Geodesic
Spatially-Nonlocal Boundary Value Problems with the generalized Samarskii--Ionkin condition for quasi-parabolic equations
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 1, pp. 110-123.

Voir la notice de l'article provenant de la source Math-Net.Ru

The work is devoted to the study of the solvability of boundary value problems for quasi-parabolic equations $$(-1)^pD^{2p+1}_tu-\frac{\partial}{\partial x}\left(a(x)u_x\right)+c(x,t)u=f(x ,t)$$ $$((x,t)\in (0,1)\times (0,T), a(x)>0, D^k_t=\frac{\partial^k}{\partial t ^k},\ p>0 - \text{integer})$$ with boundary conditions of one of the types $$u(0,t)-\beta u(1,t)=0, u_x(1,t)=0, t\in (0,T),$$ or $$u_x(0,t)-\beta u_x(1,t)=0, u(1,t)=0, t\in (0,T).$$ The problems under study can be treated as nonlocal problems with the generalized Samarskii–Ionkin condition in terms of spatial variable, for them we prove existence and uniqueness theorems for regular solutions—namely, solutions that have all generalized in the sense of S.L. Sobolev derivatives included in the corresponding equation.
Mots-clés : quasi-parabolic equations, existence
Keywords: non-local boundary value problems, generalized Samarskii–Ionkin condition, regular solutions, uniqueness. 
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A. I. Kozhanov; A. M. Abdrakhmanov. Spatially-Nonlocal Boundary Value Problems with the generalized Samarskii--Ionkin condition for quasi-parabolic equations. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 1, pp. 110-123. http://geodesic.mathdoc.fr/item/SEMR_2023_20_1_a24/

[1] J.R. Cannon, “The solution of the heat equation subject to the specification of energy”, Q. Appl. Math., 22 (1964), 155–160 | MR | Zbl

[2] L.I. Kamynin, “A boundary value problem in the theory of heat conduction with nonclassical boundary conditions”, U.S.S.R. Comput. Math. Math. Phys., 4:6 (1964), 1006–1024 | DOI | MR | Zbl

[3] N.I. Ionkin, “The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition”, Differ. Uravn., 13:2 (1977), 294–304 | MR | Zbl

[4] N.I. Yurchuk, “Mixed problem with an integral condition for certain parabolic equations”, Differ. Equations, 22 (1986), 1457–1463 | MR | Zbl

[5] N.I. Ionkin, “The stability of a problem in the theory of heat conduction with nonclassical boundary conditions”, Diff. Uravn., 15:7 (1979), 1279–1283 | MR | Zbl

[6] S.A. Beilin, “Existence of solutions for one-dimensional wave equations with nonlocal conditions”, Electron. J. Differ. Equ., 2001, 76 | MR | Zbl

[7] A.S. Berdyshev, A. Cabada, B.J. Kadirkulov, “The Samarskii-Ionkin type problem for the fourth order parabolic equation with fractional differential operator”, Comput. Math. Appl., 62:10 (2011), 3884–3893 | DOI | MR | Zbl

[8] I.A. Kaliev, M.M. Sabitova, “Problems of determining the temperature and density of heat sources from the initial and final temperatures”, Sib. Zh. Ind. Mat., 12:1 (2009), 89–97 | MR | Zbl

[9] A.A. Samarskij, “Some problems in differential equation theory”, Differ. Equations, 16 (1981), 1221–1228 | MR | Zbl

[10] N.L. La$\check{z}$etić, “On classical solutions of mixed boundary problems for one-dimensional parabolic equation of second order”, Publ. Inst. Math. Nouv. Sér., 67:81 (2000), 53–75 | MR | Zbl

[11] A.I. Kozhanov, “Solvability of boundary value problems with nonlocal and integral conditions for a parabolic equation”, Nonlinear Bound. Value Probl., 20 (2010), 54–76 | MR | Zbl

[12] A.M. Abdrakhmanov, A.I. Kozhanov, “A problem with a nonlocal boundary condition for one class of odd-order equations”, Russ. Math., 51:5 (2007), 1–10 | DOI | MR | Zbl

[13] A.M. Abdrakhmanov, “Solvability of a boundary-value problem with an integral boundary condition of the second kind for equations of odd order”, Mat. Notes, 88:2 (2010), 151–159 | DOI | MR | Zbl

[14] S.L. Sobolev, Some applications of functional analysis in mathematical physics, Amer. Math. Soc., Providence, 1991 | MR | Zbl

[15] O.A. Ladyzhenskaya, N.N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968 | MR | Zbl

[16] H. Triebel, Interpolation theory. Function spaces. Differential operators, Deutscher Verlag des Wissenschaften, Berlin, 1978 | MR | Zbl

[17] V.A. Trenogin, Functionala Analysis, Nauka, M., 1980 | MR | Zbl

[18] O.V. Besov, V.P. Il'in, S.M. Nikol'skiy, Integral representations of functions and embedding theorems, Nauka, M., 1975 | MR | Zbl

[19] A.I. Kozhanov, “On the solvability of certain spatially nonlocal boundary-value problems for linear hyperbolic equations of second order”, Math. Notes, 90:2 (2011), 238–249 | DOI | MR | Zbl