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On solvability of perturbed Sobolev type equations


Authors: V. E. Fedorov and O. A. Ruzakova
Translated by: the authors
Original publication: Algebra i Analiz, tom 20 (2008), nomer 4.
Journal: St. Petersburg Math. J. 20 (2009), 645-664
MSC (2000): Primary 34G25
DOI: https://doi.org/10.1090/S1061-0022-09-01065-6
Published electronically: June 2, 2009
MathSciNet review: 2473748
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Abstract | References | Similar Articles | Additional Information

Abstract: Linear Sobolev type equations

$\displaystyle L\dot u(t)=Mu(t)+Nu(t),\quad t\in\overline{\mathbb{R}}_+,$

are considered, with degenerate operator $ L$, strongly $ (L,p)$-radial operator $ M$, and perturbing operator $ N$. By using methods of perturbation theory for operator semigroups and the theory of degenerate semigroups, unique solvability conditions for the Cauchy problem and Showalter problem for such equations are deduced. The abstract results obtained are applied to the study of initial boundary value problems for a class of equations, the operators in which are polynomials of elliptic selfadjoint operators, including various equations of filtration theory. Perturbed linearized systems of the phase space equations and of the Navier-Stokes equations are also considered. In all cases the perturbed operators are integral or differential.


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Additional Information

V. E. Fedorov
Affiliation: Chelyabinsk State University, Ul. Br. Kashirinykh, 454021 Chelyabinsk, Russia
Email: kar@csu.ru

O. A. Ruzakova
Affiliation: Chelyabinsk State University, Ul. Br. Kashirinykh, 454021 Chelyabinsk, Russia
Email: amber@csu.ru

DOI: https://doi.org/10.1090/S1061-0022-09-01065-6
Keywords: Perturbation theory, semigroup, Cauchy problem, Sobolev type equation
Received by editor(s): April 14, 2007
Published electronically: June 2, 2009
Article copyright: © Copyright 2009 American Mathematical Society