Solvability in Hölder spaces of an inverse problem on chaotic dynamics of a polymer chain
Authors:
A. S. Fomenko and V. N. Starovoitov
Journal:
Quart. Appl. Math.
MSC (2020):
Primary 35K58, 35R30; Secondary 35Q92
DOI:
https://doi.org/10.1090/qam/1703
Published electronically:
January 7, 2025
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Additional Information
Abstract: This paper deals with an inverse problem for a quasi-linear parabolic partial differential equation that contains a nonlocal in time term that includes an integral of the solution over the entire time interval where the problem is considered. Moreover, the overdetermination condition for this inverse problem is nonlocal in space. So, the problem has a double nonlocality: in time and in space. This problem describes the chaotic dynamics of a polymer chain in a liquid. The time variable corresponds to the arc length along the chain. The time nonlocality stands in the argument of the so-called interaction potential. Its appearance is caused by the fact that the motion of each link of a chain is affected by all other links through the surrounded fluid. The solvability of the problem in Hölder spaces in the case of a bounded interaction potential is proven. Moreover, the solvability is also proven in the case of an unbounded potential with certain restrictions on its behavior on some finite interval.
References
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References
- V. N. Starovoitov and B. N. Starovoitova, Modeling the dynamics of polymer chains in water solution. Application to sensor design, J. Phys. Conf. Ser. 894 (2017), 012088. \PrintDOI{10.1088/1742-6596/894/1/012088}
- Aleksey I. Prilepko, Dmitry G. Orlovsky, and Igor A. Vasin, Methods for solving inverse problems in mathematical physics, Monographs and Textbooks in Pure and Applied Mathematics, vol. 231, Marcel Dekker, Inc., New York, 2000. MR 1748236
- Victor N. Starovoitov, Boundary value problem for a global-in-time parabolic equation, Math. Methods Appl. Sci. 44 (2021), no. 1, 1118–1126. MR 4185303, DOI 10.1002/mma.6816
- Christoph Walker, Strong solutions to a nonlocal-in-time semilinear heat equation, Quart. Appl. Math. 79 (2021), no. 2, 265–272. MR 4246493, DOI 10.1090/qam/1579
- John R. Cannon and Yan Ping Lin, Determination of a parameter $p(t)$ in some quasilinear parabolic differential equations, Inverse Problems 4 (1988), no. 1, 35–45. MR 940789
- A. I. Kozhanov, Parabolic equations with unknown time-dependent coefficients, Comput. Math. Math. Phys. 57 (2017), no. 6, 956–966. MR 3667395, DOI 10.1134/S0965542517060082
- A. Friedman, Partial differential equations of parabolic type, Robert E. Krieger Publishing Copmany, Malabar, Florida, 1983.
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
- V. N. Starovoitov, Solvability of a boundary value problem of chaotic dynamics of polymer molecule in the case of bounded interaction potential, Sib. Èlektron. Mat. Izv. 18 (2021), no. 2, 1714–1719 (Russian, with English summary). MR 4356405
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Additional Information
A. S. Fomenko
Affiliation:
Lavrentyev Institute of Hydrodynamics, Lavrentyev prospekt 15, Novosibirsk, 630090, Russia
Email:
as.fomenko1@gmail.com
V. N. Starovoitov
Affiliation:
Lavrentyev Institute of Hydrodynamics, Lavrentyev prospekt 15, Novosibirsk, 630090, Russia
MR Author ID:
224821
ORCID:
0000-0002-0392-3180
Email:
starovoitov@hydro.nsc.ru
Keywords:
Nonlocal in time parabolic equation,
inverse problem,
solvability,
Hölder spaces,
polymer chain,
chaotic dynamics
Received by editor(s):
December 3, 2024
Received by editor(s) in revised form:
December 10, 2024
Published electronically:
January 7, 2025
Additional Notes:
The first author was supported by the Russian Science Foundation, Grant No. 23-21-00261, https://rscf.ru/project/23-21-00261/. The second author was supported by the Russian Science Foundation, Grant No. 23-21-00261, https://rscf.ru/project/23-21-00261/.
Article copyright:
© Copyright 2025
Brown University
