Geodesic
The conjugation problem for pseudoparabolic and pseudohyperbolic equations
Matematičeskie zametki SVFU, Tome 23 (2016) no. 3, pp. 27-45 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study solvability of a conjugation problem for pseudoparabolic and pseudohyperbolic equations. The equations are considered as equations with discontinuous coefficients. We prove the existence and uniqueness theorem using natural parameter continuation.
Mots-clés : pseudoparabolic equation, conjugation problem.
Keywords: pseudohyperbolic equation, discontinuous coefficient 
@article{SVFU_2016_23_3_a1,
     author = {A. I. Grigorieva},
     title = {The conjugation problem for pseudoparabolic and pseudohyperbolic equations},
     journal = {Matemati\v{c}eskie zametki SVFU},
     pages = {27--45},
     year = {2016},
     volume = {23},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVFU_2016_23_3_a1/}
}
TY  - JOUR
AU  - A. I. Grigorieva
TI  - The conjugation problem for pseudoparabolic and pseudohyperbolic equations
JO  - Matematičeskie zametki SVFU
PY  - 2016
SP  - 27
EP  - 45
VL  - 23
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SVFU_2016_23_3_a1/
LA  - ru
ID  - SVFU_2016_23_3_a1
ER  - 
%0 Journal Article
%A A. I. Grigorieva
%T The conjugation problem for pseudoparabolic and pseudohyperbolic equations
%J Matematičeskie zametki SVFU
%D 2016
%P 27-45
%V 23
%N 3
%U http://geodesic.mathdoc.fr/item/SVFU_2016_23_3_a1/
%G ru
%F SVFU_2016_23_3_a1
A. I. Grigorieva. The conjugation problem for pseudoparabolic and pseudohyperbolic equations. Matematičeskie zametki SVFU, Tome 23 (2016) no. 3, pp. 27-45. http://geodesic.mathdoc.fr/item/SVFU_2016_23_3_a1/

[1] Gelfand I. M., “Some questions of analysis and differential equations”, Usp. Mat. Nauk, 14:3 (1959), 3–19 | Zbl

[2] Oleinik O. A., “Boundary value problems for linear equations of elliptic and parabolic type with discontinuous coefficients”, Izv. AN USSR, Ser. Mat., 25:1 (1961), 3–20 | MR | Zbl

[3] Ilyin V. A., “On the solvability of the Dirichlet and Neumann problems for linear elliptic operator with discontinuous coefficients”, Dokl. Akad. Nauk, 137:1 (1961), 28–30 | Zbl

[4] Ilyin V. A., “Fourier method for hyperbolic equations with discontinuous coefficients”, Dokl. Akad. Nauk, 142:1 (1961), 21–24 | Zbl

[5] Ilyin V. A. and Shishmarev I. A., “Potential method for the Dirichlet and Neumann problems in the case of equations with discontinuous coefficients”, Sib. Math. J., 2:1 (1961), 46–58 | Zbl

[6] Ladyzhenskaya O. A., Solonnikov V. A., and Uraltseva N. N.,, Linear and quasilinear parabolic equations, Nauka, Moscow, 1967 | MR

[7] Ilyin V. A. and Luferenko P. V., “Mixed problems describing the longitudinal vibrations of a rod consisting of two segments with different densities, different elastic, but the same impedances”, Dokl. Math., 428:1 (2009), 12–15 | Zbl

[8] Andropova O. A., “Spectral problems interfacing with surface dissipation of energy”, Tr. Inst. Prikl. Mat. Mekh., 19 (2009), 10–22

[9] Rogozhnikov A. M., “Study of the mixed problem, describing the process of shaft vibrations, which consists of several sections, provided match the transit time of waves in each of these areas”, Dokl. Math., 441:4 (2012), 449–451

[10] Kuleshov A. A., “Mixed problem for the equation of longitudinal vibrations of an inhomogeneous bar with a free or fixed right end, consisting of two sections of different density and elasticity”, Dokl. Math., 442:4 (2012), 451–454 | Zbl

[11] Rogozhnikov A. M., ““Study of the mixed problem, describing the process of shaft vibrations, consisting of several sections with arbitrary lengths”, Dokl. Math., 444:5 (2012), 488–491 | MR | Zbl

[12] Smirnov I. N., “Oscillations described by the telegraph equation for a system consisting of several sections of different density and elasticity”, Differ. Equ., 49:5 (2012), 643–648

[13] Bitsadze A. V., Equations of Mixed Type, Akad. Nauk SSSR, Moscow, 1959 | MR

[14] Ladyzhenskaya O. A. and Stupyalis L., “About mixed type equations”, Vestn. Leningrad.Univ., 1967, no. 18, 38–46

[15] Smirnov M. M.,, Equations of Mixed Type, Nauka, Moscow, 1970 | MR

[16] Ladyzhenskaya O. A. and Stupyalis L., “Boundary value problems for mixed type equations”, Tr. Mat. Inst. Steklova, 116 (1971), 101–136 | MR | Zbl

[17] Stupyalis L., “Boundary value problems for for elliptic-hyperbolic equations”, Tr. Mat. Inst. Steklova, 125 (1973), 211–229 | MR

[18] Tersenov S. A., Introduction to the theory of parabolic equations with a varying direction of time, Inst. Mat., Novosibirsk, 1982 | MR

[19] Juraev T. D., Boundary value problems for equations of mixed and mixed-composite types, FAN, Tashkent, 1986 | MR

[20] Moiseev E. I., Mixed Type Equations with a Spectral Parameter, MGU, Moscow, 1988 | MR

[21] Sabitov K. B. and Martemyanova N. V., “The inverse problem for an equation of elliptichyperbolic type with a nonlocal boundary condition”, Sib. Math. J., 53:3 (2012), 633–647 | MR | Zbl

[22] Kozhanov A. I., “A conjugation problem for a class of composite type equations with changing direction”, Nonclassical Equ. Math. Phys., Inst. Math., Novosibirsk, 2002, 96–109 | Zbl

[23] Bouziani A., Merazga N., “Solution to a transmission problem for quasilinear pseudoparabolic equations by the Rothe method”, Electronic Journal of Differential Equ., 2007, no. 14, 1–27 | MR

[24] Shubin V. V, “Boundary problems for third-order equations with discontinuous coefficients”, Vestn. Novosib. Gos. Univ., 12:1 (2012), 126–138 | Zbl

[25] Potapova S. V., “Boundary value problems for pseudohyperbolic equations with a variable time direction”, TWMS J. Pure Appl. Math., 3:1 (2012), 73–91 | MR

[26] Kozhanov A. I. and Sharin E. F., “A conjugation problem for some nonclassical differential equations of higher order”, Ukr. Mat. Vestn., 11:2 (2014), 181–202

[27] Kozhanov A. I. and Potapova S. V., “The Dirichlet problem for a class of composite type equations with a discontinuous coefficient of the highest derivative”, Dalnevost. Math. J., 14:1 (2014), 48–65 | MR | Zbl

[28] Antipin V. I., “The solvability of the boundary value problem for a third-order equation with changing time direction”, Mat. Zamet. YaGU, 18:1 (2011), 8–15 | Zbl

[29] Pyatkov S. G., Popov S., Antipin V. I., “On solvability of boundary value problem for kinetic operator-differential equations”, Integral Equ. Operator Theory, 80:4 (2014), 557–580 | DOI | MR | Zbl

[30] T. von Petersdorff, “Boundary integral equations for mixed Dirichlet, Neumann and transmission problems”, Math. Meth. Appl. Sci., 11 (1989), 185–213 | DOI | MR | Zbl

[31] Nikolskiy D. N., “Evolution section of various liquids in non-uniform layers”, Comput. Math. Math. Phys., 50:7 (2010), 1269–1275 | MR | Zbl

[32] Nikolskiy D. N., “The three-dimensional pollution in the border bounded by a piecewise-porous medium”, Comput. Math. Math. Phys., 51:5 (2011), 913–919 | MR | Zbl

[33] Trenogin V. A., Functional analysis, Fizmatlit, Moscow, 2007 | MR

[34] Sobolev S. L., “On a boundary problem of mathematical physics”, Izv. Akad. Nauk, 18:2 (1954), 3–50 | MR | Zbl

[35] Demidenko G. V., and Uspenskii S. V., Equations and systems unsolvable with respect to the higher derivative, Nauchn. Kniga, Novosibirsk, 1998 | MR

[36] Kozhanov A. I., Composite type equations and inverse problems, VSP, Utrecht, 1999 | MR | Zbl

[37] Sviridyuk G. A., Fedorov V. E., Linear Sobolev type equations and degenerate semigroup of operators, VSP, Utrecht, 2003 | MR

[38] Hayashi N., Kaikina E. I., Naumkin P. I., Shishmarev I. A., Asymptotic for dissipative nonlinear equations, Springer-Verl., Berlin; Heidelberg, 2006 | MR

[39] Sveshnikov A. G., Alshin A. B., Korpusov M. O., and Pletner Yu. D., Linear and non-linear equations of Sobolev type, Fizmatlit, Moscow, 2007

[40] Korpusov M. O., The destruction of solutions in the nonclassical non-local equations, Librokom, Moscow, 2011