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

Abstract

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. Then we show there is a critical cross diffusion rate δ c such that no patterns may be expected for δ ≥ δ c , while patterns are formed for δ < δ c and their stability is also derived. In particular, in 1-D, patterns are always formed whenever δ < δ c and the chemotactic coefficient is larger than an expressible bifurcation value, and there is another critical cross diffusion rate δ c < δ c such that cells with cross-diffusion rate δ ∈ (δ c , δ c ) are stable, while, for cells with δ < δ c 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.

A class of chemotaxis systems with growth source and nonlinear secretion

“…Case 1: If, for some k, the bifurcation branch C k is not compact in X × X × R, then C k extends to infinity in χ due to the elliptic regularity that any closed and bounded subset of the solution triple (u, v, χ) of our chemotaxis system (5.1) in X × X × R is compact; this can be easily shown by the Sobolev embeddings and results from [20, Chapter 3], see similar discussions in [4,50]. Clearly, in this case, we can find a sequence {χ k (u 0 )} ∞ k=1 fulfilling the statement of the theorem.…”

Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion

“…Case 1: If, for some k, the bifurcation branch C k is not compact in X ×X ×R, then C k extends to infinity in χ due to the elliptic regularity that any closed and bounded subset of the solution triple (u, v, χ) of our chemotaxis system (4.1) in X × X × R is compact; this can be easily shown by the Sobolev embeddings and results from [14,Chapter 3], see similar discussions in [29,Proposition 4.1]. Clearly, in this case, we can find a sequence {χ k (ũ)} ∞ k=1 fulfilling the statement of the theorem.…”

How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system?

“…The blow up solution or a δ function is surely connected to the phenomenon of cell aggregation; on the other hand, various mechanisms proposed to the model have manifested that the blow-up solutions is fully precluded while pattern formation arises [7,16,30,62]. Among those mechanisms (see the introduction in [62]), inclusion of a growth source of cells is a common choice. In particular, the presence of logistic source has been shown to have an effect of preventing ultimate growth of populations.…”