined by the one dimensional propagating terraces introduced by Ducrot, Giletti and Matano [DGM]. For example, we will show that in such a case, in any given direction $ν\in\mathbb{S}^{N-1}$, $u(x\cdot ν, t)$ converges to a pair of one dimensional propagating terraces, one moving in the direction of $x\cdot ν>0$, and the other is its reflection moving in the opposite direction $x\cdotν<0$. Our approach relies on the introduction of the notion "radial terrace solution", by which we mean a special solution $V(|x|, t)$ of $V_t-ΔV=f(V)$ such that, as $t\to\infty$, $V(r,t)$ converges to the corresponding one dimensional propagating terrace of [DGM]. We show that such radial terrace solutions exist in our setting, and the general solution $u(x,t)$ can be well approximated by a suitablly shifted radial terrace solution $V(|x|, t)$. These will enable us to obtain better convergence result for $u(x,t)$. We stress that $u(x,t)$ is a high dimensional solution without any symmetry. Our results indicate that the one dimensional propagating terrace is a rather fundamental concept; it provides the basic structure and ingredients for the long-time profile of solutions in all space dimensions.

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arXiv2021

Propagation dynamics of solutions to spatially periodic reaction-diffusion systems with hybrid nonlinearity

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 first-order advection term, again in a sharp contrast with scalar equations. Next we reveal the relation between the hair-trigger property of front propagation and the sign of the periodic principal eigenvalue. Finally, we discuss the linear determinacy of the propagation speed and also establish the existence of travelling waves travelling whose speeds greater than the minimal speed is also proved. We apply our results to an important class of epidemiological models with genetic mutations.

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arXiv2021

Asymptotic behavior of fronts and pulses of the bidomain model

The bidomain model is the standard model for cardiac electrophysiology. In this paper, we investigate the instability and asymptotic behavior of planar fronts and planar pulses of the bidomain Allen-Cahn equation and the bidomain FitzHugh-Nagumo equation in two spatial dimension. In previous work, it was shown that planar fronts of the bidomain Allen-Cahn equation can become unstable in contrast to the classical Allen-Cahn equation. We find that, after the planar front is destabilized, a rotating zigzag front develops whose shape can be explained by simple geometric arguments using a suitable Frank diagram. We also show that the Hopf bifurcation through which the front becomes unstable can be either supercritical or subcritical, by demonstrating a parameter regime in which a stable planar front and zigzag front can coexist. In our computational studies of the bidomain FitzHugh-Nagumo pulse solution, we show that the pulses can also become unstable much like the bidomain Allen-Cahn fronts. However, unlike the bidomain Allen-Cahn case, the destabilized pulse does not necessarily develop into a zigzag pulse. For certain choice of parameters, the destabilized pulse can disintegrate entirely. These studies are made possible by the development of a numerical scheme that allows for the accurate computation of the bidomain equation in a two dimensional strip domain of infinite extent.

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arXiv2019

Dynamics of time-periodic reaction-diffusion equations with front-like initial data on $\mathbb{R}$

This paper is concerned with the Cauchy problem $$u_t=u_{xx} +f(t,u), \,\,\, x\in\mathbb{R},\,t>0, $$ $$u(0,x)= u_0(x), \,\,\, x\in\mathbb{R},$$ where $f$ is a rather general nonlinearity that is periodic in $t$, and satisfies $f(\cdot,0)\equiv 0$ and that the corresponding ODE has a positive periodic solution $p(t)$. Assuming that $u_0$ is front-like, that is, $u_0(x)$ is close to $p(0)$ for $x\approx -\infty$ and close to $0$ for $x\approx \infty$, we aim to determine the long-time dynamical behavior of the solution $u(t,x)$ by using the notion of propagation terrace introduced by Ducrot, Giletti and Matano (2014). We establish the existence and uniqueness of propagating terrace for a very large class of nonlinearities, and show the convergence of the solution $u(t,x)$ to the terrace as $t\to\infty$ under various conditions on $f$ or $u_0$. We first consider the special case where $u_0$ is a Heaviside type function, and prove the converge result without requiring any non-degeneracy on $f$. Furthermore, if $u_0$ is more general such that it can be trapped between two Heaviside type functions, but not necessarily monotone, we show that the convergence result remains valid under a rather mild non-degeneracy assumption on $f$. Lastly, in the case where $f$ is a non-degenerate multistable nonlinearity, we show the global and exponential convergence for a much larger class of front-like initial data.

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arXiv2019

Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type

We investigate spreading properties of solutions of a large class of two-component reaction-diffusion systems, including prey-predator systems as a special case. By spreading properties we mean the long time behaviour of solution fronts that start from localized (i.e. compactly supported) initial data. Though there are results in the literature on the existence of travelling waves for such systems, very little has been known-at least theoretically-about the spreading phenomena exhibited by solutions with compactly supported initial data. The main difficulty comes from the fact that the comparison principle does not hold for such systems. Furthermore, the techniques that are known for travelling waves such as fixed point theorems and phase portrait analysis do not apply to spreading fronts. In this paper, we first prove that spreading occurs with definite spreading speeds. Intriguingly, two separate fronts of different speeds may appear in one solution-one for the prey and the other for the predator-in some situations.

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arXiv2019

Existence and uniqueness of propagating terraces

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 non-linearities.

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arXiv2018

Generation of fine transition layers and their dynamics for the stochastic Allen--Cahn equation

We study an $\ep$-dependent stochastic Allen--Cahn equation with a mild random noise on a bounded domain in $\mathbb{R}^n$, $n\geq 2$. Here $\ep$ is a small positive parameter that represents formally the thickness of the solution interface, while the mild noise $ξ^\ep(t)$ is a smooth random function of $t$ of order $\mathcal O(\ep^{-γ})$ with $0<γ<1/3$ that converges to white noise as $\ep\rightarrow 0^+$. We consider initial data that are independent of $\ep$ satisfying some non-degeneracy conditions, and prove that steep transition layers---or interfaces---develop within a very short time of order $\ep^2|\ln\ep|$, 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 $\ep\rightarrow 0^+$. Furthermore, we prove that the thickness of the interface for $\ep$ small is indeed of order $\mathcal O(\ep)$ 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 destroy the solution profile near the interface as long as the noise is spatially uniform. Our results on the motion of interface improve the earlier results of Funaki (1999) and Weber (2010) by considerably weakening the requirements for the initial data and establishing the robustness of the solution profile near the interface that has not been known before.

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arXiv2018

Dynamics of Time-Periodic Reaction-Diffusion Equations with Compact Initial Support on R

This paper is concerned with the asymptotic behavior of bounded solutions of the Cauchy problem \begin{equation*} \left\{ \begin{array}{ll} u_t=u_{xx} +f(t,u), & x\in\mathbb{R},\,t>0,\\ u(x,0)= u_0, & x\in\mathbb{R}, \end{array}\right. \end{equation*} where $u_0$ is a nonnegative bounded function with compact support and $f$ is a rather general nonlinearity that is periodic in $t$ and satisfies $f(\cdot,0)=0$. In the autonomous case where $f=f(u)$, the convergence of every bounded solution to an equilibrium has been established by Du and Matano (2010). However, the presence of periodic forcing makes the problem significantly more difficult, partly because the structure of time periodic solutions is much less understood than that of steady states. In this paper, we first prove that any $ω$-limit solution is either spatially constant or symmetrically decreasing, even if the initial data is not symmetric. Furthermore, we show that the set of $ω$-limit solutions either consists of a single time-periodic solution or it consists of multiple time-periodic solutions and heteroclinic connections among them. Next, under a mild non-degenerate assumption on the corresponding ODE, we prove that the $ω$-limit set is a singleton, which implies the solution converges to a time-periodic solution. Lastly, we apply these results to equations with bistable nonlinearity and combustion nonlinearity, and specify more precisely which time-periodic solutions can possibly be selected as the limit.

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arXiv2015

On the large time behavior of the solutions of a nonlocal ordinary differential equation with mass conservation

We consider an initial value problem for a nonlocal differential equation with a bistable nonlinearity in several space dimensions. The equation is an ordinary differential equation with respect to the time variable t, while the nonlocal term is expressed in terms of spatial integration. We discuss the large time behavior of solutions and prove, among other things, the convergence to steady-states. The proof that the solution orbits are relatively compact is based upon the rearrangement theory.

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arXiv2012

Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations

We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic nonlinearities (including multistable 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.

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arXiv2011

On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data

Formal asymptotic expansions have long been used to study the singularly perturbed Allen-Cahn type equations and reaction-diffusion systems, including in particular the FitzHugh-Nagumo system. Despite their successful role, it has been largely unclear whether or not such expansions really represent the actual profile of solutions with rather general initial data. By combining our earlier result and known properties of eternal solutions of the Allen-Cahn equation, we prove validity of the principal term of the formal expansions for a large class of solutions.

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arXiv2010

Maximizing the spreading speed of KPP fronts in two-dimensional stratified media

We consider the equation $u_t=u_{xx}+u_{yy}+b(x)f(u)+g(u)$, $(x,y)\in\mathbb R^2$ with monostable nonliearity, where $b(x)$ is a nonnegative measure on $\mathbb R$ that is periodic in $x.$ In the case where $b(x)$ is a smooth periodic function, there exists a pulsating travelling wave that propagates in the direction $(\cosθ,\sinθ)$ -- with average speed $c$ if and only if $c\geq c^*(θ,b),$ where $c^*(θ,b)$ is a certain positive number depending on $b.$ Moreover, the quantity $w(θ;{b})=\min_{|θ-ϕ|<\fracπ{2}}c^*(ϕ;{b})/\cos(θ-ϕ)$ is called the spreading speed. This theory can be extended by showing the existence of the minimal speed $c^*(θ,b)$ for any nonnegative measure $b$ with period $L.$ We then study the question of maximizing $c^*(θ,b)$ under the constraint $\int_{[0,L)}b(x)dx=αL,$ where $α$ is an arbitrarily given positive constant. We prove that the maximum is attained by periodically arrayed Dirac's delta functions $h(x)=αL\sum_{k\in\mathbb Z}δ(x+kL)$ for any direction $θ$. Based on these results, for the case that $b=h$ we also show the monotonicity of the spreading speedsin $θ$ and study the asymptotic shape of spreading fronts for large $L$ and small $L$ . Finally, we show that for general 2-dimensional periodic equation $u_t=u_{xx}+u_{yy}+b(x,y)f(u)+g(u)$, $(x,y)\in\mathbb R^2$, the similar conclusions do not hold.

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arXiv2010

A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations

We consider the equation $u_t=u_{xx}+b(x)u(1-u),$ $x\in\mathbb R,$ where $b(x)$ is a nonnegative measure on $\mathbb R$ that is periodic in $x.$ In the case where $b(x)$ is a smooth periodic function, it is known that there exists a travelling wave with speed $c$ for any $c\geq c^*(b),$ where $c^*(b)$ is a certain positive number depending on $b.$ Such a travelling wave is often called a \lq\lq pulsating travelling wave" or a \lq\lq periodic travelling wave", and $c^*(b)$ is called the \lq\lq minimal speed". In this paper, we first extend this theory by showing the existence of the minimal speed $c^*(b)$ for any nonnegative measure $b$ with period $L.$ Next we study the question of maximizing $c^*(b)$ under the constraint $\int_{[0,L)}b(x)dx=αL,$ where $α$ is an arbitrarily given constant. This question is closely related to the problem studied by mathematical ecologists in late 1980's but its answer has not been known. We answer this question by proving that the maximum is attained by periodically arrayed Dirac's delta functions $αL\sum_{k\in\mathbb Z}δ(x+kL).$

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