On the biharmonic heat equation on complete Riemannian manifolds

TL;DR

This paper investigates the biharmonic heat equation on complete Riemannian manifolds, providing decay estimates, uniqueness criteria, and conservation laws under geometric assumptions.

Contribution

It introduces new decay estimates and uniqueness criteria for solutions of the biharmonic heat equation on manifolds with specific curvature conditions.

Findings

Exponential decay estimates for the biharmonic heat kernel.

A proven uniqueness criterion for the Cauchy problem.

Conservation law and uniform L-infinity bounds for solutions.

Abstract

We study entire solutions of the biharmonic heat equation on complete Riemannian manifolds without boundary. We provide exponential decay estimates for the biharmonic heat kernel under assumptions on the lower bound of Ricci curvature and noncollapsing of unit balls. And we prove a uniqueness criteria for the Cauchy problem. As corollaries we prove the conservation law for the biharmonic heat kernel and a uniform L-infinite estimate for entire solutions starting with bounded initial data.