Geodesic
On the heat integral identity for unbounded functions
Problemy analiza, Tome 7 (2018) no. 3, pp. 3-11 Cet article a éte moissonné depuis la source Math-Net.Ru

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The well known heat integral identity in an unbounded strip is extended to a class of unbounded functions both at $x$ near infinity and $t$ near zero. Continuity of derivatives are relaxed to differentiability in the $L^1_{loc}$-Sobolev sense.
Keywords: heat operator, heat kernel. 
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A. Biryuk; A. Svidlov; E. Silchenko. On the heat integral identity for unbounded functions. Problemy analiza, Tome 7 (2018) no. 3, pp. 3-11. http://geodesic.mathdoc.fr/item/PA_2018_7_3_a0/

[1] Biryuk A., Svidlov A., Silchenko E., “Use of Holder Norms for Derivatives Bounds for Solutions of Evolution Equations”, International Conference “Geometrical analysis and its applications” (Volgograd, May, 30 – June, 3, 2016), 27–29 (in Russian)

[2] Mikhailov V. P., Partial Differential Equations, Second edition, Nauka, M., 1983 (in Russian) | MR