This article is cited in 7 papers

On Carleman-type formulas for solutions to the heat equation

Ilya A. Kurilenko, Alexander A. Shlapunov

Institute of Mathematics and Computer Science, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041, Russia

Abstract: We apply the method of integral representations to study the ill-posed Cauchy problem for the heat equation. More precisely we recover a function, satisfying the heat equation in a cylindrical domain, via its values and the values of its normal derivative on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural (anisotropic) spaces (Sobolev and Hölder spaces, etc). Finally, we obtain a uniqueness theorem for the problem and a criterion of its solvability and a Carleman-type formula for its solution.

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

UDC: 517.955

Received: 28.02.2019
Received in revised form: 11.03.2019
Accepted: 20.04.2019

Language: English

DOI: 10.17516/1997-1397-2019-12-4-421-433

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