On solvability of perturbed Sobolev type equations

Fedorov, V.; Ruzakova, O.

doi:10.1090/S1061-0022-09-01065-6

On solvability of perturbed Sobolev type equations
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by V. E. Fedorov and O. A. Ruzakova
Translated by: the authors

St. Petersburg Math. J. 20 (2009), 645-664

DOI: https://doi.org/10.1090/S1061-0022-09-01065-6

Published electronically: June 2, 2009

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Abstract:

Linear Sobolev type equations \[ L\dot u(t)=Mu(t)+Nu(t),\quad t\in \overline {\mathbb R}_+,\] 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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