DCDS
Preface
Peter Bates Danielle Hilhorst Hiroshi Matano Yoshihisa Morita
Discrete & Continuous Dynamical Systems - A 2017, 37(2): i-iii doi: 10.3934/dcds.201702i
Professor Paul Chase Fife was born in Cedar City, Utah, on February 14, 1930. After undergraduate studies at the University of Chicago, he obtained a Master's degree in physics from the University of California Berkeley where he also received a Phi Beta Kappa Award. He then completed a PhD in Applied Mathematics at New York University, Courant Institute, in June 1959. While at NYU he met Jayne Winters, and they married on December 22, 1959. They then moved to Palo Alto, California, where Paul joined the Department of Mathematics at Stanford University.

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CPAA
Preface
Jacques Demongeot Danielle Hilhorst Hiroshi Matano Masayasu Mimura
Communications on Pure & Applied Analysis 2012, 11(1): i-i doi: 10.3934/cpaa.2012.11.1i
This volume deals with the mathematical modeling and the analysis of reaction-diffusion systems, as well as their applications in a number of different fields. It grew from a workshop organized by ReaDiLab, a Japan-France research collaboration unit of CNRS (Laboratoire International Associé du CNRS). This workshop took place at the University of Paris-Sud in June, 2009, bringing together many members of ReaDiLab with researchers from other French and Japanese laboratories. ReaDiLab is composed of 33 Japanese and 36 French researchers in the fields of mathematics, biology, medicine, and chemistry. Its goal is to develop mathematical modeling, analysis and numerical methods for reaction-diffusion systems arising in all those fields.

In order to understand the problems occurring in these areas of application, one should not only apply known methods, but also develop novel mathematical tools. Because of this, many results corresponding to new approaches are given in the main topics of this CPAA Special Volume, including demography and travelling waves in epidemics modelling, structured populations growth, propagation in inhomogeneous media, ecology and dry land vegetation, formation of stationary spatio-temporal patterns in reaction-diffusion systems both from a mathematical and an experimental view point, spatio-temporal dynamics of cooperation, cell migration and bacterial suspensions. This issue also includes more mathematically oriented topics such as interface dynamics, stability of non-constant stationary solutions, heterogeneity-induced spot dynamics, boundary spikes, appearance of anomalous singularities in parabolic equations, finite time blow-up, a multi-parameter inverse problem, and the numerical approximation of parabolic equations and chemotactic systems. We hope these advanced results will be useful to the community of researchers working in the domain of partial differential equations, and that they will serve as examples of mathematical modelling to those working in the different areas of application mentioned above.
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DCDS
Connecting equilibria by blow-up solutions
Marek Fila Hiroshi Matano
Discrete & Continuous Dynamical Systems - A 2000, 6(1): 155-164 doi: 10.3934/dcds.2000.6.155
We study heteroclinic connections in a nonlinear heat equation that involves blow-up. More precisely we discuss the existence of $L^1$ connections among equilibrium solutions. By an $L^1$-connection from an equilibrium $\phi^{-1}$ to an equilibrium $\phi^+$ we mean a function $u$($.,t$) which is a classical solution on the interval $(-\infty,T)$ for some $T\in \mathbb R$ and blows up at $t=T$ but continues to exist in the space $L^1$ in a certain weak sense for $t\in [T,\infty)$ and satisfies $u$($.,t$)$\to \phi^\pm$ as $t\to\pm\infty$ in a suitable sense. The main tool in our analysis is the zero number argument; namely to count the number of intersections between the graph of a given solution and that of various specific solutions.
keywords: zero number. connecting orbits blow­up Semilinear parabolic equation nonlinear heat equation
DCDS
Global existence and uniqueness of a three-dimensional model of cellular electrophysiology
Hiroshi Matano Yoichiro Mori
Discrete & Continuous Dynamical Systems - A 2011, 29(4): 1573-1636 doi: 10.3934/dcds.2011.29.1573
We study a three-dimensional model of cellular electrical activity, which is written as a pseudodifferential equation on a closed surface $\Gamma$ in $\R^3$ coupled with a system of ordinary differential equations on $\Gamma$. Previously the existence of a global classical solution was not known, due mainly to the lack of a uniform $L^\infty$ bound. The main difficulty lies in the fact that, unlike the Laplace operator that appears in traditional models, the pseudodifferential operator in the present model does not satisfy the maximum principle. We overcome this difficulty by introducing the notion of "quasipositivity principle" and prove a uniform $L^\infty$ bound of solutions -- hence the existence of global classical solutions -- for a large class of nonlinearities including the FitzHugh-Nagumo and the Hodgkin-Huxley kinetics. We then study the asymptotic behavior of solutions to show that the system possesses a finite dimensional global attractor consisting entirely of smooth functions despite the fact that the system is only partially dissipative. We also show that ordinary differential equation models without spatial extent, often used in modeling studies, can be obtained from the present model in the small-cell-size limit.
keywords: global attractor 3D cable model asymptotic smoothing pseudodifferential operator. electrophysiology quasipositivity partially dissipative systems
DCDS
The global attractor of semilinear parabolic equations on $S^1$
Hiroshi Matano Ken-Ichi Nakamura
Discrete & Continuous Dynamical Systems - A 1997, 3(1): 1-24 doi: 10.3934/dcds.1997.3.1
We study the global attractor of semilinear parabolic equations of the form

$u_t=u_{x x}+f(u,u_x),\ x\in\mathbb{R}$/$\mathbb{Z}, \ t>0.$

Under suitable conditions on $f$, the equation generates a global semiflow on a suitable function space. The general theory of inertial manifolds does not apply to this equation due to lack of the so-called spectral gap condition. Using a totally different method, we show that the global attractor is the graph of a continuous mapping of finite dimension. We also show that this dimension is equal to $2[N$/$2]+1$, where $N$ is the maximal value of the generalized Morse index of equilibria and periodic solutions. Note that we do not make any assumption regarding the hyperbolicity of those solutions. We further prove that there exists no homoclinic orbit nor heteroclinic cycle.

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DCDS
Monotonicity and convergence results in order-preserving systems in the presence of symmetry
Toshiko Ogiwara Hiroshi Matano
Discrete & Continuous Dynamical Systems - A 1999, 5(1): 1-34 doi: 10.3934/dcds.1999.5.1
This paper deals with various applications of two basic theorems in order- preserving systems under a group action -- monotonicity theorem and convergence theorem. Among other things we show symmetry properties of stable solutions of semilinear elliptic equations and systems. Next we apply our theory to traveling waves and pseudo-traveling waves for a certain class of quasilinear diffusion equa- tions and systems, and show that stable traveling waves and pseudo-traveling waves have monotone profiles and, conversely, that monotone traveling waves and pseudo- traveling waves are stable with asymptotic phase. We also discuss pseudo-traveling waves for equations of surface motion.
keywords: traveling waves and pseudo-traveling waves quasilinear diffusion equations. semilinear elliptic equations Order-preserving systems
DCDS
Preface
Danielle Hilhorst Hiroshi Matano
Discrete & Continuous Dynamical Systems - A 2009, 25(1): i-ii doi: 10.3934/dcds.2009.25.1i
Professor Masayasu Mimura, known to his friends as ''Mayan", was born on Shikoku Island, Japan, on October 11th, 1941. He studied at Kyoto University, where he prepared his doctoral thesis under the supervision of Professor Masaya Yamaguti. Thanks to his advisor's acquaintance with a large circle of scientists, Mayan could keep close relations with a group of biologists and biophysicists - Ei Teramoto and Nanako Shigesada, among others - since his very early days.

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DCDS-B
On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data
Matthieu Alfaro Hiroshi Matano
Discrete & Continuous Dynamical Systems - B 2012, 17(6): 1639-1649 doi: 10.3934/dcdsb.2012.17.1639
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.
keywords: Singular perturbation reaction-diffusion system. asymptotic expansion front profile
NHM
Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit
Hiroshi Matano Ken-Ichi Nakamura Bendong Lou
Networks & Heterogeneous Media 2006, 1(4): 537-568 doi: 10.3934/nhm.2006.1.537
We study a curvature-dependent motion of plane curves in a two-dimensional cylinder with periodically undulating boundary. The law of motion is given by $V=\kappa + A$, where $V$ is the normal velocity of the curve, $\kappa$ is the curvature, and $A$ is a positive constant. We first establish a necessary and sufficient condition for the existence of periodic traveling waves, then we study how the average speed of the periodic traveling wave depends on the geometry of the domain boundary. More specifically, we consider the homogenization problem as the period of the boundary undulation, denoted by $\epsilon$, tends to zero, and determine the homogenization limit of the average speed of periodic traveling waves. Quite surprisingly, this homogenized speed depends only on the maximum opening angle of the domain boundary and no other geometrical features are relevant. Our analysis also shows that, for any small $\epsilon>0$, the average speed of the traveling wave is smaller than $A$, the speed of the planar front. This implies that boundary undulation always lowers the speed of traveling waves, at least when the bumps are small enough.
keywords: front propagation periodic traveling wave homogenization. curve-shortening

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