Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models (Q2321128)

The authors of the paper under review study the existence of weak solutions and the upper bound on the blow-up time for \textit{time discrete parabolic-elliptic Keller-Segel models for chemotaxis} in \(\mathbb{R}^2\). The paper can be considered as a continuation of the program initiated by the first author and collaborators, on ``translating mathematical techniques from continuous to discrete solutions''. The parabolic-elliptic Keller-Segel model mathematically can be described as the Cauchy problem associated to the system of PDEs \[ \left\{ \begin{array}{l} \partial_t n = \operatorname{div}(\nabla n - n\nabla c),\\ -\Delta c + \alpha c = n, \end{array} \right. \] where \(n\) describes the cell density of some bacteria that are attracted by a chemical substance, and so \(c\) stands for the density of the chemo-attractant. \(\alpha\ge 0\) is a given parameter that measures the degradation rate of the chemical substance. The previous system received lots of attention in the literature in various settings. It is well-known that if the initial mass of the bacteria population is above a certain threshold, namely \(8\pi\), the solutions blow up in finite time (this time depends on the critical mass, on the value \(\alpha\) and on the second moment of the initial bacteria density). The main results of this paper aim to reproduce such blow-up phenomena for solutions to various cases of time-discrete equations (in particular these are associated to numerical schemes, the implicit Euler and Runge-Kutta ones). It worth mentioning that the blow-up results obtained for the discrete solutions are the same as for their continuous counterparts, showing also that the continuous methods carry over to semi-discrete cases.