Prevention of blow-up by fast diffusion in chemotaxis
The authors consider the following coupled parabolic system \[ \begin{cases} u_t=\nabla\cdot \left(\frac{1}{(1-u)^{\alpha}}\nabla u-\chi(u,v)\nabla v\right),\\ v_t=d\Delta v+f(u)-\beta v,\\ \frac{\partial u}{\partial \nu}|_{\partial\Omega}=0,\quad \frac{\partial v}{\partial \nu}|_{\partial\Omega}=0,\\ u|_{t=0}=u_0,\quad v|_{t=0}=v_0,\quad 0\leq u_0(x)<1,\quad v_0(x)\geq 0,\end{cases}\tag{1} \] where \(\Omega\subset\mathbb{R}^n\) is a bounded \(C^{2+\gamma}\) domain for some \(\gamma\in (0,1)\), \(\alpha>0\), \(\beta\geq 0\), \(d>0\) are constants, while positive functions \(\chi\) and \(f\) are given smooth functions representing the cross diffusion coefficient and a source term of the second species, respectively. The system (1) is a modified Keller-Segel model which describes directed cell movement in response to chemical concentration gradient. \(u\) represents the cell density and \(v\) accounts for the chemical (external signal) concentration. The diffusion coefficient \(\frac{1}{(1-u)^{\alpha}}\) goes to infinity when \(u\) tends to \(1\). In this article, they want to show that such a ``fast diffusion'' here leads to global existence of solutions in bounded domains for any given initial data irrespective of the spatial dimension. In fact, they show this conclusion for \(n=1\), \(\alpha>1\) and \(n\geq 2\), \(\alpha\geq 2\). The key estimates that make this possible have been obtained by a technique that uses ideas from Moser's iterations. And the conclusion for the cases \(n=1\), \(\alpha\in (0,1]\) and \(n\geq 2\), \(\alpha\in (0,2)\) are still open, but they give some numerical experiments which implies the global existence of solutions for \(n=1\), \(\alpha\in (0,1]\).
scientific article; zbMATH DE number 5643993