Energy Non-collapsing and Refined Blowup for a Semilinear Heat Equation

TL;DR

This paper investigates the structure of blowup solutions in a semilinear heat equation, establishing conditions for non-collapsing blowup, characterizing blowup types, and estimating the blowup set’s Hausdorff dimension.

Contribution

It provides new results on the blowup set structure, including conditions for non-collapsing blowup and bounds on the Hausdorff dimension, along with examples of collapsing and non-collapsing blowups.

Findings

Blowup set is empty for non-collapsing blowup in subcritical case.

All finite-time non-collapsing blowups are type II in the critical case.

Hausdorff dimension of the blowup set is bounded by N-2-4/(p-1) for certain parameters.

Abstract

Refined structures of blowup for non-collapsing maximal solution to a semilinear parabolic equation are studied. We will prove that the blowup set is empty for non-collapsing blowing-up in subcritical case, and all finite time non-collapsing blowing-up must be type II in critical case. When p>p_S=(N+2)/(N-2) for N>=3, the Hausdorff dimension of the blowup set for maximal solution whose energy is non-collapsing is shown to be no greater than N-2-4/(p-1), which answers a question proposed in [7] positively. At the end of this paper, we also present some new examples of collapsing and non-collapsing blowups.