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

Ilya A. Kurilenko ; Alexander A. Shlapunov

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 2, pp. 194-203

Résumé

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 $n\geqslant 1$, $m\geqslant 1$ 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.

Détail

Keywords: the polyharmonic heat equation, ill-posed problems, integral representation method.