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On the Solvability of the Dirichlet Problem for the Heat Equation in a Degenerating Domain

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Abstract

A domain, degenerating at the initial moment of time, is considered. A boundary value problem of heat conduction in this domain is studied. By virtue of the isotropy property, the solvability theorems for given boundary value problem are established in weight spaces of essentially bounded functions. The proof of the theorems is based on the solvability conditions of a nonhomogeneous integral equation of the third kind. Using the Fourier series method, the problem splits into families of boundary value problems. The method of representation of the solution to the boundary value problem in the form of sum of constructed thermal potentials is used. The given problem is reduced to the problems of solvability of integral equations. In addition, the solvability theorems for the boundary value problems are proved also for the case, when the axial symmetry property is absent.

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Funding

This work was supported by the Committee of Science of the Ministry of Education and Sciences RK (Grant nos. AP09259780, 2021-2023; AP0885372, 2020-2022).

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Authors and Affiliations

  1. Buketov Karaganda University, 470074, Karaganda, Kazakhstan

    M. I. Ramazanov & M. T. Kosmakova

  2. Karaganda Technical University, 470075, Karaganda, Kazakhstan

    Zh. M. Tuleutaeva

Authors
  1. M. I. Ramazanov
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  2. M. T. Kosmakova
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  3. Zh. M. Tuleutaeva
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Corresponding authors

Correspondence to M. I. Ramazanov, M. T. Kosmakova or Zh. M. Tuleutaeva.

Additional information

(Submitted by T. K. Yuldashev)

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Ramazanov, M.I., Kosmakova, M.T. & Tuleutaeva, Z.M. On the Solvability of the Dirichlet Problem for the Heat Equation in a Degenerating Domain. Lobachevskii J Math 42, 3715–3725 (2021). https://doi.org/10.1134/S1995080222030179

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  • DOI: https://doi.org/10.1134/S1995080222030179

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