On a class of Keller–Segel chemotaxis systems with cross-diffusion

In this paper, we study a class of Keller–Segel chemotaxis systems with cross-diffusion. By using the entropy dissipation method and assuming mainly the chemotactic sensitivity separates the cell density and the chemical signal, we first establish the existence of global weak solutions with the effects of cross diffusion included in ≤3-D≤3-D. Then we show there is a critical cross diffusion rate δcδc such that no patterns may be expected for δ≥δcδ≥δc, while patterns are formed for δ<δcδ<δc and their stability is also derived. In particular, in 1-D, patterns are always formed whenever δ<δcδ<δc and the chemotactic coefficient is larger than an expressible bifurcation value, and there is another critical cross diffusion rate δc<δcδc<δc such that cells with cross-diffusion rate δ∈(δc,δc)δ∈(δc,δc) are stable, while, for cells with δ<δcδ<δc to be stable, their degradation rate must be less than a threshold value. Hence, in some sense, cross-diffusion is harmful to enable pattern formation, while it is helpful to stabilize the cells once patterns are formed. Finally, we show that the cross diffusion plays a role in regularizing the cell aggregation phenomenon for large chemotactic coefficient. Our results provide global dynamics and insights on how the biological parameters, especially, the cross diffusion, affect pattern formations.