Blow-up Sets and Fujita Type Curves for a Degenerate Parabolic System with Nonlinear Boundary Conditions F ERNANDO QUIR ´ OS & J ULIO D. ROSSI ABSTRACT. We study nonnegative solutions of two porous medium equations with nonlinear coupled boundary conditions and nonnegative nontrivial compactly supported initial data. We describe them in terms of the different parameters appearing in the problem, when solutions blow up in a finite time, and when they exist globally in time. We find three regions, bounded by two curves. One of them is a Fujita type curve. In the first region every solution is global, in the second every nontrivial solution blows up, and in the last one both behaviours coexist. In the blow-up case we find the blow-up rates and the blow-up sets. In particular, we prove that under certain conditions the blow-up sets of the two components of the system are different. This is the first known example of a nontrivially coupled parabolic sys- tem showing such a behaviour. 1. I NTRODUCTION In this paper we deal with the following degenerate parabolic system,  u t = (u m ) xx , v t = (v n ) xx , x> 0, 0 <t<T, (1.1) where m, n> 1. Thus, we are dealing with two porous medium equations. These equations are complemented with nonlinear coupled boundary conditions,  −(u m ) x (0,t) = v p (0,t), −(v n ) x (0,t) = u q (0,t), 0 <t<T, (1.2) 629 Indiana University Mathematics Journal c , Vol. 50, No. 1 (2001)