ISSN 2079-6900 (Print)
ISSN 2587-7496 (Online)

#### Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.24.202203.297-303

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 58J35, 47D06, 65M12, 35K15, 35C20, 58D25

#### $L_p$-approximations for solutions of parabolic differential equations on manifolds

A. S. Smirnova

Higher School of Economics (Nizhny Novgorod, Russian Federation)

Abstract. 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.

Key Words: parabolic equation on manifold, Cauchy problem, representation of solutions, approximation of solutions, manifold of bounded geometry, semigroup of operators

For citation: A. S. Smirnova. $L_p$-approximations for solutions of parabolic differential equations on manifolds. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 24:3(2022), 297–303. DOI: https://doi.org/10.15507/2079-6900.24.202203.297-303

Submitted: 01.07.2022; Revised: 10.08.2022; Accepted: 24.08.2022