Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption

Abstract

This paper deals with the Neumann initial-boundary value problem for a classical chemotaxis system with signal consumption in a disk. In contrast to previous studies which have established a comprehensive theory of global classical solutions for suitably regular nonnegative initial data, the focus in the present work is on the question to which extent initially prescribed singularities can be regularized despite the presence of the nonlinear cross-diffusive interaction. The main result in this paper asserts that at least in the framework of radial solutions immediate regularization occurs under an essentially optimal condition on the initial distribution of the population density. More precisely, it will turn out that for any radially symmetric initial data belonging to the space of regular signed Borel measures for the population density and to L2 for the signal density, there exists a classical solution to the Neumann initial-boundary value problem, which is smooth and approaches the given initial data in an appropriate trace sense.