# One Goursat problem in a Sobolev space

Article

First Online:

- 48 Downloads
- 2 Citations

## Abstract

In this paper we consider a hyperbolic-type differential equation with *L* _{ p }-coefficients in a three-dimensional space. For this equation we study the Goursat problem with nonclassical boundary constraints not requiringmatched conditions. We prove the equivalence of these boundary conditions to classical ones in the case when one seeks for a solution to the stated problem in an anisotropic space introduced by S. L. Sobolev. In addition, we prove the correct solvability of the Goursat problem by the method of integral equations.

## Keywords and phrases

hyperbolic equation three-dimensional Goursat problem equations with L_{p}-coefficients

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.V. I. Zhegalov and E. A. Utkina, “The Goursat Problem for a Three-Dimensional Equation with Higher Derivative,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 11, 77–81 (2001) [Russian Mathematics (Iz. VUZ)
**45**(11) 74–78 (2001)].Google Scholar - 2.E. A. Utkina, “A Certain Boundary-Value Problem with Shifts in a Four-Dimensional Space,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 4, 50–55 (2009) [Russian Mathematics (Iz. VUZ)
**53**(4) 40–44 (2009)].Google Scholar - 3.E. A. Utkina, “A Three-Dimensional Goursat Problem,” Vestn. Samarsk. Gos. Univ., Ser. Fiz.-Mat. Nauk, No. 12, 30–35 (2001).Google Scholar
- 4.O. M. Jokhadze, “The Three-Dimensional Generalized Goursat Problem for a Third-Order Equation and Related General Two-Dimensional Volterra Integral Equations of the First Kind,” Differents. Uravneniya,
**42**(3), 385–394 (2006).Google Scholar - 5.B. Midodashvili, “Generalized Goursat Problem for a Spatial Fourth Order Hyperbolic Equation with Dominated Low Terms,” Proceedings of A. Razmadze Math. Inst.
**138**, 43–54 (2005).zbMATHMathSciNetGoogle Scholar - 6.O. A. Koshcheeva, “Construction of the Riemann Function for the Bianchi Equation in an
*n*-Dimensional Space,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 9, 40–46 (2008) [Russian Mathematics (Iz. VUZ)**52**(9) 35–40 (2008)].Google Scholar - 7.A. V. Berezin, A. S. Vorontsov, M. B. Markov, and B. D. Plyushchenkov, “On the Conclusion and Decision of Maxwell’s Equations for the Problems with Given Wavefront,” Matem. Modelirovanie
**18**(4), 43–60 (2006).zbMATHMathSciNetGoogle Scholar - 8.V. I. Zhegalov and E. A. Utkina, “On a Pseudoparabolic Equation of the Third Order,” Izv. Vyssh. Uchebn. Zaved., Mat. №10, 73–76 (1999) [Russian Mathematics (Iz. VUZ)
**43**(10) 70–73 (1999)].Google Scholar - 9.I. G. Mamedov, “A Fundamental Solution to the Cauchy Problem for a Fourth-Order Pseudoparabolic Equation,” Zhurn. Vychisl. Matem. i Matem. Fiz.
**49**(1), 99–110 (2009).Google Scholar - 10.A. I. Kozhanov, “On a Nonlocal Boundary-Value Problem with Variable Coefficients for the Heat Equation and the Aller Equation,” Differents. Uravneniya
**40**(6), 763–764 (2004).MathSciNetGoogle Scholar - 11.V. A. Vodakhova, “A Boundary-Value Problem with Nakhushev Nonlocal Condition for a Certain Pseudo-Parabolic Water Transfer Equation,” Differents. Uravneniya,
**18**(2), 280–285 (1982).zbMATHMathSciNetGoogle Scholar - 12.M. Kh. Shkhanukov, “Boundary Value Problems for a Third-Order Equation Occurring in the Modeling of Water Filtration in Porous Mediua,” Differents. Uravneniya
**18**(4), 689–699 (1982).zbMATHMathSciNetGoogle Scholar - 13.A.M. Nakhushev,
*Equations of Mathematical Biology*(Vysshaya Shkola, Moscow, 1995) [in Russian].zbMATHGoogle Scholar - 14.V. I. Zhegalov, “A Three-Dimensional Analog of the Goursat Problem,” in
*Non-Classic Equations and Equations of Mixed Type*(Inst. Matem. SO AN SSSR, Novosibirsk, 1990), pp. 94–98.Google Scholar - 15.V. I. Zhegalov, “On the Three-Dimensional Riemann Function,” Sib. Matem. Zhurn.
**36**(5), 1074–1079 (1997).MathSciNetGoogle Scholar - 16.I. G. Mamedov, “On Correct Solvability of a Problem with Loaded Boundary Conditions for a Fourth Order Pseudoparabolic Equation,” in
*Memoirs on Differ. Equat. and Math. Phys.*(Georgian Academy of Sciences A. Razmadze Mathematical Institute, 2008), Vol. 43, pp. 107–118.zbMATHGoogle Scholar - 17.I. G. Mamedov, “Generalization of Multipoint Boundary-Value Problems of Bitsadze-Samarski and Samarski-Ionkin Type for Fourth Order Loaded Hyperbolic Integro-Differential Equations and Their Operator Generalization,” Proceedings of Inst. Math. and Mech.
**23**, 77–84 (2005).zbMATHGoogle Scholar - 18.I. G. Mamedov, “Integral Presentations of Functions in One Anisotropic S. L. Sobolev Space with a Dominating Mixed Derivative,” in
*Proceedings of International Conference on Mathematics and Mechanics dedicated to the 50th Anniversary of Institute of Mathematics and Mechanics of Azerbaijan Academy of Sciences*(Baku, 2009), pp. 198–199.Google Scholar - 19.Yu. M. Berezanskii and Ya. A. Roitberg, “A Theorem on Homeomorphisms and the Green Function for General Elliptic Boundary-Value Problems,” Ukr. Matem. Zhurn.
**19**(5), 3–32 (1967).MathSciNetGoogle Scholar - 20.N. V. Zhitarashu, “Theorems on the Complete Set of Isomorphisms in the
*L*_{2}-Theory of Model Initial Parabolic Boundary-Value Problems,” Matem. Issledovaniya, No. 88, 40–59 (1986).Google Scholar - 21.I. G. Mamedov, “The Local Boundary Value Problem for an Integro-Differential Equation,” Proceedings of Inst. Math. and Mech.
**17**, 96–101 (2002).Google Scholar

## Copyright information

© Allerton Press, Inc. 2011