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

Abstract

We study solutions of the focusing energy-critical nonlinear heat equation $u_t=\Delta u-|u{|}^{2}u$ 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.

41 pages

References in corpus (2)

• Blow-up of critical Besov norms at a potential Navier-Stokes singularity
• Well-posedness of the supercritical Lane-Emden heat flow in Morrey spaces