Dirichlet problem for mixed type equation with characteristic degeneration

Yu. K. Sabitova

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 4, pp. 622-645.

Résumé

For a mixed elliptic-hyperbolic type equation with characteristic degeneration, the first boundary value problem in a rectangular region is investigated. The criterion for the uniqueness of the solution of the problem is established. Earlier, in proving the uniqueness of solutions of boundary value problems for equations of mixed type, the extremum principle or the method of integral identities was used. The uniqueness of the solution to this problem is established on the basis of the completeness of the system of eigenfunctions of the corresponding one-dimensional spectral problem. The solution of the problem is constructed as a sum of a series in the system of eigenfunctions. When we proved the convergence of the obtained series, the problem of small denominators of a more complicated structure than in other known works arose. These denominators contain a parameter depending on the lengths of the sides of the rectangle in the hyperbolic part of the domain and the exponent of the degree of degeneration. In this connection, estimates are established about separation from zero with the corresponding asymptotics, in cases where this parameter is a natural, rational and algebraic irrational number of degree two. If this parameter is not an algebraic irrational number of degree two, then the solution of the problem as a sum of a series does not exist. Using the obtained estimates, the uniform convergence of the constructed series in the class of regular solutions is justified under certain sufficient conditions with respect to the boundary functions. The stability of the solution of the problem with respect to the boundary functions in the norms of the space of summable functions and in the space of continuous functions is also proved.

Détail

Keywords: equation of mixed type with characteristic degeneration, Dirichlet problem, criterion of uniqueness, small denominator.
Mots-clés : existence

Bibliographie

[1] Keldysh M. V., “On some cases of degeneration of an equation of elliptic type on the domain boundary”, Dokl. Akad. Nauk SSSR, 77:2 (1951), 181–183 (In Russian)

[2] Karol I. L., “On a boundary value problem for an equation of mixed elliptic-hyperbolic type”, Dokl. Akad. Nauk SSSR, 88:2 (1953), 197–200 (In Russian) | MR | Zbl

[3] Isamukhamedov S. S., “The Tricomi boundary value problem for a mixed type equation of the second kind”, Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, 1970, no. 4, 9–12 (In Russian) | MR | Zbl

[4] Krikunov Yu. M., Kraevye zadachi dlia model'nykh uravnenii smeshannogo tipa [Boundary Value Problems for Model Mixed-Type Equations], Kazan' State Univ., Kazan', 1986, 150 pp. (In Russian)

[5] Khairullin R. S., “The Tricomi problem for an equation of mixed type of the second kind in the case of a normal domain”, Differ. Equ., 26:8 (1990), 1031–1039 | MR

[6] Sokhadze R. I., “The first boundary value problem for an equation of mixed type in a rectangle”, Differ. Uravn., 19:1 (1983), 127–134 (In Russian) | MR | Zbl

[7] Sokhadze R. I., “The first boundary value problem for an equation of mixed type with weighted glueing conditions along a line of parabolic degeneration”, Differ. Uravn., 17:1 (1981), 150–156 (In Russian) | MR | Zbl

[8] Sabitov K. B., Suleimanova A. Kh., “The Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain”, Russian Math. (Iz. VUZ), 51:4 (2007), 42–50 | DOI | MR | Zbl

[9] Khairullin R. S., “On the Dirichlet problem for a mixed-type equation of the second kind with strong degeneration”, Differ. Equ., 49:4 (2013), 510–516 | DOI | DOI | MR | Zbl

[10] Sabitov K. B., “The Dirichlet problem for equations of mixed type in a rectangular domain”, Dokl. Math., 75:2 (2007), 193–196 | DOI | MR | Zbl

[11] Khairullin R. S., “Solvability of the Dirichlet problem for a mixed-type equation of the second kind”, Diff Equ., 53:5 (2017), 677–685 | DOI | DOI | MR | Zbl

[12] Kozhanov A. I., “Boundary value problems for ultraparabolic and quasi-ultraparabolic equations with a varying direction of evolution”, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics”, Kabardino-Balkaria, Nalchik, May 17–21, 2017, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 149, VINITI, Moscow, 2018, 56–63 (In Russian) | MR

[13] Kozhanov A. I., Potapova S. V., “Boundary value problems for odd order forward-backward-type differential equations with two time variables”, Siberian Math. J., 59:5 (2018), 870–884 | DOI | DOI | MR | Zbl

[14] Egorov I. E., “Application of the modified Galerkin method for the first boundary problem for mixed type equation”, Math. Notes NEFU, 22:3 (2015), 3–10 (In Russian) | Zbl

[15] Egorov I. E., Efimova E. S., Tikhonova I. M., “On Fredholm solvability of first boundary value problem for mixed-type second-order equation with spectral parameter”, Math. Notes NEFU, 25:1 (2018), 15–24 (In Russian) | DOI | Zbl

[16] Sabitova Yu. K., “Boundary-value problem with nonlocal integral condition for mixed-type equations with degeneracy on the transition line”, Math. Notes, 98:3 (2015), 454–465 | DOI | DOI | MR | Zbl

[17] Sabitova Yu. K., “Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation”, Russian Math. (Iz. VUZ), 53:12 (2009), 41–49 | DOI | MR | Zbl

[18] Sabitova Yu. K., “Criterion for the uniqueness of a solution of a nonlocal problem for a degenerate equation of the mixed type in a rectangular domain”, Differ. Equ., 46:8 (2010), 1215–1218 | MR | Zbl

[19] Smirnov M. M., Uravneniia smeshannogo tipa [Mixed Type Equations], Nauka, Moscow, 1970, 296 pp. (In Russian)

[20] Bitsadze A. V., Nekotorye klassy uravnenii v chastnykh proizvodnykh [Some Classes of Partial Differential Equations], Nauka, Moscow, 1981, 448 pp. (In Russian)

[21] Sabitov K. B., K teorii uravnenii smeshannogo tipa [On the Theory of Equations of Mixed Type], Fizmatlit, Moscow, 2014, 304 pp. (In Russian)

[22] Arnol'd V. I., “Small denominators. I. Mapping the circle onto itself”, Izv. Akad. Nauk SSSR Ser. Mat., 25:1 (1961), 21–86 (In Russian) | MR | Zbl

[23] Arnol'd V. I., “Small denominators and problems of stability of motion in classical and celestial mechanics”, Russian Math. Surveys, 18:6 (1963), 85–191 | DOI | MR | Zbl

[24] Keldysh M. V., “On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations”, Dokl. Akad. Nauk SSSR, 77:5 (1951), 11–14 (In Russian) | Zbl

[25] Bateman H., Erdélyi A., Vysshie transtsendentnye funktsii [Higher Transcendental Functions], v. 2, Bessel functions, parabolic cylinder functions, orthogonal polynomials, Nauka, Moscow, 1966, 296 pp. (In Russian) | Zbl

[26] Sabitov K. B., Safin E. M., “The inverse problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain”, Russian Math. (Iz. VUZ), 54:4 (2010), 48–54 | DOI | MR | Zbl

[27] Khinchin A. Ya., Tsepnye drobi [Continued Fraction], Nauka, Moscow, 1978, 112 pp. (In Russian) | MR

[28] Bukhshtab A. A., Teoriia chisel [Number Theory], Lan', St. Petersburg, 2008, 384 pp. (In Russian)

[29] Sabitov K. B., Uravneniia matematicheskoi fiziki [Equations of Mathematical Physics], Fizmatlit, Moscow, 2013, 312 pp. (In Russian)