Model of chemotaxis with threshold density and singular diffusion
The existence of classical and weak solutions is studied for a chemotaxis model including a volume-filling effect (the density of cells cannot exceed a given threshold), a singular diffusion (the cells diffusion becomes infinite as the density of cells approaches its threshold value), and a vanishing velocity in the regions where the density of cells vanishes or attains its threshold value. If \(u\) and \(v\) denote the cell density and the concentration of chemoattractant, respectively, and the threshold value is normalised to one, the model reads \[ \begin{aligned} \partial_{t} u & = \nabla\cdot \left( D(u) \nabla u - u h(u) \nabla v \right),\\ \partial_{t} v & = \Delta v + \gamma u - \sigma v , \end{aligned} \] in \((0,\infty)\times\Omega\), supplemented with homogeneous Neumann boundary conditions and initial data \(0\leq u_{0}< 1\) and \(v_{0}\geq 0\). Here, \(\Omega\) is an open bounded subset of \(\mathbb R^{N}\), \(\gamma>0\) and \(\sigma\geq 0\) are constants, and the functions \(D\) and \(h\) are \(C^2\)-smooth in \([0,1)\), and satisfy \[ D(r)\geq C_D\;(1-r)^{-\alpha}, \quad 0<h(r)\leq C_h\;(1-r)^\beta, \quad r\in [0,1), \] for some \(\alpha>0\) and \(\beta\geq 0\). It is first shown that, if \(\beta\geq (2-\alpha)/2\), then there is a unique global classical solution \((u,v)\) and it satisfies \(\|u(t)\|_\infty<1\) for all \(t\geq 0\). Next, if \(\alpha\in (0,1)\) and \(D\) is additionally bounded from above by \(C^D\;(1-r)^{-\alpha}\), the existence of a weak solution is proved, and uniqueness as well under some additional assumptions on \(D\) and \(h\). Steady states and their properties are also studied.
scientific article; zbMATH DE number 5720163