No abstract

Spiral waves are commonly observed in biological and chemical systems. Representing each wave front by a single curve, Brazhnik, Davydov, and Mikhailov introduce a kinematic model equation. The aim of this paper is to provide a detailed analysis for the steady state solutions of these equations. The existence of asymptotically Archimedean solutions is analytically shown.

We consider the curvature-driven motion of an interface on a bounded domain that contacts with the boundary at the right angle and has triple junctions with prescribed angles. We derive a linearized system at a stationary interface, and obtain a characteristic function whose zeros correspond to the eigenvalues of the linearized operator. From the characteristic function, it is shown that the unstable dimension is not relevant to the topology of the stationary interface but depends mainly on the curvature of the boundary.

An approximate method is proposed and numerically confirmed for a time-dependent Fokker-Planck equation (FPE) modeling the Brownian motor. The situation addressed in this paper is that the potential is spatially asymmetric and multiplicatively modulated. The approximate solution is composed of Sturm-Liouville eigenfunctions. While some time-independent partial differential equations and those perturbed around them have been known to be reduced to ordinary differential equations with a few degrees of freedom, the time-dependent FPE dealt with in this paper is also found to be reduced to an ordinary differential equation.

It is shown in a rigorous way that propagation speeds of disturbances are bounded for a class of reaction-diffusion systems. It turns out that solutions for various initial states are confined by traveling waves. A new technique is developed for the construction of the comparison functions. The technique is based on the operator-splitting methodology, which is known as a numerical computation method. By using an exact solution of the Fisher equation we can make a simple proof. The upper bounds of the speeds are given explicitly in terms of the diffusion constants and the Lipschitz norms of the reaction terms.