Spikes and diffusion waves in one-dimensional model of chemotaxis

TL;DR

This paper analyzes a one-dimensional viscous transport equation with nonlocal velocity, demonstrating global solutions, their asymptotic behavior, and phenomena like spike creation relevant to chemotaxis models.

Contribution

It establishes existence of solutions, characterizes their large-time asymptotics, and reveals spike formation phenomena in a nonlocal chemotaxis-inspired model.

Findings

Existence of global nonnegative solutions

Asymptotic solutions are either linear or nonlinear diffusion waves

Spike creation phenomena observed for certain kernels

Abstract

We consider the one-dimensional initial value problem for the viscous transport equation with nonlocal velocity $u_t = u_{xx} - \left(u (K^\prime \ast u)\right)_{x}$ with a given kernel $K'\in L^1(\R)$. We show the existence of global-in-time nonnegative solutions and we study their large time asymptotics. Depending on $K'$, we obtain either linear diffusion waves ({\it i.e.}~the fundamental solution of the heat equation) or nonlinear diffusion waves (the fundamental solution of the viscous Burgers equation) in asymptotic expansions of solutions as $t\to\infty$. Moreover, for certain aggregation kernels, we show a concentration of solution on an initial time interval, which resemble a phenomenon of the spike creation, typical in chemotaxis models.