Spikes and diffusion waves in a one-dimensional model of chemotaxis

We consider the one-dimensional initial value problem for the viscous transport equation with nonlocal velocity u(t) = u(xx) - (u(K' * u)) x with a given kernel K' is an element of 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 (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 -> infinity. 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.