Global large-data generalized solutions in a chemotactic movement with rotational flux caused by two stimuli

The main purpose of this paper is to study the global existence of large-data solutions to the following chemotactic model with general rotational sensitivity caused by two stimuli: {u(t) = Delta u - del . (uS(1)(x, u, v, w)del v) + del . (uS(2)(x, u, v, w)del w), v(t) = Delta v - uv, w(t) = Delta w - uw in a bounded domain Omega subset of R-n with smooth boundary under suitable initial-boundary conditions. Systems of this type arise in mathematical biology as models for the evolution of Escherichia coli suspensions in a vertical cylindrical cell by letting the bacteria be uniformly distributed in an oxygen -saturated medium with a glucose concentration step gradient at the mid height of the cell. For the two-dimensional case, the first author and Li (2016) showed that for suitably regular initial data (u(0), v(0), w(0)) fulfilling a smallness condition on the L-infinity-norm of v(0) and w(0), the initial-boundary value problem of this system possesses a global bounded classical solution. In this paper, we will remove such a smallness assumption to show the global existence of generalized solutions with general large initial data by using a new method developed by Winkler (2015). Our result holds in arbitrary dimension n >= 1. (C) 2017 Elsevier Ltd. All rights reserved.