Type II blowup of solutions to a simplified Keller-Segel system in two dimensional domains
The following problem to the Keller-Segel system \[ \begin{aligned} \left.\begin{aligned} u_t&=\nabla(\nabla u-u \nabla u)\\ 0&=\Delta v-v+u\end{aligned} \right\} \quad &\text{in }\Omega,\\ \frac{\partial u}{\partial \nu }= \frac{\partial v}{\partial \nu }=0 \quad &\text{on } \partial \Omega,\\ u(\cdot,0)=u_0 \quad &\text{in } \Omega,\end{aligned} \] is considered. Here \(\Omega\) is a bounded domain in \(\mathbb R^2\) with smooth boundary \(\partial \Omega, \nu =\nu (x)\) in the unit normal outer vector on \(\partial \Omega\), and \(u_0\) is smooth and non-negative in \(\overline{\Omega}\). The author investigates whether finite time blowup solutions to the system exhibit Type II blowup or not. It is shown existing of the radial solution whose blowup speed is faster than that of backward self similar solution.
scientific article; zbMATH DE number 5142234