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

TL;DR

This paper constructs the biharmonic heat kernel on edge manifolds, analyzes its properties, and applies it to establish short-time existence for certain nonlinear fourth-order equations like the Cahn-Hilliard equation.

Contribution

It introduces a microlocal approach to the biharmonic heat kernel on edge manifolds and applies it to nonlinear PDEs with singularities.

Findings

Established mapping properties of the biharmonic heat operator

Proved short-time existence for semi-linear fourth-order equations

Derived asymptotic behavior of solutions near singularities

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.