Geodesic
A similar for $\Delta_1$ problem for the second-order hyperbolic equation in the 3D Euclidean space
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 19 (2015) no. 4, pp. 697-709.

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

The second-order hyperbolic type equation is considered in the 3D Euclidean space. Boundary value problem is posed in the infinite cylindrical region bounded by the characteristic surfaces of this equation with data on the related characteristic surfaces of the equation and with conditions mates on the internal non-descriptive plane. The solution is also assumed to be zero when $z\to\infty$ with derivative by variable $z$. By the Fourier transform method the problem reduced to the corresponding planar problem $\Delta_1$ for hyperbolic equation, which in characteristic coordinates is the generalized Euler–Darboux equation with a negative parameter. Authors obtained estimates of the plane problem solution and its partial derivatives up to the second order inclusive. This, in turn, provided an opportunity to impose the conditions to given boundary functions ensuring the existence of a classical solution of the problem in the form of the Fourier transform.
Keywords: integral equations, boundary value problems, second-order hyperbolic type equations. 
@article{VSGTU_2015_19_4_a8,
     author = {I. N. Rodionova and V. M. Dolgopolov},
     title = {A similar for $\Delta_1$ problem for the second-order hyperbolic equation in the {3D} {Euclidean} space},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {697--709},
     publisher = {mathdoc},
     volume = {19},
     number = {4},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2015_19_4_a8/}
}
TY  - JOUR
AU  - I. N. Rodionova
AU  - V. M. Dolgopolov
TI  - A similar for $\Delta_1$ problem for the second-order hyperbolic equation in the 3D Euclidean space
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2015
SP  - 697
EP  - 709
VL  - 19
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2015_19_4_a8/
LA  - ru
ID  - VSGTU_2015_19_4_a8
ER  - 
%0 Journal Article
%A I. N. Rodionova
%A V. M. Dolgopolov
%T A similar for $\Delta_1$ problem for the second-order hyperbolic equation in the 3D Euclidean space
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2015
%P 697-709
%V 19
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2015_19_4_a8/
%G ru
%F VSGTU_2015_19_4_a8
I. N. Rodionova; V. M. Dolgopolov. A similar for $\Delta_1$ problem for the second-order hyperbolic equation in the 3D Euclidean space. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 19 (2015) no. 4, pp. 697-709. http://geodesic.mathdoc.fr/item/VSGTU_2015_19_4_a8/

[1] Bitsadze A. V., “On the problem of equations of mixed type in multidimensional domains”, Dokl. Akad. Nauk SSSR, 110:6 (1956), 901–902 (In Russian) | Zbl

[2] Nakhushev A. M., “On an Three-Dimensional Analog of the Gellerstedt Problem”, Differ. Uravn., 4:1 (1968), 52–62 (In Russian) | Zbl

[3] Pul'kin S. P., “On the question of formulating the Tricomi problem in space”, Uchenye zapiski Kuib. ped. in-ta, 1956, no. 14, 63–77 (In Russian)

[4] Dolgopolov V. M., Dolgopolov M. V., Rodionova I. N., “Construction of special classes of solutions for some differential equations of hyperbolic type”, Dokl. Math., 80:3 (2009), 860–866 | DOI | MR | Zbl

[5] Dolgopolov M. V., Rodionova I. N., “Problems involving equations of hyperbolic type in the plane or three-dimensional space with conjugation conditions on a characteristic”, Izv. Math., 75:4 (2011), 681–689 | DOI | DOI | MR | Zbl

[6] Dolgopolov V. M., Rodionova I. N., “Extremal properties of solutions of special classes of a hyperbolic-type equation”, Math. Notes, 92:4 (2012), 490–496 | DOI | DOI | Zbl