When are solutions to the biharmonic heat equation unique?

What is it about?

In this article we find a new way to guarantee that solutions to the biharmonic heat equation in Euclidean space are unique. We do this by proving some new local estimates.

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Why is it important?

Uniqueness for the biharmonic heat equation is a difficult problem. We know that without any restrictions on the solution uniqueness does not hold. This can be seen by explicit construction of Tychonoff-style solutions. (We do this also in the paper.) More important however than giving a new condition that guarantees uniqueness is the proof, and this relies on local estimates. These local estimates in turn follow from energy estimates, that are reasonably elementary, and a blowup argument, which is also fairly straightforward. The blowup argument has been previously used by the first author in the context of Ricci flow and the heat equation.

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