Backward uniqueness for the heat equation in cones

TL;DR

This paper investigates the conditions under which solutions to the heat equation in conical domains are uniquely determined by their zero values at a certain time, extending known results from half-spaces to wider cones.

Contribution

It establishes backward uniqueness for the heat equation in cones with opening angles larger than 110 degrees, including cases with bounded measurable lower-order terms.

Findings

Backward uniqueness holds for cones with opening angles >110°

The result applies to heat equations with bounded measurable lower-order coefficients

The property fails for cones with opening angles <90°

Abstract

It is known that a bounded solution of the heat equation in a half-space which becomes zero at some time must be identically zero, even though no assumptions are made on the boundary values of the solutions. In a recent example, Luis Escauriaza showed that this statement fails if the half-space is replaced by cones with opening angle smaller than 90 degrees. Here we show the result remains true for cones with opening angle larger than 110 degrees. The proof covers heat equations having lower-order terms with bounded measurable coefficients.