Global solutions in a high-dimensional two-species chemotaxis model with Lotka-Volterra competitive kinetics

This paper deals with the two-species chemotaxis system with logistic source { u(t) = Delta u-chi(1)del.(u del w)+mu(1)u(1-u-a(1)upsilon), x is an element of Omega, t > 0, v(t) = Delta u-chi(2)del.(u del w)+mu(2)u(1-a(2)u-upsilon), x is an element of Omega, t > 0, w(t) = Delta w-lambda w + alpha u + beta upsilon, x is an element of Omega, t > 0, under homogeneous Neumann boundary condition in a smooth bounded domain Omega subset of R-n (n >= 1). It is proved that in convex domains the problem possesses a unique global bounded solution if mu(1) and mu(2) are large enough. Moreover, we establish the existence of global weak solution for any mu(1) > 0 and mu(2) > 0. (C) 2018 Elsevier Inc. All rights reserved.