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 $$x\in \Omega $$.