Representation formulas for some 1-dimensional linearized eigenvalue problems
Tohru Wakasa Shoji Yotsutani
Communications on Pure & Applied Analysis 2008, 7(4): 745-763 doi: 10.3934/cpaa.2008.7.745
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.
keywords: exact solution linearized eigenvalue problem Nonlinear eigenvalue problem linearized stability eigenvalue asymptotic formula. eigenfunction
Exact multiplicity of stationary limiting problems of a cell polarization model
Tatsuki Mori Kousuke Kuto Tohru Tsujikawa Shoji Yotsutani
Discrete & Continuous Dynamical Systems - A 2016, 36(10): 5627-5655 doi: 10.3934/dcds.2016047
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
Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization
Tatsuki Mori Kousuke Kuto Masaharu Nagayama Tohru Tsujikawa Shoji Yotsutani
Conference Publications 2015, 2015(special): 861-877 doi: 10.3934/proc.2015.0861
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.
keywords: Exact solution Bifurcation diagram Elliptic integral. Reaction diffusion model
On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows
Hideo Ikeda Koji Kondo Hisashi Okamoto Shoji Yotsutani
Communications on Pure & Applied Analysis 2003, 2(3): 381-390 doi: 10.3934/cpaa.2003.2.381
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.
keywords: Nonlocal boundary value problem Navier-Stokes flow.
Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems
Tohru Wakasa Shoji Yotsutani
Communications on Pure & Applied Analysis 2010, 9(2): 539-561 doi: 10.3934/cpaa.2010.9.539
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.
keywords: Nonlinear eigenvalue problem eigenfunction linearized eigenvalue problem asymptotic formula. linearized stability eigenvalue exact solution
On a limiting system in the Lotka--Volterra competition with cross-diffusion
Yuan Lou Wei-Ming Ni Shoji Yotsutani
Discrete & Continuous Dynamical Systems - A 2004, 10(1&2): 435-458 doi: 10.3934/dcds.2004.10.435
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.
keywords: existence asymptotic behavior cross-diffusion parameter representation.
Global bifurcation structure on a shadow system with a source term - Representation of all solutions-
Hideaki Takaichi Izumi Takagi Shoji Yotsutani
Conference Publications 2011, 2011(Special): 1344-1350 doi: 10.3934/proc.2011.2011.1344
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.
keywords: parameter representation reaction di usion stationary
A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions
Satoshi Kosugi Yoshihisa Morita Shoji Yotsutani
Communications on Pure & Applied Analysis 2005, 4(3): 665-682 doi: 10.3934/cpaa.2005.4.665
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.
keywords: Ginzburg-Landau equation secondary bifurcation. periodic boundary conditions bifurcation diagram
Pattern formation in a cross-diffusion system
Yuan Lou Wei-Ming Ni Shoji Yotsutani
Discrete & Continuous Dynamical Systems - A 2015, 35(4): 1589-1607 doi: 10.3934/dcds.2015.35.1589
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.
keywords: existence competition stability Density-dependent diffusion steady states.
Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems
Y. Kabeya Eiji Yanagida Shoji Yotsutani
Communications on Pure & Applied Analysis 2002, 1(1): 85-102 doi: 10.3934/cpaa.2002.1.85
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.
keywords: Semilinear elliptic equations radial solutions canonical forms.

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