Solvability and completeness of solutions of parabolic differential-operator equations

Author M. M. Mamedov
mamadali@yahoo.com
Sumgait State University, Azerbaijan

Abstract We consider an abstract Cauchy problem for parabolic differential-operator equations in Hilbert spaces. Initial boundary value problems for parabolic equations are reduced to the Cauchy problem for a system of parabolic differential equations. It is proved that the solution of an initial boundary value problem for partial parabolic equation can be approximated by linear combinations of elementary solutions. Completeness of elementary solutions is also proved for differential-operator equations in abstract Hilbert spaces. The obtained abstract results are applied to differential equations.
Keywords Cauchy problem for parabolic differential-operator equations; boundary value problem
Reference 1. Yakubov S.Y. Linear differential-operator equations and their applications. – Baku, Elm, 1985.

 2. Ashyralyev A., Sozen Y., Sobolevskii P.E. A note on the parabolic differential and difference equations, Abstr. Appl. Anal. – 2007, Art. ID 61659. – 16p.

 3. Amann H. Linear and quasilinear parabolic problems, Birkhauser – Basel, 1995.

 4. Krein S.G. Linear differential equations in Banach space – M. Nauka, 1967.

 5. Yakubov S.Y., Yakubov Y.Y. Differential-Operator Equations, Ordinary and Partial Differential Equations, – Chapmen and Hall /CRC, Boca Raton, 2000.

 6. Triebel H. Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften – Berlin, 1978. – 528p.

 7. Shakhmurov V.B. Coercive boundary value problems for regular degenerate differential-operator equations// J. Math. Anal. Appl. – 2004. – V.292, ¹2. – P. 605–620.
Pages 77-85
Volume 36
Issue 1
Year 2011
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML