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

TL;DR

The paper addresses the ill-posed Cauchy problem for the polyharmonic heat operator $\mathcal{L}_m=\partial_t+(-\Delta)^m$ in a cylinder $\Omega_T$, seeking recovery of a solution from Cauchy data on a portion of the lateral boundary. It deploys an integral representation approach based on the fundamental solution $\Phi_m$ and Green's formula to define parabolic potentials $I_{\Omega,T_1}$, $G_{\Omega,T_1}$, and $V^{(j)}_{S,T_1}$, and establishes a uniqueness result alongside a solvability criterion expressed via an auxiliary smooth function $F$ with $\mathcal{L}_m F=0$ on a larger domain $D_T$ and boundary matching on $\Omega_T^+$. The main contributions include a sharp solvability condition, a jump-relations lemma for boundary potentials, and a discussion of extensions to anisotropic Sobolev spaces and Carleman-type representations. These results connect ill-posed parabolic Cauchy problems to real-analytic continuation of parabolic potentials, providing a framework for constructive solution representations and potential numerical methods.

Abstract

We consider the ill-posed Cauchy problem for the polyharmonic heat equation on recovering a function, satisfying the equation $(\partial _t + (- Δ)^m) u=0$ in a cylindrical domain in the half-space ${\mathbb R}^n \times [0,+\infty)$, where $n\geq 1$, $m\geq 1$ and $Δ$ 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.

Theorems & Definitions (10)

• Lemma 2.1
• proof
• Theorem 2.2: A Uniqueness Theorem
• proof
• Theorem 2.3: Solvability criterion
• Lemma 2.4
• Lemma 2.5
• proof
• Corollary 2.6
• proof