Multiple existence of traveling waves of a free boundary problem describing cell motility
Harunori Monobe Hirokazu Ninomiya
Discrete & Continuous Dynamical Systems - B 2014, 19(3): 789-799 doi: 10.3934/dcdsb.2014.19.789
In this paper we consider a free boundary problem describing cell motility, which is a simple model of Umeda (see [11]). This model includes a non-local term and the interface equation with curvature. We prove that there exist at least two traveling waves of the model. First, we rewrite the problem into a fixed-point problem for a continuous map $T$ and then show that there exist at least two fixed points for the map $T$.
keywords: free boundary problems cell crawling. Traveling waves
Traveling wave solutions with convex domains for a free boundary problem
Harunori Monobe Hirokazu Ninomiya
Discrete & Continuous Dynamical Systems - A 2017, 37(2): 905-914 doi: 10.3934/dcds.2017037

In this paper, a free boundary problem related to cell motility is discussed. This free boundary problem consists of an interface equation for the domain evolution and a parabolic equation governing actin concentration in the domain. In [10] the existence of traveling wave solutions with disk-shaped domains were shown in a special situation where a polymerization rate is specified. In this paper, by relaxing the condition for the polymerization rate, the previous result is extended to the existence of traveling wave solutions with convex domains.

keywords: Traveling waves free boundary problems cell crawling
Global stability of traveling curved fronts in the Allen-Cahn equations
Hirokazu Ninomiya Masaharu Taniguchi
Discrete & Continuous Dynamical Systems - A 2006, 15(3): 819-832 doi: 10.3934/dcds.2006.15.819
This paper is concerned with the global stability of a traveling curved front in the Allen-Cahn equation. The existence of such a front is recently proved by constructing supersolutions and subsolutions. In this paper, we introduce a method to construct new subsolutions and prove the asymptotic stability of traveling curved fronts globally in space.
keywords: curved front global stability. Allen-Cahn equation Traveling wave
Relative compactness in $L^p$ of solutions of some 2m components competition-diffusion systems
Danielle Hilhorst Masato Iida Masayasu Mimura Hirokazu Ninomiya
Discrete & Continuous Dynamical Systems - A 2008, 21(1): 233-244 doi: 10.3934/dcds.2008.21.233
We consider a class of $2m$ components competition-diffusion systems which involve $m$ parabolic equations as well as $m$ ordinary differential equation, and prove the strong convergence in $L^p$ of a subsequence of each component as the reaction coefficient tends to infinity. In the special case of $4$ components the solution of this system converges to that of a Stefan problem.
keywords: reaction-diffusion system singular limit relative compactness in $L^p$ Frechet-Kolmogoroff theorem.
Traveling wave solutions for a bacteria system with density-suppressed motility
Roger Lui Hirokazu Ninomiya
Discrete & Continuous Dynamical Systems - B 2019, 24(2): 931-940 doi: 10.3934/dcdsb.2018213

In 2011, Liu et. al. proposed a three-component reaction-diffusion system to model the spread of bacteria and its signaling molecules (AHL) in an expanding cell population. At high AHL levels the bacteria are immotile, but diffuse with a positive diffusion constant at low distributions of AHL. In 2012, Fu et. al. studied a reduced system without considering nutrition and made heuristic arguments about the existence of traveling wave solutions. In this paper we provide rigorous proofs of the existence of traveling wave solutions for the reduced system under some simple conditions of the model parameters.

keywords: Cell motility degenerate reaction-diffusion equations traveling wave equations pattern formations nonhomogeneous linear differential equations
Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems
Yan-Yu Chen Yoshihito Kohsaka Hirokazu Ninomiya
Discrete & Continuous Dynamical Systems - B 2014, 19(3): 697-714 doi: 10.3934/dcdsb.2014.19.697
In this study, we consider the traveling spots that were observed in the photosensitive Belousov-Zhabotinsky reaction experiment conducted by Mihailuk et al. in 2001. First, we introduce the interface equation by the singular limit analysis of a FitzHugh--Nagumo-type reaction-diffusion system. Then, we obtain the profile of the support of the solution. Next, we prove the uniqueness of the traveling spot by studying ordinary differential equations that describe its front and back. In addition, we provide an upper bound for the width of the spot. Furthermore, we compare the singular limit problem with the wave front interaction model proposed by Zykov and Showalter in 2005 and obtain traveling fingers.
keywords: FitzHugh-Nagumo system singular limit analysis back. Traveling spot front interface equation
Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems
Marek Fila Hirokazu Ninomiya Juan-Luis Vázquez
Discrete & Continuous Dynamical Systems - A 2006, 14(1): 63-74 doi: 10.3934/dcds.2006.14.63
This paper examines the following question: Suppose that we have a reaction-diffusion equation or system such that some solutions which are homogeneous in space blow up in finite time. Is it possible to inhibit the occurrence of blow-up as a consequence of imposing Dirichlet boundary conditions, or other effects where diffusion plays a role? We give examples of equations and systems where the answer is affirmative.
keywords: Blow-up reaction-diffusion Dirichlet conditions prevent blow-up.
Existence of a rotating wave pattern in a disk for a wave front interaction model
Jong-Shenq Guo Hirokazu Ninomiya Chin-Chin Wu
Communications on Pure & Applied Analysis 2013, 12(2): 1049-1063 doi: 10.3934/cpaa.2013.12.1049
We study the rotating wave patterns in an excitable medium in a disk. This wave pattern is rotating along the given disk boundary with a constant angular speed. To study this pattern we use the wave front interaction model proposed by Zykov in 2007. This model is derived from the FitzHugh-Nagumo equation and it can be described by two systems of ordinary differential equations for wave front and wave back respectively. Using a delicate shooting argument with the help of the comparison principle, we derive the existence and uniqueness of rotating wave patterns for any admissible angular speed with convex front in a given disk.
keywords: Rotating wave pattern back front angular speed.
Imperfect bifurcations in nonlinear elliptic equations on spherical caps
C. Bandle Y. Kabeya Hirokazu Ninomiya
Communications on Pure & Applied Analysis 2010, 9(5): 1189-1208 doi: 10.3934/cpaa.2010.9.1189
We consider elliptic boundary value problems on large spherical caps with parameter dependent power nonlinearities. In this paper we show that imperfect bifurcation occurs as in the work [13]. When the domain is the whole sphere, there is a constant solution. In the case where the domain is a spherical cap, however, the constant solution disappears due to the boundary condition. For large spherical caps we construct solutions which are close to the constant solution in the whole n-dimensional sphere, using the eigenvalues of the linearized problem in the whole sphere and fixed point arguments based on a Lyapunov-Schmidt type reduction. Numerically there is a surprising similarity between the diagrams of this problem and the ones obtained in [18], also [5], for a Brezis-Nirenberg type problem on spherical caps.
keywords: Nonlinear elliptic equations spherical coordinates fixed point theorems. bifurcation

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