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

Abstract

We study blow-up of radially symmetric solutions of the nonlinear heat equation u t = Δ u + | 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 > p S : = N + 2 N − 2 . Our first goal is to analyze a threshold behavior for solutions with initial data u 0 = λ v , 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 λ ⩾ λ ⁎ . We show that there exist at most finitely many exceptional values λ 1 = λ ⁎ < λ 2 < ⋯ < λ k such that, for all λ > λ ⁎ with λ ≠ λ j ( 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.