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

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 $n\ge 1$, $m\ge 1$ 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.

arXiv admin note: text overlap with arXiv:1904.06797