Solvability of a Keller-Segel system with signal-dependent sensitivity and essentially sublinear production

TL;DR

This paper proves the global existence and boundedness of solutions for a chemotaxis system with signal-dependent sensitivity and sublinear production, ensuring no cell collapse occurs in a bounded domain.

Contribution

It establishes the solvability and boundedness of solutions for a Keller-Segel chemotaxis system with general sensitivity and sublinear chemical production, extending previous results.

Findings

No chemotactic collapse occurs for the cell distribution.

Global and uniformly bounded solutions exist for all initial data.

Numerical simulations illustrate diverse dynamics of the system.

Abstract

In this paper we consider the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_{t}=\Delta u-\nabla \cdot (u \chi(v)\nabla v) & \textrm{in}\quad \Omega\times (0,\infty), \\ 0=\Delta v-v+g(u) & \textrm{in}\quad \Omega\times (0,\infty),\\ \end{equation*} in a smooth and bounded domain $\Omega$ of $\mathbb{R}^2$. The chemotactic sensitivity $\chi$ is a general nonnegative function from $C^1((0,\infty))$ whilst $g$, the production of the chemical signal $v$, belongs to $C^1([0,\infty))$ and satisfies $\lambda_1\leq g(s)\leq \lambda_2(1+s)^\beta$, for all $s\geq 0$, $0\leq\beta\leq \frac{1}{2}$ and $0<\lambda_1\leq \lambda_2.$ It is established that no chemotactic collapse for the cell distribution $u$ occurs in the sense that any arbitrary nonnegative and sufficiently regular initial data $u(x,0)$ emanates a unique pair of global and uniformly bounded functions $(u,v)$ which classically solve the corresponding initial-boundary value problem. Finally, we illustrate the range of dynamics present within the chemotaxis system by means of numerical simulations.