Abstract:
Linear Sobolev type equations
L˙u(t)=Mu(t)+Nu(t),t∈¯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 the cases the perturbed operators are integral or differential.
Keywords:
Perturbation theory, semigroup, Cauchy problem, Sobolev type equation.
Citation:
V. E. Fedorov, O. A. Ruzakova, “On solvability of perturbed Sobolev type equations”, Algebra i Analiz, 20:4 (2008), 189–217; St. Petersburg Math. J., 20:4 (2009), 645–664
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\by V.~E.~Fedorov, O.~A.~Ruzakova
\paper On solvability of perturbed Sobolev type equations
\jour Algebra i Analiz
\yr 2008
\vol 20
\issue 4
\pages 189--217
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\transl
\jour St. Petersburg Math. J.
\yr 2009
\vol 20
\issue 4
\pages 645--664
\crossref{https://doi.org/10.1090/S1061-0022-09-01065-6}
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Linking options:
https://www.mathnet.ru/eng/aa526
https://www.mathnet.ru/eng/aa/v20/i4/p189
This publication is cited in the following 8 articles:
V. E. Fedorov, O. A. Stakheeva, “On the Local Existence of Solutions of Equations with Memory not Solvable with Respect to the Time Derivative”, Math. Notes, 98:3 (2015), 472–483
V. E. Fedorov, E. A. Omel'chenko, “Linear equations of the Sobolev type with integral delay operator”, Russian Math. (Iz. VUZ), 58:1 (2014), 60–69
M. V. Plekhanova, V. E. Fedorov, “On control of degenerate distributed systems”, Ufa Math. J., 6:2 (2014), 77–96
V. E. Fedorov, N. D. Ivanova, Yu. Yu. Fedorova, “On a time nonlocal problem for inhomogeneous evolution equations”, Siberian Math. J., 55:4 (2014), 721–733
V. E. Fedorov, L. V. Borel', “Solvability of weighted linear evolution equations with degenerate operator at the derivative”, St. Petersburg Math. J., 26:3 (2015), 487–497
V. E. Fedorov, E. A. Omel'chenko, “Inhomogeneous degenerate Sobolev type equations with delay”, Siberian Math. J., 53:2 (2012), 335–344
V. E. Fedorov, B. Shklyar, “Exact null controllability of degenerate evolution equations with scalar control”, Sb. Math., 203:12 (2012), 1817–1836
M. V. Plekhanova, E. S. Zorina, “Optimalnoe upravlenie polulineinymi
sistemami sobolevskogo tipa v zadachakh bez ucheta zatrat na upravlenie”, Vestnik ChelGU, 2012, no. 15, 80–89