Global, decaying solutions of a focusing energy-critical heat equation in $\mathbb{R}^4$

TL;DR

This paper proves that solutions to a focusing energy-critical nonlinear heat equation in four dimensions with initial data below a certain threshold are global and decay to zero, using concentration-compactness and rigidity methods.

Contribution

It establishes the global existence and decay of solutions below the stationary solution's energy and norm, extending the understanding of the equation's long-term behavior.

Findings

Solutions with sub-threshold energy decay to zero.

Finite-time blow-up is ruled out for these solutions.

The approach combines concentration-compactness with backward uniqueness.

Abstract

We study solutions of the focusing energy-critical nonlinear heat equation $u_t = \Delta u - |u|^2u$ in $\mathbb{R}^4.$ We show that solutions emanating from initial data with energy and $\dot{H}^1-$norm below those of the stationary solution $W$ are global and decay to zero, via the "concentration-compactness plus rigidity" strategy of Kenig-Merle. First, such global solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza, Seregin and Sverak in an argument similar to that of Kenig and Koch for the Navier-Stokes equations.