GLOBAL SOLVABILITY AND LARGE TIME BEHAVIOR IN A TWO-SPECIES CHEMOTAXIS-CONSUMPTION MODEL

In this work, we study global existence, eventual smoothness and large time behavior of positive solutions for the following two-species chemo-taxis consumption model: { ut = Delta u - X1 del.(u del w), x is an element of Omega, t > 0, vt = Delta v - X2 del. (v del w), x is an element of Omega, t > 0, wt = Delta w - (alpha u + beta v)w, x is an element of Omega, t > 0, in a bounded and smooth domain Omega subset of R-n(n = 2, 3, 4,5) with nonnegative initial data u0, v0, w0 and homogeneous Neumann boundary data. Here, the parameters X1, X2 are positive and alpha, beta are nonnegative. In such setup, for all reasonably regular initial data and for all parameters, we show global existence and uniform-in-time boundedness of classical solutions in 2D, global existence of weak solutions in nD (n = 3, 4, 5), and, finally, we show eventual smoothness and uniform convergence of global weak solutions in 3D convex domains. Our 2D boundedness removes a smallness condition required in [50] and other findings improve and extend the existing knowledge about one-species chemotaxis-consumption models in the literature.