Geodesic
Global solvability of a kernel determination problem in 2D heat equation with memory
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 18 (2025) no. 1, pp. 14-24.

Voir la notice de l'article provenant de la source Math-Net.Ru

The inverse problem of determining two dimensional kernel in the integro-differential heat equation is considered in this paper. The kernel depends on the time variable $t$ and space variable $x$. Assuming that kernel function is given, the direct initial-boundary value problem with Neumann conditions on the boundary of a rectangular domain is studied for this equation. Using the Green's function, the direct problem is reduced to integral equation of the Volterra-type of the second kind. Then, using the method of successive approximation, the existence of a unique solution of this equation is proved. The direct problem solution on the plane $y = 0$ is used as an overdetermination condition for inverse problem. This problem is replaced by an equivalent auxiliary problem which is more suitable for further study. Then the last problem is reduced to the system of integral equations of the second order with respect to unknown functions. Applying the fixed point theorem to this system in the class of continuous in time functions with values in the Hölder spaces with exponential weight norms, the main result of the paper is proved. It consists of the global existence and uniqueness theorem for inverse problem solution.
Keywords: integro-differential heat equation, inverse problem, Banach theorem, uniqueness.
Mots-clés : existence 
@article{JSFU_2025_18_1_a1,
     author = {Durdumurod K. Durdiev and Zhavlon Z. Nuriddinov},
     title = {Global solvability of a kernel determination problem in {2D} heat equation with memory},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {14--24},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {2025},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2025_18_1_a1/}
}
TY  - JOUR
AU  - Durdumurod K. Durdiev
AU  - Zhavlon Z. Nuriddinov
TI  - Global solvability of a kernel determination problem in 2D heat equation with memory
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2025
SP  - 14
EP  - 24
VL  - 18
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2025_18_1_a1/
LA  - en
ID  - JSFU_2025_18_1_a1
ER  - 
%0 Journal Article
%A Durdumurod K. Durdiev
%A Zhavlon Z. Nuriddinov
%T Global solvability of a kernel determination problem in 2D heat equation with memory
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2025
%P 14-24
%V 18
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2025_18_1_a1/
%G en
%F JSFU_2025_18_1_a1
Durdumurod K. Durdiev; Zhavlon Z. Nuriddinov. Global solvability of a kernel determination problem in 2D heat equation with memory. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 18 (2025) no. 1, pp. 14-24. http://geodesic.mathdoc.fr/item/JSFU_2025_18_1_a1/

[1] L.Pandolfi, Systems with Persistent Memory, Springer Nature Switzerland, 2021 | MR | Zbl

[2] D.K.Durdiev, Z.D.Totieva, Kernel Determination Problems in Hyperbolic Integro-Differential Equations, Springer, Singapore, 2023 | MR | Zbl

[3] S.Avdonin, S.Ivanov, J.Wang, “Inverse problems for the heat equation with memory”, Inverse problems and imaging, 13:1 (2019), 31–38 | DOI | MR | Zbl

[4] D.Guidetti, “Reconstruction of a convolution kernel in a parabolic problem with a memory term in the boundary conditions”, Bruno Pini Mathematical Analysis Seminar, 4:1 (2013), 47–55 | DOI | MR

[5] C.Cavaterra, D.Guidetti, “Identification of a convolution kernel in a control problem for the heat equation with a boundary memory term”, Annali di Matematica, 193 (2014), 779–816 | DOI | MR | Zbl

[6] D.K.Durdiev, J.J.Jumaev, “Memory kernel reconstruction problems in the integro-differential equation of rigid heat conductor”, Mathematical Methods in the Applied Sciences, 45 (2022), 8374–8388 | DOI | MR | Zbl

[7] D.K.Durdiev, Zh.Zh.Zhumaev, “One-dimensional inverse problems of finding the kernel of the integro-differential heat equation in a bounded domain”, Ukrains'kyi Matematychnyi Zhurnal, 73:11 (2021), 1492–1506 | DOI | MR

[8] D.K.Durdiev, Zh.Zh.Zhumaev, “On determination of the coefficient and kernel in an integro-differential equation of parabolic type”, Eurasian Journal of Mathematical and Computer Applications, 11:1 (2023), 49–65 | DOI

[9] D.K.Durdiev, Z.Z.Nuriddinov, “Determination of a multidimensional kernel in some parabolic integro-differential equation”, Journal of Siberian Federal University. Mathematics and Physics, 14:1 (2021), 117–127 | DOI | MR | Zbl

[10] D.K.Durdiev, Z.Z.Nuriddinov, “Uniqueness of the Kernel Determination Problem in a Integro-Differential Parabolic Equation with Variable Coefficients”, Russian Mathematics, 67:11 (2023), 1–11 | DOI | MR | Zbl

[11] S.Avdonin, S.Ivanov, J.Wang, “Inverse problems for the heat equation with memory”, Inverse problems and imaging, 13:1 (2019), 31–38 | DOI | MR | Zbl

[12] D.Guidetti, “Reconstruction of a convolution kernel in a parabolic problem with a memory term in the boundary conditions”, Bruno Pini Mathematical Analysis Seminar, 4:1 (2013), 47–55 | DOI | MR

[13] D. Guidetti, “Some inverse problems of identification for integrodifferential parabolic systems with a boundary memory term”, Discrete $\$ Continuous Dynamical Systems – S, 8:4 (2015), 749–756 | DOI | MR | Zbl

[14] A.L.Karchevsky, A.G. Fatianov, “Numerical solution of the inverse problem for a system of elasticity with the aftereffect for a vertically inhomogeneous medium”, Sib. Zh. Vychisl. Mat., 4:3 (2001), 259–268 | Zbl

[15] U.D.Durdiev, “Numerical method for determining the dependence of the dielectric permittivity on the frequency in the equation of electrodynamics with memory”, Sib. Elektron. Mat. Izv., 17 (2020), 179–189 | DOI | MR | Zbl

[16] Z.R.Bozorov, “Numerical determining a memory function of a horizontally-stratified elastic medium with aftereffect”, Eurasian journal of mathematical and computer applications, 8:2 (2020), 4–16 | DOI

[17] S.I.Kabanikhin, A.L.Karchevsky, A.Lorenzi, “Lavrent'ev Regularization of Solutions to Linear Integro-differential Inverse Problems”, J. Inverse Ill-Posed Probl., 2:1 (1993), 115–140 | MR | Zbl

[18] A.Ladyzhenskaya, V.A.Solonnikov, N.N.Ural'tseva, Linear and quasilinear equations of parabolic type, Nauka, M., 1967 | MR | Zbl