The oriented movement of biological individuals sensitive to a gradient of chemical substance (this movement is named chemotaxis) is studied using some initial boundary value problems for the Keller-Segal systems \[ \left. \begin{aligned}{\partial u\over\partial t} &=\nabla\cdot(\nabla u- xu\nabla v)\\ {\partial v\over\partial t} &=\Delta v-\gamma v+\alpha u\end{aligned} \right\}\quad\text{in }\Omega,\quad t>0, \]
\[ {\partial u\over\partial u}= {\partial v\over\partial u}= 0\quad\text{on }\partial\Omega, \]
\[ u|_{t=0}= u_0,\quad v|_{t= 0}= v_0\quad \text{in }\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(u\) is the density of the population, \(v\) is the concentration of the chemical, \(\chi\), \(\tau\), \(\gamma\), \(\alpha\) are positive constants; \(u_0\), \(v_0\) are nonnegative smooth functions on \(\overline\Omega\). One of the features is the finite-time blow-up of \(u\), having singularities of the \(\delta\)-functions at the blow-up time (such a blow-up phenomenon is referred to as chemotactic collapse). The chemotactic collapse may depend on the space dimension \(n\).
In this paper the author reviews mathematical results, in the case \(n=2\), concerning certain models of chemotaxis which are versions of Keller-Segal systems.
For the entire collection see [Zbl 0943.00054].