On the ill-posed Cauchy problem for the polyharmonic heat equation

TL;DR

This paper addresses the ill-posed Cauchy problem for the polyharmonic heat equation, establishing uniqueness and solvability criteria based on analytic continuation of associated potentials in a cylindrical domain.

Contribution

It provides a uniqueness theorem and solvability conditions for the ill-posed problem involving the polyharmonic heat equation with partial boundary data.

Findings

Proved a uniqueness theorem for the problem.

Established a solvability criterion via analytic continuation.

Connected the problem's solvability to properties of parabolic potentials.

Abstract

We consider the ill-posed Cauchy problem for the polyharmonic heat equation on recovering a function, satisfying the equation $(\partial _t + (- \Delta)^m) u=0$ in a cylindrical domain in the half-space ${\mathbb R}^n \times [0,+\infty)$, where $n\geq 1$, $m\geq 1$ and $\Delta$ is the Laplace operator, via its values and the values of its normal derivatives up to order $(2m-1)$ on a given part of the lateral surface of the cylinder. We obtain a Uniqueness Theorem for the problem and a criterion of its solvability in terms of the real-analytic continuation of parabolic potentials, associated with the Cauchy data.