Сызықты емес шекаралық шартты псевдогиперболалық теңдеудiң шешiмдiлiгi
DOI:
https://doi.org/10.26577/JMMCS.2020.v108.i4.03Кілттік сөздер:
псевдогиперболалық теңдеулер, сызықты емес шекаралық шарттар, Галеркин әдiсi, шешiмнiң бар болуы, шешiмнiң жалғыздығыАннотация
Бұл жұмыс жеткiлiктi тегiс шекарасы бар квазисызықты псевдогиперболалық теңдеу үшiн (Соболев типтi теңдеулер деп те аталатын) бастапқы-шеттiк есептiң шешiлуiн зерттеудiң iргелi мәселесiне арналған. ұсынылған жұмыста Нейман-Дирихле сызықты емес шекаралық шарттары бар псевдогиперболалық типтi квазисызықты теңдеуiне арналған бастапқы-шеттiк есеп зерттеледi. Мақалада Галеркин әдiсi арқылы шектелген облыста квазисызықты псевдопараболалық теңдеудiң әлсiз шешiмiнiң бар болуы дәлелденедi. Соболевтың енгiзу теоремаларын қолдану арқылы шешiмнiң априорлы бағалары алынды. Галеркиннiң жуықтауларын қолдану шешiмнiң бар болу уақытынан тыс бағалауларды алуға мүмкiндiк бередi. Әлсiз жалпыланған шешiмнiң бар болу жөнiндегi жергiлiктi теорема дәлелдендi. Қарастырылған шеттiк есептiң iзделiндi шешiмiнiң бар болуын дәлелдеу кезiнде априорлы бағалаулар мен Реллих-Кондрашов теоремасы қолданылады. Псевдопараболалық типтi квазисызықты теңдеу үшiн бастапқы-шеттiк есептiң әлсiз жалпыланған шешiмiнiң жалғыздығы алынған априорлы бағалаулар мен Гронуолл-Беллман леммасы негiзiнде дәлелденедi. Квазисызықты псевдогиперболалық теңдеу үшiн бастапқы-шеттiк есептердi қарастыру және зерттеу қажеттiлiгi практикалық мұқтаждықтардан туындайды. Мысалы, физикалық әдерiстердi модельдейтiн дифференциалдық теңдеулердi шешу кезiнде таңдалған модель мен нақты объект арасында бiр жақсы сәйкестiк болуы қажеттi әрi маңызды.
Библиографиялық сілтемелер
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[3] Ting T.W., "Parabolic and pseudoparabolic partial differential equations"„ J. Math. Soc. Japan. 14(1969), 1-26.
[4] Benjamin Т.В., "Lectures on nonlinear wave motion", Ltd. Appl. Math. Vol. Amer. Math. Soc: Providence; R.I. 15(1974), 3-7.
[5] Benjamin T.B., Bona J.L., Mahony J.J., "Model equations for long waves in nonlinear dispersive systems", Pliilos. Trans. Roy. Soc. 272:1220(1972), 47-78.
[6] Showalter R.E., "Existence and representation theorems for a semilinearSobolev equation in Banachspace", SIAM J. Math. Anal. 3:3(1972), 527-543.
[7] Showalter R.E., Ting T. W., "Pseudoparabolic partial differential equations", SIAM J. Math. Anal. 1:1(1970), 1-26.
[8] Pokhozhaev S.I., "Ob odnom klasse kvazilineynykh giperbolicheskikh uravneniy.[On a class of quasilinear hyperbolic equations]"Mat. Sb. 25:1(1975), 145-158 [in Russian].
[9] Oskolkov A.P., Nachal’no-krayevyye zadachi dlya uravneniy dvizheniya zhidkostey Kel’vina-Foygta i zhidkostey Oldroyta [Initialboundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids] (Tr. Mat. in-ta im. V. A. Steklova AN SSSR, 179(1988), 126-164) [in Russian].
[10] Oskolkov A.P., "Nelokal’nyye problemy dlya odnogo klassa nelineynykh operatornykh uravneniy, voznikayushchikh v teorii uravneniy tipa S. L. Soboleva. [Nonlocal problems for a class of nonlinear operator equations arising in the theory of equations of the Sobolev type]", Zap. nauch. semin. LOMI. 198(1991), 31-48 [in Russian].
[11] Gabov S.L.,Sveshnikov A.G., Lineynyye zadachi teorii nestatsionarnykh vnutrennikh voln.[Linear problems in the theory of nonstationary internal waves.] (M.: Nauka, 1990) [in Russian].
[12] Shishmarev I.A., "Ob odnom nelineynom uranenii tipa Soboleva. [On a nonlinear uranium of the Sobolev type]", Differ. Equ. 41:1(2000), 1-3 [in Russian].
[13] Korpusov M.O., Sveshnikov A.G., "O razreshimosti sil’no nelineynogo uravneniya psevdoparabolicheskogo tipa s dvoynoy nelineynost’yu. [On the solvability of a strongly nonlinear equation of pseudoparabolic type with double nonlinearity]", ZH. vychisl. mat. mat. fiz. 43:7(2003), 944-962 [in Russian].
[14] Sveshnikov G., Al’shin A. B., Korpusov M. O., PletnerYU.D., Lineynyye i nelineynyye uravneniya sobolevskogo tipa [Pletner, Linear and Nonlinear Equations of Sobolev Type] (M.: Fizmatlit, 2007) [in Russian].
[15] Samarskiy A.A., Galaktionov V.A., Kurdyumov S.P., Mikhaylov A.P., Rezhimy s obostreniyem v zadachakh dlya kvazilineynykh parabolicheskikh uravneniy [Regimes with Peaking in Problems for Quasilinear Parabolic Equations] (M.: Nauka, 1987) [in Russian].
[16] S.N. Antontsev, Kh. Khompysh, "Kelvin-Voigt equation with p-Laplacian and damping term: Existence, uniqueness and blowup", Mathematical Analysis and Applications 446(2017), 1255-1273.
[17] S.N. Antontsev, Kh. Khompysh, "Generalized Kelvin–Voigt equations with p-Laplacian and source/absorption terms", Mathematical Analysis and Applications 456(2017), 99-116.
[18] S.N. Antontsev, H.B. de Oliveira, Kh. Khompysh, "Kelvin-Voigt equations perturbed by anisotropic relaxation, diffusion and damping", Mathematical Analysis and Applications 473(2019), 1122-1154.
[19] S.N. Antontsev, H.B. de Oliveira, Kh. Khompysh, "Generalized Kelvin-Voigt equations for nonhomogeneous and incompressible fluids", Communications in Mathematical Sciences 17:7(2019), 1915-1948.
[20] А.I. Kozhanov, N.S. Popov, "O razreshimosti nekotorykh zadach so smeshcheniyem dlya psevdoparabolicheskikh uravneniy [On the solvability of some displacement problems for pseudoparabolic equations]", Vestnik NGU. Ser. matem., mekh., inform. 1:3(2010), 46-62 [in Russian].
[21] Sh. Аmirov, А., I. Kozhanov, "Razreshimost’ smeshannoy zadachi dlya nekotorykh sil’no nelineynykh uravneniy sobolevskogo tipa vysokogo poryadka [Solvability of the mixed problem for some highly nonlinear high-order sobolev type equations]", Sib. zhurn. industr. matem. 17:4(2014), 14-30 [in Russian].
[22] А.I. Kozhanov, "Krayevyye zadachi dlya uravneniy sobolevskogo tipa s neobratimym operatorom pri starshey proizvodnoy [Boundary value problems for Sobolev type equations with an irreversible operator for the highest derivative]", Itogi nauki i tekhn. Ser. Sovrem. mat. i yeye pril. 167(2019), 34-41 [in Russian].
[23] Demidovich B.P., Lektsii po matematicheskoy teorii ustoychivosti [Lectures on the mathematical theory of stability] (M.: Nauka, 1967) [in Russian].
[24] Lions ZH.-L., Nekotoryye metody resheniya nelineynykh krayevykh zadach [Some methods for solving nonlinear boundary value problems] (M.: Nauka, 1972, 588 pp.) [in Russian].
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Как цитировать
Aitzhanov, S. E., Bekenaeva К. S., & Zhumagul G. О. (2020). Сызықты емес шекаралық шартты псевдогиперболалық теңдеудiң шешiмдiлiгi. Қазұу Хабаршысы. Математика, механика, информатика сериясы, 108(4), 26–37. https://doi.org/10.26577/JMMCS.2020.v108.i4.03
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