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

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