The riemann problem fora nonlinear non-strictly hyperbolic system arising in biology

Abstract We study the Riemann problem for the following system v t − u x =0 u t − (uv) x = 0 which describes the macroscopic behaviour of some (so-called ‘chemotactic’) bacterial populations, which are attracted by a chemical substate. This system is elliptic for v 2 + 4 u ⩽ 0, and hyperbolic elsewhere. Here, since u is the concentration of the bacteria, we solve this problem in the half-plane { u ⩾ 0}, in which (1) is strictly hyperbolic, except at the origin. Moreover, each eigenvlue is linearly degenerate along one half of the axis { u =0}, and the Hugoniot locus of a point on this axis is unusual.