Geodesic
On Carleman-type formulas for solutions to the heat equation
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 4, pp. 421-433

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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
Mots-clés : Carleman formulas. 
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     title = {On {Carleman-type} formulas for solutions to the heat equation},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
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Ilya A. Kurilenko; Alexander A. Shlapunov. On Carleman-type formulas for solutions to the heat equation. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 4, pp. 421-433. http://geodesic.mathdoc.fr/item/JSFU_2019_12_4_a2/