Abstract :
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.
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