Toshitaka Nagai

Traveling waves in a chemotactic model

A model for chemotaxis in a bacteria-substrate mixture introduced by Keller and Segel, which is described by nonlinear partial differential equations, is studied analytically. The existence of traveling waves is shown for the system in which the substrate diffusion is taken into account and the chemotactic coefficient is greater than the motility one, and the instability of traveling waves is discussed.

Asymptotic Behavior for a Nonlinear Degenerate Diffusion Equation in Population Dynamics

We consider a spatially aggregating population model which provides the homogenizing process due to density-dependent diffusion and the dehomogenizing one due to a certain long-range transport. The result asserts that, by a balance between two processes, an initial distribution of populations forms itself into a traveling solitary wave pattern for large time, which exhibits phenomenologically a kind of aggregation of a species.

BREZIS–MERLE INEQUALITIES AND APPLICATION TO THE GLOBAL EXISTENCE OF THE CAUCHY PROBLEM OF THE KELLER–SEGEL SYSTEM

We discuss the existence of the global solution for two types of nonlinear parabolic systems called the Keller–Segel equation and attractive drift–diffusion equation in two space dimensions. We show that the system admits a unique global solution in

Stability of localized stationary solutions

L^{\infty}_{\rm loc}(0, \infty \, {;}\, L^{\infty}(\mathbb{R}^2))

On the interfaces in a nonlocal quasilinear degenerate equation arising in population dynamics

. The proof is based upon the Brezis–Merle type inequalities of the elliptic and parabolic equations. The proof can be applied to the Cauchy problem which is describing the self-interacting system.

Stability properties of traveling pulse solutions of the higher dimensional FitzHugh-Nagumo equations

The present paper is devoted to the study of the stablity properties of localized stationary solutions of a nonlinear degenerate diffusion equation involving a nonlocally convective term. The equation, which is related to population dynamics, has various stationary solutions. The paper shows that the most fundamental stationary solution is stable in a sense. The asymptotic stability is also proved.

Asymptotic Behavior of the interfaces to a nonlinear degenerate diffusion equation in population dynamics

AbstractWe study regularity and propagation properties of interfaces separating regions where nonnegative weak solutions of the Cauchy problem for the equation

Propagation of excited state and its failure in a simple model of myelinated nerve axons

u_t = (u^m )xx + \left[ {u\left( {\int_{ - \infty }^x {u(y,t)dy} - \int_x^\infty {u(y,t)dy} } \right)} \right]_x , m > 1,