МІШАНА ЗАДАЧА ДЛЯ НЕЛІНІЙНИХ ПАРАБОЛІЧНИХ РІВНЯНЬ ВИЩИХ ПОРЯДКІВ ЗІ ЗМІННИМИ ПОКАЗНИКАМИ НЕЛІНІЙНОСТІ В НЕОБМЕЖЕНИХ ОБЛАСТЯХ БЕЗ УМОВ НА НЕСКІНЧЕННОСТІ

  • M. M. Bokalo Львівський національний університет імені Івана Франка, Львів, Україна
Ключові слова: мішана задача, початково-крайова задача, параболічне рівняння вищого порядку, нелінійне параболічне рівняння, узагальнений розв'язок

Анотація

У даній роботі доведено однозначну розв'язність мішаної задачі для деяких анізотропних параболічних рівнянь вищих порядків зі змінними показниками нелінійності в необмежених областях без умов нескінченності. Також отримано апріорну оцінку узагальнених розв'язків цієї задачі.

Завантаження

Дані завантаження ще не доступні.

Посилання

[1] Bernis F. Elliptic and parabolic semilinear parabolic problems without conditions at infinity. Arch. Rational Mech. Anal. 1989, 106 (3), 217–241.
[2] Benilan Ph., Grandall M.G., Pierre M. Solutions of the porous medium equations in Rn under optimal conditions on initial values. Indiana Univ. Math. J. 1984, 33 (1), 51–87.
[3] Boccardo L., Gallouët Th., Vazquez J.L. Solutions of nonlinear parabolic equations without growth restrictions on the data, Electronic J. Diff. Eq. 2001, 60, 1–20.
[4] Bokalo M.M. Boundary value problems for semilinear parabolic equations in unbounded domains without conditions at infinity. Siberian Math. J. 1996, 37 (5), 860–867.
[5] Bokalo N.M. The well-posedness of the first boundary value problem and the Cauchy problem for some quasilinear parabolic systems without conditions at infinity. J. Math. Sci. 2006, 135 (1), 2625–2636.
[6] Бокало М.М., Паучок І.Б. Коректність задачі Фур'є для нелінійних параболічних рівнянь вищих порядків зі змінними показниками нелінійності. Математичні студії 2006, 24 (1), 25–48.
[7] Bokalo M.M., Buhrii O.M., Mashiyev R.A. Unique solvability of initial-boundary-value problems for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity. J. Nonl. Evol. Eq. Appl. 2013, 6, 67–87.
[8] Bokalo M., Buhrii O., Hryadil N. Initial-boundary value problems for nonlinear elliptic-parabolic equations with variable exponents of nonlinearity in unbounded domains without conditions at infinity. Nonlinear Analysis. Elsevier. USA, 2020, 192, 1–17.
[9] Bokalo M. Initial-boundary value problems for anisotropic parabolic equations with variable exponents of the nonlinearity in unbounded domains with conditions at infinity. Journal of optimization, differential equations and their applications (JODEA) 2022, 30 (1), 98-121. doi 10.15421/142205.
[10] Brézis H. Semilinear equations in R^N without conditions at infinity. Appl. Math. Optim. 1984, 12 (3), 271–282.
[11] Buhrii O., Buhrii N. Nonlocal in time problem for anisotropic parabolic equations with variable exponents of nonlinearities. J. Math. Anal. Appl. 2019, 473, 695–711.
[12] Diening L., Harjulehto P., Hästö P., Růžička M. Lebesgue and Sobolev spaces with variable exponents. Springer, Heidelberg, 2011.
[13] Gladkov A., Guedda M. Diffusion-absorption equation without growth restrictions on the data at infinity. J. Math. Anal. Appl. 2002, 274 (1), 16–37.
[14] Івасишен С.Д., Пасічник Г.С. Зображення розв'язкiв рiвняння типу Колмогорова зi зростаючими коефiцiєнтами та виродженнями на початковiй гiперплощинi. Буковинський математичний журнал 2021. 9 (1), 189–199. doi.org/10.31861/bmj2021.01.16.
[15] Kováčik O., Rákosníc J. On spaces L^(p(x)) and W^(k; p(x)). Czechoslovak Mathematical Journal 1991, 41 (116), 592–618.
[16] Kováčik O. Parabolic equations in generalized Sobolev spaces W^(k; p(x)). Fasciculi Mathematici. 1995, 25, 87–94.
[17] Lions J.-L. Quelques méthodes de résolution des problémes aux limites non linéaires. Paris (France): Dunod Gauthier-Villars, 1969.
[18] Mashiyev R. A., Buhrii O. M. Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity. Journal of Mathematical Analysis and Applications 2011, 377, 450–463.
[19] Marchi C., Tesei A. Higher-order parabolic equations without conditions at infinity. J. Math. Anal. Appl. 2002, 269, 352–368.
[20] Oleinik O.A., Iosifyan G.A. An analog of Saint-Venant principle and uniqueness of the solutions of the boundary-value problems in unbounded domains for parabolic equations. Usp. Mat. Nauk 1976, 31 (6), 142–166. (in Russian)
[21] Rădulescu V., Repovš D., Partial differential equations with variable exponents: variational methods and qualitative analysis. CRC Press, Boca Raton, London, New York, 2015.
[22] Růžička M. Electroreological fluids: modeling and mathematical theory. Springer-Verl., Berlin, 2000.
[23] Samokhin V. N. On a class of equations that generalize equations of polytropic filtration. Diff. Equat. 1996, 32 (5), 648–657. (in Russian)
[24] Shishkov A.E. The solvability of the boundary-value problems for quasilinear elliptic and parabolic equations in unbounded domains in the classes of functions growing at the infinity. Ukr. Math. J. 1985, 47 (2), 277–289. (in Russian)
[25] Tikhonov A.N. Théoremes d’unicité pour l’équation de la chaleur. Mat. Sb. 1935, 42 (2), 199–216.
References
[1] Bernis F. Elliptic and parabolic semilinear parabolic problems without conditions at infinity. Arch. Rational Mech. Anal. 1989, 106 (3), 217–241.
[2] Benilan Ph., Grandall M.G., Pierre M. Solutions of the porous medium equations in Rn under optimal conditions on initial values. Indiana Univ. Math. J. 1984, 33 (1), 51–87.
[3] Boccardo L., Gallouët Th., Vazquez J.L. Solutions of nonlinear parabolic equations without growth restrictions on the data, Electronic J. Diff. Eq. 2001, 60, 1–20.
[4] Bokalo M.M. Boundary value problems for semilinear parabolic equations in unbounded domains without conditions at infinity. Siberian Math. J. 1996, 37 (5), 860–867.
[5] Bokalo N.M. The well-posedness of the first boundary value problem and the Cauchy problem for some quasilinear parabolic systems without conditions at infinity. J. Math. Sci. 2006, 135 (1) , 2625–2636.
[6] Bokalo M.M., Pauchok I.B. On the well-posedness of a Fourier problem for nonlinear parabolic equations of higher order with variable exponents of nonlinearity. Matematychni Studii 2006, 26 (1), 25–48. (in Ukrainian)
[7] Bokalo M.M., Buhrii O.M., Mashiyev R.A. Unique solvability of initial-boundary-value problems for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity. J. Nonl. Evol. Eq. Appl. 2013, 6, 67–87.
[8] Bokalo M., Buhrii O., Hryadil N. Initial-boundary value problems for nonlinear elliptic-parabolic equations with variable exponents of nonlinearity in unbounded domains without conditions at infinity. Nonlinear Analysis. Elsevier. USA, 2020, 192, 1–17.
[9] Bokalo M. Initial-boundary value problems for anisotropic parabolic equations with variable exponents of the nonlinearity in unbounded domains with conditions at infinity. Journal of optimization, differential equations and their applications (JODEA) 2022, 30 (1), 98-121. doi 10.15421/142205.
[10] Brézis H. Semilinear equations in R^N without conditions at infinity. Appl. Math. Optim. 1984, 12 (3), 271–282.
[11] Buhrii O., Buhrii N. Nonlocal in time problem for anisotropic parabolic equations with variable exponents of nonlinearities. J. Math. Anal. Appl. 2019, 473, 695–711.
[12] Diening L., Harjulehto P., Hästö P., Růžička M. Lebesgue and Sobolev spaces with variable exponents. Springer, Heidelberg, 2011.
[13] Gladkov A., Guedda M. Diffusion-absorption equation without growth restrictions on the data at infinity. J. Math. Anal. Appl. 2002, 274 (1), 16–37.
[14] Ivasyshen S. D., Pasichnyk H. S. Representation of solutions of Kolmogorov type equations with increasing coefficients and degenerations on the initial hyperplane Bukovinian. Math. J. 2021, 9 (1), 189–199. doi.org/10.31861/bmj2021.01.16. (in Ukrainian)
[15] Kováčik O., Rákosníc J. On spaces L^(p(x)) and W^(k; p(x)). Czechoslovak Mathematical Journal 1991, 41 (116), 592–618.
[16] Kováčik O. Parabolic equations in generalized Sobolev spaces W^(k; p(x)). Fasciculi Mathematici. 1995, 25, 87–94.
[17] Lions J.-L. Quelques méthodes de résolution des problémes aux limites non linéaires. Paris (France): Dunod Gauthier-Villars, 1969.
[18] Mashiyev R. A., Buhrii O. M. Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity. Journal of Mathematical Analysis and Applications 2011, 377, 450–463.
[19] Marchi C., Tesei A. Higher-order parabolic equations without conditions at infinity. J. Math. Anal. Appl. 2002, 269, 352–368.
[20] Oleinik O.A., Iosifyan G.A. An analog of Saint-Venant principle and uniqueness of the solutions of the boundary-value problems in unbounded domains for parabolic equations. Usp. Mat. Nauk 1976, 31 (6), 142–166. (in Russian)
[21] Rădulescu V., Repovš D., Partial differential equations with variable exponents: variational methods and qualitative analysis. CRC Press, Boca Raton, London, New York, 2015.
[22] Růžička M. Electroreological fluids: modeling and mathematical theory. Springer-Verl., Berlin, 2000.
[23] Samokhin V. N. On a class of equations that generalize equations of polytropic filtration. Diff. Equat. 1996, 32 (5), 648–657. (in Russian)
[24] Shishkov A.E. The solvability of the boundary-value problems for quasilinear elliptic and parabolic equations in unbounded domains in the classes of functions growing at the infinity. Ukr. Math. J. 1985, 47 (2), 277–289. (in Russian)
[25] Tikhonov A.N. Théoremes d’unicité pour l’équation de la chaleur. Mat. Sb. 1935, 42 (2), 199–216.
Опубліковано
2023-01-13
Як цитувати
[1]
Bokalo, M. 2023. МІШАНА ЗАДАЧА ДЛЯ НЕЛІНІЙНИХ ПАРАБОЛІЧНИХ РІВНЯНЬ ВИЩИХ ПОРЯДКІВ ЗІ ЗМІННИМИ ПОКАЗНИКАМИ НЕЛІНІЙНОСТІ В НЕОБМЕЖЕНИХ ОБЛАСТЯХ БЕЗ УМОВ НА НЕСКІНЧЕННОСТІ. Буковинський математичний журнал. 10, 2 (Січ 2023), 59-76. DOI:https://doi.org/10.31861/bmj2022.02.05.