A dispersive model equation is considered, which has been proposed by Whitham [Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974] as a shallow water model, and which can also be seen as a caricature of two-species Euler--Poisson problems. A number of formal properties as well as similarities to other dispersive equations are derived. A travelling wave analysis and some numerical tests are carried out. The equation features wave breaking in finite time. A local existence result for smooth solutions and a global existence result for weak entropy solutions are proved. Finally, a small dispersion limit is carried out for situations where the solution of the limiting equation is smooth.

RESUMEN

The Keller--Segel model is the classical model for chemotaxis of cell populations. It consists of a drift-diffusion equation for the cell density coupled to an equation for the chemoattractant. Here a variant of this model is studied in one-dimensional position space, where the chemotactic drift is turned off for a limiting cell density by a logistic term and where the chemoattractant density solves an elliptic equation modeling a quasi-stationary balance of reaction and diffusion with production of the chemoattractant by the cells. The case of small cell diffusivity is studied by asymptotic and numerical methods. On a time scale characteristic for the convective effects, convergence of solutions to weak entropy solutions of the limiting nonlinear hyperbolic conservation law is proven. Numerical and analytic evidence indicates that solutions of this problem converge to irregular patterns of cell aggregates separated by entropic shocks from vacuum regions as time tends to infinity. Close to each of these patterns an "almost" stationary solution of the full parabolic problem can be constructed up to an exponentially small (in terms of the cell diffusivity) residual. Based on a metastability hypothesis, the methodsof exponential asymptotics are used to derive systems of ordinary differential equations approximating the long-time behavior of the parabolic problem on exponentially large time scales. The observed behavior is a coarsening process reminiscent of phase change models. A hybrid asymptotic-numerical approach for the simulation of the system is introduced, and the accuracy of this new approach is shown by comparison to numerical simulations of the full problem.