Blowup behavior of solutions to the rescaled Jäger-Luckhaus system.
The Jäger-Luckhaus system was introduced by Jäger and Luckhaus for the purpose of modeling chemotaxis. The Jäger-Luckhaus system consists of partial differential equations of parabolic and elliptic types, and the behavior of its solutions depends on a parameter \(\lambda\), which in turn is related to the \(L^1\)-norm of the initial datum. When \(\lambda\ll 1\), the solutions of the Jäger-Luckhaus system exist for all time. In contrast, its solutions associated with \(\lambda\gg 1\) blow up in a finite time. In this paper, the authors rescale the Jäger-Luckhaus system around the blowup points and study the infinite time blowup behavior of the rescaled system. They show that the solutions of the rescaled system converge to a sum of functions with support concentrated at the blowup points and with integral equal to \(8\pi\) (for blowup points inside the domain) or \(4\pi\) (for blowup points on the boundary). The interpretation of this result is that the quantization of chemotactic collapse occurs. The proof is very technical and is divided into a major proposition and several lemmas.
scientific article; zbMATH DE number 2002549