Solvability of a Pseudohyperbolic Equation with a Nonlinear Boundary Condition
DOI:
https://doi.org/10.26577/JMMCS.2020.v108.i4.03Keywords:
pseudohyperbolic equations, nonlinear boundary conditions, Galerkin method, existence of a solution, uniqueness of a solutionAbstract
This paper is devoted to the fundamental problem of investigating the solvability of an initial-boundary value problem for a quasi-linear pseudo-hyperbolic equation (also called Sobolev type equations) with a sufficiently smooth boundary. In this work, we study an initial-boundary value problem for a quasi-linear pseudo-hyperbolic equation with a nonlinear Neumann-Dirichlet boundary condition. The paper uses the Galerkin method to prove the existence of a weak solution of a quasi-linear pseudo-hyperbolic equation in a bounded domain. Using Sobolev embedding theorems, priori estimates of the solution are obtained. The use of Galerkin approximations allows us to obtain an overtime estimate of the solution’s existence. A local theorem on the existence of a weak generalized solution is proved. A priori estimates and the Rellich-Kondrashov theorem are used to prove the existence of the desired solution to the boundary value problem under consideration. The uniqueness of a weak generalized solution to the initial boundary value problem of a quasi-linear pseudo-hyperbolic equation is proved on the basis of the obtained a priori estimates and the application of the Gronwall-Bellman Lemma. The need to consider and study such initial-boundary value problems for a quasi-linear pseudo-hyperbolic equation follows from practical needs. For example, when solving differential equations that model physical processes, it is important that there is a good match between the selected model and the real object.
References
[2] Barenblatt G.I., Zheltov Yu.P., Konina I.N., "Ob osnovnykh predstavleniyakh teorii fil’tratsii v treshchinnovatykh sredakh. [On the basic concepts of the theory of filtration in fractured media]", Prikl. matem. i mekhan. 24:5(1960), 73-58 [in Russian].
[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].
