Some local estimates and a uniqueness result for the entire biharmonic heat equation

TL;DR

This paper establishes local estimates and a uniqueness result for smooth solutions to the biharmonic heat equation on Euclidean space under a specific boundedness condition involving the Laplacian.

Contribution

It introduces new local estimates for solutions with bounded Laplacian squared and proves their implications for the uniqueness of such solutions.

Findings

Local space-time estimates for solutions

Uniqueness of smooth solutions under boundedness condition

Boundedness condition involving the Laplacian squared

Abstract

We consider smooth solutions to the biharmonic heat equation on Euclidean space for which the square of the Laplacian at time t is globally bounded from above by k/t for some k in R, for all t in [0,T]. We prove local, in space and time, estimates for such solutions. We explain how these estimates imply uniqueness of smooth solutions in this class.