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- Advances in Mathematics of Communications
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PROC

We are interested in wave-pinning in a reaction-diffusion model for cell polarization
proposed by Y.Mori, A.Jilkine and L.Edelstein-Keshet.
They showed interesting bifurcation diagrams and stability results
for stationary solutions for a limiting equation by numerical computations.
Kuto and Tsujikawa showed several mathematical bifurcation results
of stationary solutions of this problem.
We show exact expressions of all the solution
by using the Jacobi elliptic functions and complete elliptic integrals.
Moreover, we construct a bifurcation sheet
which gives bifurcation diagram.
Furthermore, we show numerical results of the stability of stationary solutions.

CPAA

We consider a parameterized, nonlocally constrained boundary-value problem, whose solutions are known to yield exact solutions, called Oseen's spiral flows, of the Navier-Stokes equations.
We represent all solutions explicitly in terms of elliptic functions, and clarify completely the structure of the set of all the global branches of the solutions.

CPAA

We are interested in the asymptotic profiles of
all eigenfunctions for 1-dimensional
linearized eigenvalue problems
to nonlinear boundary value problems
with a diffusion coefficient $\varepsilon$.
For instance, it seems that
they have simple and beautiful properties
for sufficiently small $\varepsilon$
in the balanced bistable nonlinearity case.
As the first step to give rigorous proofs
for the above case,
we study the case $f(u)=\sin u$ precisely.
We show that two special eigenfunctions completely
control the asymptotic profiles of other eigenfunctions.

CPAA

We study nontrivial stationary solutions to a nonlinear boundary value
problem with parameter $\varepsilon>0$
and the corresponding linearized eigenvalue problem.
By using a particular
solution of a linear ordinary differential equation of the third order,
we give expressions of all eigenvalues and eigenfunctions to the
linearized problems.
They are completely determined by a characteristic
function which consists of complete elliptic
integrals.
We also show asymptotic formulas of eigenvalues
with respect to sufficiently small $\varepsilon$.
These results give important information for profiles of
corresponding eigenfunctions.

DCDS

We show existence, nonexistence, and exact multiplicity for stationary limiting problems of a cell polarization model
proposed by Y. Mori, A. Jilkine and L. Edelstein-Keshet.
It is a nonlinear boundary value problem with total mass constraint.
We obtain exact multiplicity results by investigating a global bifurcation sheet which we constructed by using complete elliptic integrals in a previous paper.

keywords:
exact solution
,
nonlocal term
,
Reaction diffusion model
,
elliptic integral.
,
bifurcation
,
level set analysis

DCDS

In this paper we investigate a limiting system that arises from
the study of steady-states of the Lotka-Volterra competition model with
cross-diffusion. The main purpose here is to understand

*all possible*solutions to this limiting system, which consists of a nonlinear elliptic equation and an integral constraint. As far as existence and non-existence in one dimensional domain are concerned, our knowledge of the limiting system is nearly complete. We also consider the qualitative behavior of solutions to this limiting system as the remaining diffusion rate varies. Our basic approach is to convert the problem of solving the limiting system to a problem of solving its "representation" in a different parameter space. This is first done*without*the integral constraint, and then we use the integral constraint to find the "solution curve" in the new parameter space as the diffusion rate varies. This turns out to be a powerful method as it gives fairly precise information about the solutions.
PROC

As the first step to understand the Gierer-Meinhardt system with
source term, it is important to know the global bifurcation diagram of a shadow
system. For the case without source term, it is well-understood. However, for
the case with source term, the shadow system has a nonlocal term. Thus
standard methods do not work, and there are a few partial results even for
one-dimensional case. We give explicit representations of all solutions in terms
of elliptic functions. They play crucial roles to clarify the global bifurcation
diagram.

CPAA

We consider the Ginzburg-Landau equation with a positive parameter, say lambda,
and solve all equilibrium solutions with periodic boundary conditions.
In particular we reveal a complete bifurcation diagram of the equilibrium solutions
as lambda increases.
Although it is known that this equation allows bifurcations from not only a trivial
solution but also secondary bifurcations as lambda varies,
the global structure of the secondary branches was open.
We first classify all the equilibrium solutions
by considering some configuration of the solutions.
Then we formulate the problem to find a solution which bifurcates
from a nontrivial solution
and drive a reduced equation for the solution in terms of complete elliptic integrals
involving useful parametrizations.
Using some relations between the integrals,
we investigate the reduced equation.
In the sequel we obtain a global branch of the bifurcating solution.

DCDS

In this paper we study the Shigesada-Kawasaki-Teramoto model [17]
for two competing species
with cross-diffusion. We prove the existence of spectrally stable non-constant positive steady
states for high-dimensional
domains when one of the cross-diffusion coefficients is sufficiently large
while the other is equal to zero.

CPAA

We propose a method to investigate the structure of positive radial
solutions to semilinear elliptic problems with various boundary conditions.
It is already shown that the boundary value problems can be reduced to a
canonical form by a suitable change of variables. We show structure theorems
to canonical forms to equations with power nonlinearities and various boundary
conditions. By using these theorems, it is possible to study the properties of
radial solutions of semilinear elliptic equations in a systematic way, and make
clear unknown structure of various equations.

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