On the global solvability of the diffusion equation of Satsuma and Mimura (Q1174113)

The author proves a conjecture by \textit{J. Satsuma} and \textit{M. Mimura} [J. Phys. Soc. Japan, 894-900 (1985)] concerning the global solvability of a nonlinear diffusion equation on \(\mathbb{R}\times\mathbb{R}_ +\) with a singular integral term, which arises in population dynamics: There is a critical value \(I_ 0\) of the (constant) total population \(I\) such that the solution exists globally (with respect to time) whenever \(I\leq I_ 0\), but blows up in finite time whenever \(I>I_ 0\). The first part of the conjecture (global existence for \(I\leq I_ 0)\) was originally proved by \textit{L. von Wolfersdorf} [Math. Nachr. 145, 243-254 (1990)] who introduced a change of variables providing a one-to-one correspondence between the solutions \(u\) of the Satsuma-Mimura equation and the non- vanishing solutions \(w\) of a complex heat equation; finite-time blow-up of \(u\) corresponds to the vanishing of \(w\) at some point \((x_ 0,t_ 0)\in\mathbb{R}\times\mathbb{R}_ +\). In the paper at hand, the author proves both parts of the conjecture by means of an elegant geometric argument based on the maximum principle.