Geodesic
$L_p$-approximations for solutions of parabolic differential equations on manifolds
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 24 (2022) no. 3, pp. 297-303

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The paper considers the Cauchy problem for a parabolic partial differential equation in a Riemannian manifold of bounded geometry. A formula is given that expresses arbitrarily accurate (in the $L_p$-norm) approximations to the solution of the Cauchy problem in terms of parameters - the coefficients of the equation and the initial condition. The manifold is not assumed to be compact, which creates significant technical difficulties – for example, integrals over the manifold become improper in the case when the manifold has an infinite volume. The presented approximation method is based on Chernoff theorem on approximation of operator semigroups.
Keywords: parabolic equation on manifold, Cauchy problem, representation of solutions, approximation of solutions, manifold of bounded geometry, semigroup of operators. 
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     author = {A. S. Smirnova},
     title = {$L_p$-approximations for solutions of parabolic differential equations on manifolds},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {297--303},
     publisher = {mathdoc},
     volume = {24},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2022_24_3_a3/}
}
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A. S. Smirnova. $L_p$-approximations for solutions of parabolic differential equations on manifolds. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 24 (2022) no. 3, pp. 297-303. http://geodesic.mathdoc.fr/item/SVMO_2022_24_3_a3/