The effect of second terms on a nonlinear hyperbolic system describing bacterial dynamics

Abstract

We consider a general model describing the collective bacterial movement due to a chemical stimulus which is driven by sources terms, i.e., the growth of bacteria cells, in one spatial dimension ⎧ ⎨ ⎩ ut + ∇χ(s)u∇s = f(u, s) ,x ∈ R ,t >0 st = g(u, s) The aim of this work is to investigate the entropy solution of generalized Riemann problem with source terms, in which initial conditions consist of two smooth functions separated by a small discontinuity at the origin. The method consists on an asymptotic approximation of the solution at cell interfaces in the same state-expansion approach as in [17]. The specific structure allows for a possibility to determine the spatial derivatives at the origin and identify some properties of the solution.