Global classical solvability in a food metric chemotaxis system under zero boundary conditions at infinity

Abstract

We consider the nutrient–chemotaxis model, derived by a food metric, on the real line. The model consists of the equation for the nutrient density of ordinary differential equation type and that for the microorganism density of parabolic type, in which the diffusion coefficient is singular at the zero nutrient density. We establish a global existence result of bounded classical solutions for a large class of initial data under zero boundary conditions at infinity. To this end, we propose a transformation related to the food metric that transforms the original system into a parabolic–hyperbolic system with linear diffusion. We obtain a proper global well-posedness result for the transformed system using standard estimates, including the heat kernel estimates. We convert this to the solution of the original equation via an inverse transformation.