Infinite time blow-up and slow decay for the six dimensional energy-critical heat equation with self-similarly decaying initial data

TL;DR

This paper investigates the six-dimensional energy-critical heat equation, demonstrating solutions with infinite-time blow-up and slow decay, highlighting the complex dynamics beyond classical self-similar behavior.

Contribution

It constructs solutions with infinite-time blow-up and slow decay, showing the non-uniqueness of blow-up and decay rates depending on initial data.

Findings

Existence of sign-changing solutions with infinite-time blow-up.

Nonnegative solutions decay more slowly than self-similar rate.

Blow-up and decay rates depend on initial data construction.

Abstract

We consider the six dimensional energy-critical semilinear heat equation with self-similarly decaying initial data. Our main result shows the existence of sign-changing solutions that exhibit infinite-time blow-up and nonnegative solutions that decay strictly more slowly than the self-similar rate. Moreover, the blow-up and decay rates are not uniquely determined by the decay rate of the initial data, but exhibit a certain flexibility depending on the construction. The proof is based on gluing suitably rescaled bubbles to forward self-similar solutions.