Prescribed Signal Concentration on the Boundary: Radial Solutions to a Chemotaxis System with Proliferation and Nonlinear Consumption

The chemotaxis model

$$\begin{aligned} \left\{ \begin{array}{l} \begin{aligned} & u_t = \Delta u-\chi \nabla \cdot (g(u)\nabla v)+ku-\mu u^l, & x\in \Omega ,\ t>0& ,\\ & v_t=\Delta v-g(u)v, & x\in \Omega ,\ t>0& \\ \end{aligned} \end{array} \right. \end{aligned}$$

is considered under the boundary conditions \(\frac{\partial u}{\partial \nu }-\chi g(u)\frac{\partial v}{\partial \nu }=0\) and \(v=v_*\) on \(\partial \Omega\), where \(\Omega \subset {\mathbb {R}}^n\) (\(n\in \{2,3\}\)) is a ball and \(v_*\) is a given positive constant. Here, the parameters \(\chi ,k,\mu\) are positive and the function \(g\in C^1([0,\infty ))\) satisfies \(0\le g(u)\le u^{\beta }\) with \(\frac{5}{6}\le \beta <1\). For all suitably regular initial data, the present work provides a result on global boundedness of the radially symmetric classical solution in two dimensions when \(\beta =\chi\) and \(1<l<\frac{5}{3}\), while the global existence of the radially symmetric weak solution is established in three-dimensional settings.