Spikes and diffusion waves in one-dimensional model of chemotaxis

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 $K^{\prime }\in L^1(\mathbb{R})$. We show the existence of global-in-time nonnegative solutions and we study their large time asymptotics. Depending on $K^{\prime }$, 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.

References in corpus (2)

• The fractional Keller-Segel model
• Blowup of solutions to a diffusive aggregation model

Cited by in corpus (3)

• Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic equations
• Decay rates for a class of diffusive-dominated interaction equations
• A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation