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

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.