Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems
This paper is devoted to the solvability of an initial-boundary value problem for second-order parabolic equations in divergence form with variable order of nonlinearity. The characteristic feature of the considered class of Cauchy-Neumann parabolic problem is the fact that the variable exponent p(t...
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| Format: | Article |
| Language: | English |
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Oles Honchar Dnipro National University
2022-09-01
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| Series: | Journal of Optimization, Differential Equations and Their Applications |
| Subjects: | |
| Online Access: | https://model-dnu.dp.ua/index.php/SM/article/view/172 |
| _version_ | 1797953621276491776 |
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| author | Peter Kogut Yaroslav Kohut Nataliia Parfinovych |
| author_facet | Peter Kogut Yaroslav Kohut Nataliia Parfinovych |
| author_sort | Peter Kogut |
| collection | DOAJ |
| description | This paper is devoted to the solvability of an initial-boundary value problem for second-order parabolic equations in divergence form with variable order of nonlinearity. The characteristic feature of the considered class of Cauchy-Neumann parabolic problem is the fact that the variable exponent p(t, x) and the anisotropic diffusion tensor D(t, x) are not well predefined a priori, but instead these characteristic depend on a solution of this problem, i.e., p = p(t, x, u) and D = D(t, x, u). Recently, it has been shown that the similar models appear in a natural way as the optimality conditions for some variational problems related to the image restoration technique. However, from practical point of view, in this case some principle difficulties can appear because of the absence of the corresponding rigorous mathematically substantiation. Thus, in this paper, we study the solvability issues of the Cauchy-Neumann parabolic boundary value problem for which the corresponding principle operator is strongly non-linear, non-monotone, has a variable order of nonlinearity, and satisfies a nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a weak solution, we apply the technique of passing to the limit in a special approximation scheme and the Schauder fixed-point theorem. |
| first_indexed | 2024-04-10T23:04:58Z |
| format | Article |
| id | doaj.art-cb6390434e4f448fb7e427d94917be86 |
| institution | Directory Open Access Journal |
| issn | 2617-0108 2663-6824 |
| language | English |
| last_indexed | 2024-04-10T23:04:58Z |
| publishDate | 2022-09-01 |
| publisher | Oles Honchar Dnipro National University |
| record_format | Article |
| series | Journal of Optimization, Differential Equations and Their Applications |
| spelling | doaj.art-cb6390434e4f448fb7e427d94917be862023-01-13T12:20:59ZengOles Honchar Dnipro National UniversityJournal of Optimization, Differential Equations and Their Applications2617-01082663-68242022-09-01302194810.15421/142207164Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising ProblemsPeter Kogut0Yaroslav Kohut1Nataliia Parfinovych2Department of Differential Equations, Oles Honchar Dnipro National UniversityDepartment of Mathematical Analysis and Theory of Functions, Oles Honchar Dnipro National UniversityDepartment of Mathematical Analysis and Theory of Functions, Oles Honchar Dnipro National UniversityThis paper is devoted to the solvability of an initial-boundary value problem for second-order parabolic equations in divergence form with variable order of nonlinearity. The characteristic feature of the considered class of Cauchy-Neumann parabolic problem is the fact that the variable exponent p(t, x) and the anisotropic diffusion tensor D(t, x) are not well predefined a priori, but instead these characteristic depend on a solution of this problem, i.e., p = p(t, x, u) and D = D(t, x, u). Recently, it has been shown that the similar models appear in a natural way as the optimality conditions for some variational problems related to the image restoration technique. However, from practical point of view, in this case some principle difficulties can appear because of the absence of the corresponding rigorous mathematically substantiation. Thus, in this paper, we study the solvability issues of the Cauchy-Neumann parabolic boundary value problem for which the corresponding principle operator is strongly non-linear, non-monotone, has a variable order of nonlinearity, and satisfies a nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a weak solution, we apply the technique of passing to the limit in a special approximation scheme and the Schauder fixed-point theorem.https://model-dnu.dp.ua/index.php/SM/article/view/172weak solutionparabolic equationvariable order of nonlinearitynoncoercive problemcompensated compactness technique |
| spellingShingle | Peter Kogut Yaroslav Kohut Nataliia Parfinovych Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems Journal of Optimization, Differential Equations and Their Applications weak solution parabolic equation variable order of nonlinearity noncoercive problem compensated compactness technique |
| title | Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems |
| title_full | Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems |
| title_fullStr | Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems |
| title_full_unstemmed | Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems |
| title_short | Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems |
| title_sort | solvability issues for some noncoercive and nonmonotone parabolic equations arising in the image denoising problems |
| topic | weak solution parabolic equation variable order of nonlinearity noncoercive problem compensated compactness technique |
| url | https://model-dnu.dp.ua/index.php/SM/article/view/172 |
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