To what extent does the consideration of positive total flux influence the dynamics of Keller–Segel-type models?

Since the introduction of the Keller-Segel model in 1970 to describe chemotaxis (the interactions between cell distributions u and chemical distributions v), there has been a significant proliferation of research articles exploring various extensions and modifications of this model within the scientific community. From a technical standpoint, the totality of results concerning these variants are characterized by the assumption that the total flux, involving both distributions, of the model under consideration is zero. This research aims to present a novel perspective by focusing on models with a positive total flux. Specifically, by employing Robin-type boundary conditions for u and v, we seek to gain insights into the interactions between cells and their environment, uncovering important dynamics such as how variations in boundary conditions influence chemotactic behavior. In particular, the choice of the boundary conditions is motivated by real-world phenomena and by the fact that the related analysis reveals some interesting properties of the system.

Mathematically, or $h,\chi >0$ and $\alpha \in (0,1]$ we investigate Keller–Segel-type models with positive total flux $u_{\nu }-\chi u{v}_{\nu }=\chi \alpha huv$, reading as(⊕)$\{\begin{array}{cc}{u}_t=\Delta u-\chi \nabla \cdot \left(u\nabla v\right)& \text{in}\Omega \times (0,T_{\text{max}}),\\ \tau v_t=\Delta v-v+u& \text{in}\Omega \times (0,T_{\text{max}}),\\ u_{\nu }=(\alpha -1)\chi huv,v_{\nu }=-hv& \text{on}\partial \Omega \times (0,T_{\text{max}}),\end{array}$ where $\Omega \subset R^n$, $n\ge 1$ is a bounded and smooth domain, $\tau \in \{0,1\}$, $T_{\text{max}}>0$ and ν denoting the outward normal vector to the boundary of Ω, ∂Ω.

We aim at emphasizing how the inclusion of the incoming flowing flow makes the overall analysis more complex. The technical difficulties are essentially tied to the lack of the crucial property of the mass conservation, which in this case is replaced by an increase in the mass itself. Such behavior of the mass cannot be circumvented by merely including classical logistics of the form $au-b{u}^2$ (with $a,b>0$); an additional dissipative term involving gradient nonlinearities is required. But there is another indication that suggests how the total positive flux significantly alters the dynamics of taxis models with zero-flux. Indeed:

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  for mechanisms with positive flux ($\alpha >0$), global (i.e. $T_{\text{max}}=\infty$) and bounded solutions are obtained as long as an essential strong logistic of the form $au-b{u}^2-c|\nabla u{|}^2$ ($a,b,c>0$) is included in the model;
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  for phenomena with zero flux ($\alpha =0$), analogies with the classical Keller–Segel models, without logistic, endowed with homogeneous Neumann boundary conditions, specially for blow-up scenarios (i.e., $T_{\text{max}}<\infty$), can be observed.