The biharmonic heat operator on edge manifolds and non-linear fourth order equations

Abstract

We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping properties of the corresponding biharmonic heat operator on certain Banach spaces. This yields short time existence for a class of semi-linear equations of fourth order, including for example the Cahn–Hilliard equation. We also obtain asymptotics of the solutions near the edge singularity.