Eventual smoothness and stabilization of renormalized radial solutions in a chemotaxis consumption system with bounded chemotactic sensitivity

Abstract

In this paper, a chemotaxis model with bounded chemotactic sensitivity and signal absorption

is considered under homogeneous Neumann boundary conditions in the ball $\Omega =B_R(0)\subset {\mathbb {R}}^n$, where $R>0$ and $n\ge 2$. Here, S is a scalar function with $S(s,t)\in C^2([0,\infty )\times [0,\infty ))$. Moreover, for some positive constant K, $|S(s,t)|\le K$ for all $s,t\in [0,\infty )$. For all appropriately regular and radially symmetric initial data ($u_0,v_0$) fulfilling $u_0\ge 0$ and $v_0>0$, the present paper shows that there is a globally defined pair (u, v) of radially symmetric functions which are continuous in $({{\overline{\Omega }}} \backslash \{0\}) \times [0, \infty )$ and smooth in $({{\overline{\Omega }}} \backslash \{0\}) \times (0, \infty )$, and which solve the corresponding initial-boundary value problem for ($\star$) with $(u(\cdot , 0), v(\cdot , 0))=\left( u_{0}, v_{0}\right)$ in an appropriate generalized sense. Moreover, in the two-dimensional setting, it is shown that these solutions are global mass-preserving in the flavor of the identity

$$\begin{aligned} \int \limits _\Omega u(x,t)=\int \limits _\Omega u_0(x)\quad \text {for all }t>0 \end{aligned}$$

and any nontrivial of these globally defined solutions eventually becomes smooth and satisfies

$$\begin{aligned} u(\cdot , t) \rightarrow \frac{1}{|\Omega |}\int \limits _\Omega u_{0},\quad \text { and } \quad v(\cdot , t) \rightarrow 0 \quad \text { as } t \rightarrow \infty \end{aligned}$$

uniformly with respect to $x\in \Omega$.