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
Information about the author:
Anna S. Smirnova, Postgraduate Student, Department of Fundamental Mathematics, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID: https://orcid.org/0000-0003-4172-2811, firstname.lastname@example.org
The author have read and approved the final manuscript.
Conflict of interest: The author declare no conflict of interest.