In this paper we consider a free boundary problem describing cell motility,
which is a simple model of Umeda (see ).
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 $T$.
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  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.
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.
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.
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.
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.
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.
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.
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 . 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 , also ,
for a Brezis-Nirenberg type problem on spherical caps.