Threshold and generic type I behaviors for a supercritical nonlinear heat equation

We study blow-up of radially symmetric solutions of the nonlinear heat equation ut = u + |u|p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume that N 3 and p > pS := N+2 N−2 . Our first goal is to analyze a threshold behavior for solutions with initial data u0 = λv, where v ∈ C ∩ H1 and v 0, v ≡ 0. It is known that there exists λ ∗ > 0 such that the solution converges to 0 as t→∞if 0 < λ < λ ∗, while it blows up in finite time if λ λ ∗. We show that there exist at most finitely many exceptional values λ1 = λ ∗ < λ2 < · · · < λk such that, for all λ > λ ∗ with λ = λj (j = 1, 2, . . . , k), the blow-up is complete and of type I with a flat local profile. Our method is based on a combination of the zero-number principle and energy estimates. In the second part of the paper, we employ the very same idea to show that the constant solution κ attains the smallest rescaled energy among all non-zero stationary solutions of the rescaled equation. Using this result, we derive a sharp criterion for no blow-up. © 2011 Elsevier Inc. All rights reserved.