Geodesic
On the Approximation of Solutions to the Heat Equation in the Lebesgue Class $L^2$ by More Regular Solutions
Matematičeskie zametki, Tome 111 (2022) no. 5, pp. 778-794.

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A criterion for the approximability of all solutions of the heat equation in a bounded cylindrical domain that belong to the Lebesgue class by more regular (e.g., Sobolev) solutions of the same equation in a bounded cylindrical domain with larger base is obtained. Namely, the complement of the smaller base to the larger one must have no (nonempty connected) compact components. As an important corollary, we prove a theorem on the existence of a doubly orthogonal basis for the corresponding pair of Hilbert spaces.
Keywords: heat equation, approximation theorem. 
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A. A. Shlapunov. On the Approximation of Solutions to the Heat Equation in the Lebesgue Class $L^2$ by More Regular Solutions. Matematičeskie zametki, Tome 111 (2022) no. 5, pp. 778-794. http://geodesic.mathdoc.fr/item/MZM_2022_111_5_a10/

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