Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition Arturo de Pablo Departamento de Matem´aticas, U. Carlos III de Madrid 28911 Legan´ es, Spain. Fernando Quir ´ os Departamento de Matem´aticas, U. Aut´onoma de Madrid 28049 Madrid, Spain. Julio D. Rossi Departamento de Matem´atica, F.C.E y N., UBA (1428) Buenos Aires, Argentina. Abstract We study nonnegative solutions of the porous medium equation with a source and a nonlinear flux boundary condition,    u t =(u m ) xx + u p in (0, ∞) × (0,T ), −(u m ) x (0,t)= u q (0,t) t ∈ (0,T ), u(x, 0) = u 0 (x) in (0, ∞), where m> 1, p, q > 0 are parameters. For every fixed m we prove that there are two critical curves in the (p, q) plane: (i) the critical existence curve, separating the region where every solution is global from the region where there exist blowing up solutions, and (ii) the Fujita curve, separating a region of parameters in which all solutions blow up from a region where both global in time solutions and blowing up solutions exist. In the case of blow-up we find the blow-up rates, the blow-up sets and the blow-up profiles, showing that there is a phenomenon of asymptotic simplification. If 2q<p + m the asymptotics are governed by the source term. On the other hand, if 2q>p + m the evolution close to blow-up is ruled by the boundary flux. If 2q = p + m both terms are of the same order. AMS Subject Classification: 35B40, 35B33, 35K65. Keywords and phrases: blow-up, porous medium equation, nonlinear boundary condition. 1