Open Access Research

Well-posedness of a boundary value problem for a class of third-order operator-differential equations

Araz R Aliev12 and Ahmed L Elbably13*

Author Affiliations

1 Baku State University, 23 Z. Khalilov St., Baku, 1148, Azerbaijan

2 Institute of Mathematics and Mechanics of NAS of Azerbaijan, 9 B. Vahabzadeh St., Baku, 1141, Azerbaijan

3 Helwan University, Ain Helwan, Cairo, 11795, Egypt

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Boundary Value Problems 2013, 2013:140  doi:10.1186/1687-2770-2013-140

Published: 30 May 2013

Abstract

This paper investigates the well-posedness of a boundary value problem on the semiaxis for a class of third-order operator-differential equations whose principal part has multiple real characteristics. We obtain sufficient conditions for the existence and uniqueness of the solution of a boundary value problem in the Sobolev-type space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1">View MathML</a>. These conditions are expressed in terms of the operator coefficients of the investigated equation. We find relations between the estimates of the norms of intermediate derivatives operators in the subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1">View MathML</a> and the solvability conditions. Furthermore, we calculate the exact values of these norms. The results are illustrated with an example of the initial-boundary value problems for partial differential equations.

MSC: 34G10, 47A50, 47D03, 47N20.

Keywords:
well-posed and unique solvability; operator-differential equation; multiple characteristic; self-adjoint operator; the Sobolev-type space; inter-mediate derivatives operators; factorization of pencils