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arXiv · 1008.0020

Spikes and diffusion waves in one-dimensional model of chemotaxis

Grzegorz Karch·Kanako Suzuki
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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.

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Grzegorz Karch, Kanako Suzuki. 2010-07-30. Spikes and diffusion waves in one-dimensional model of chemotaxis. https://doi.org/10.1088/0951-7715%2F23%2F12%2F007

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