Asymptotic simplification for a reaction–diffusion problem with a nonlinear boundary condition

We study non‐negative solutions of the porous medium equation with a source and a nonlinear flux boundary condition, ut = (u^m)xx + u^p in (0, ∞), × (0, T); − (u^m)x (0, t) = u^q (0, t) for t ∈ (0, T); u (x, 0) = u0 (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.