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

摘要

We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|~(p-1)u either on R~N 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 u_0=λv, where v∈C∩H~1 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 Δ≥λ*. We show that there exist at most finitely many exceptional values λ1=λ*<2<λ_2<···<λ_k 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.