Existence of Weak Solutions for a Hyperbolic Model of Chemosensitive Movement

Abstract

A hyperbolic model for chemotaxis and chemosensitive movement in one space dimension is considered. In contrast to parabolic models for chemotaxis the hyperbolic model allows us to take the dependence of the particle speed on external stimuli explicitly into account. This qualitatively covers recent experiments on chemotaxis in which it has been shown that particles adapt their speed to the surrounding environment. The model presented here consists of two hyperbolic differential equations of first order coupled with an elliptic equation. We assume that the speed depends on the external stimulus only (and not on its gradients). In that case solutions with steep gradients are expected which have the interpretation of moving swarms. A notion of weak solutions for this hyperbolic chemotaxis model is presented and the global existence of weak solutions is shown. The proof relies on the vanishing viscosity method; i.e., we obtain the weak solution as the limit of classical solutions of an associated parabolically regularized problem for vanishing viscosity parameter. Numerical simulations demonstrate phenomena like swarming behaviour and formation of steep gradients.