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=Δu+|u|^p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume that N?3 and . 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^?<λ₂λk 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.