Global solvability and large time behavior in a two-species chemotaxis-consumption model

Abstract

In this work, we study global existence, eventual smoothness and large time behavior of positive solutions for the following two-species chemotaxis consumption model:$ \left\{ \begin{array}{lll} &u_t = \Delta u-\chi_1\nabla \cdot ( u\nabla w), &\quad x\in \Omega, t>0, \\[0.2cm] & v_t = \Delta v-\chi_2\nabla \cdot (v\nabla w), &\quad x\in \Omega, t>0, \\[0.2cm] & w_t = \Delta w -(\alpha u+\beta v)w, &\quad x\in \Omega, t>0, \end{array}\right. $in a bounded and smooth domain $ \Omega\subset \mathbb{R}^n (n = 2,3,4,5) $ with nonnegative initial data $ u_0, v_0, w_0 $ and homogeneous Neumann boundary data. Here, the parameters $ \chi_1,\chi_2 $ 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 $ n $D $ (n = 3,4,5) $, and, finally, we show eventual smoothness and uniform convergence of global weak solutions in $ 3 $D 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.