# Nonclassical analog of the Goursat problem for a three-dimensional equation with highest derivative

Article

First Online:

- 32 Downloads
- 1 Citations

## Abstract

In the present paper, we study the Goursat problem for a three-dimensional equation with highest derivative of fifth order with *L* _{ p }-coefficients and establish a homeomorphism between certain pairs of Banach spaces by reducing this problem to the equivalent Volterra integral equation.

## Keywords

three-dimensional equation with highest derivative of fifth order Goursat problem Volterra integral equation Sobolev space## 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 Math. (Iz. VUZ)
**45**(11), 74–78 (2001)].Google Scholar - 2.E. A. Utkina, “On a boundary-value problem with shifts in a four-dimensional space,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 4, 50–55 (2009) [Russian Math. (Iz. VUZ)
**53**(4), 40–44 (2009)].Google Scholar - 3.E. A. Utkina, “On a three-dimensional Goursat problem,” in
*Vestnik Sam. Gos. Tekhn. Univ. Fiz.-Mat. Nauki*(SamGTU, Samara, 2001), Vol. 12, pp. 30–35 [in Russian].Google Scholar - 4.O. M. Dzhokhadze, “The three-dimensional generalized Goursat problem for a third-order equation and related general two-dimensional Volterra integral equations of the first kind,” Differ. Uravn.
**42**(3), 385–394 (2006) [Differ. Equations**42**(3), 412–421 (2006)].MathSciNetGoogle Scholar - 5.B. Midodashvili, “Generalized Goursat problem for a spatial fourth order hyperbolic equation with dominated low terms,” Proc. 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 Math. (Iz. VUZ)
**52**(9), 35–40 (2008)].Google Scholar - 7.A. N. Mironov, “On the construction of the Riemann function for an equation with leading fifth partial derivative,” Differ.Uravn.
**46**(2), 266–272 (2010) [Differ. Equations**46**(2), 270–276 (2010)].MathSciNetGoogle Scholar - 8.V. I. Zhegalov and V. A. Sevast’yanov, “The Goursat problem in the four-dimensional space,” Differ. Uravn.
**32**(10), 1429–1430 (1996) [Differ. Equations**32**(10), 1427–1428 (1996)].MathSciNetGoogle Scholar - 9.V. A. Sevast’yanov, “The Riemann method for a three-dimensional hyperbolic equation of third order,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 69–73 (1997) [Russian Math. (Iz. VUZ)
**41**(5), 66–70 (1997)].Google Scholar - 10.E. A. Utkina, “Problem with displacements for the three-dimensional Bianchi equation,” Differ. Uravn.
**46**(4), 535–539 (2010) [Differ. Equations**46**(4), 538–542 (2010)].MathSciNetGoogle Scholar - 11.V. I. Zhegalov and A. N. Mironov,
*Differential Equations with Highest Partial Derivatives*(Kazan Mat. Obshch., Kazan, 2001) [in Russian].Google Scholar - 12.A. P. Soldatov and M. Kh. Shkhanukov, “Boundary value problems with A. A. Samarskii’s general nonlocal condition for higher-order pseudoparabolic equations,” Dokl. Akad. Nauk SSSR
**297**(3), 547–552 (1987) [SovietMath. Dokl.**36**(3), 507–511 (1987)].MathSciNetGoogle Scholar - 13.O. M. Dzhokhadze, “The influence of lower terms on the well-posedness of the formulation of characteristic problems for third-order hyperbolic equations,” Mat. Zametki
**74**(4), 517–528 (2003) [Math. Notes**74**(4), 491–501 (2003)].CrossRefMathSciNetGoogle Scholar - 14.V. I. Zhegalov and E. A. Utkina, “Pseudoparabolic equation of the third order,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 73–76 (1999) [Russian Math. (Iz. VUZ)
**43**(10), 70–73 (1999)].Google Scholar - 15.I.G. Mamedov, “A fundamental solution to the Cauchy problemfor a fourth-order pseudoparabolic equation,” Zh.Vychisl. Mat. i Mat. Fiz.
**49**(1), 99–110 (2009) [Comput. Math. andMath. Phys.**49**(1), 93–104 (2009)].zbMATHGoogle Scholar - 16.M. Kh. Shkhanukov, “On some boundary value problems for an equation of third order arising from simulation of filtration of a fluid in porous media,” Differ.Uravn.
**18**(4), 689–699 (1982).zbMATHMathSciNetGoogle Scholar - 17.A. M. Nakhushev, “A nonlocal problem and the Goursat problem for a loaded equation of hyperbolic type, and their applications to the prediction of ground moisture,” Dokl. Akad. Nauk SSSR
**242**(5), 1008–1011 (1978) [Soviet Math. Dokl.**19**(5), 1243–1247 (1978)].MathSciNetGoogle Scholar - 18.V. A. Vodakhova, “A boundary value problem with non-local Nakhushev condition for a certain pseudoparabolic moisture transfer equation,” Differ. Uravn.
**18**(2), 280–285 (1982).zbMATHMathSciNetGoogle Scholar - 19.A. M. Nakhushev,
*Equations of Mathematical Biology*(Vyssh. Shkola, Moscow, 1995) [in Russian].zbMATHGoogle Scholar - 20.I. G. Mamedov, “One Goursat problem in a Sobolev space,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 2, 54–64 (2011) [Russian Math. (Iz. VUZ)
**55**(2), 46–55 (2011)].Google Scholar - 21.I. G. Mamedov, “Generalization multipoint boundary-value problems of Bitsadze-Samarski and Samarski-Ionkin type for fourth order loaded hyperbolic integro-differential equations and their operator generalization,” Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb.
**23**, 77–84 (2005).zbMATHMathSciNetGoogle Scholar - 22.Yu. M. Berezanskii and Ya. A. Roitberg, “A theorem on homeomorphisms and Green’s function for general elliptic boundary problems,” Ukrain. Mat. Zh.
**19**(5), 3–32 (1967).MathSciNetGoogle Scholar - 23.S. S. Akhiev, “Fundamental solutions of some local and nonlocal boundary value problems and their representations,” Dokl. Akad. Nauk SSSR
**271**(2), 265–269 (1983) [Soviet Math. Dokl.**28**(2), 52–56 (1983)].MathSciNetGoogle Scholar - 24.N. V. Zhitarashu, “A theorem on the complete collection of isomorphisms in the
*L*_{2}-theory of model parabolic initial-boundary value problems,” in*Differential Equations and Their Invariants*,*Mathematical Studies*(Shtiintsa, Kishinev, 1986), Vol. 88, pp. 40–59 [in Russian].Google Scholar

## Copyright information

© Pleiades Publishing, Ltd. 2014