Existence of a separating norm for the semilinear heat equation

Abstract

We consider the initial value problem of the semilinear heat equation in the Fujita supercritical range for which global-in-time solutions may exist. The main result shows the existence of a separating norm $\|\cdot\|_X$ in the sense that, for nonnegative initial data $u_0\in L^\infty(\mathbf{R}^n)$, the smallness of $\|u_0\|_X$ implies global-in-time existence, whereas the largeness of $\|u_0\|_X$ implies finite-time blow-up. The separating norm is unique up to equivalence on nonnegative data.