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St. Petersburg Mathematical Journal

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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
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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  • 1. G. A. Sviridyuk, On the general theory of operator semigroups, Uspekhi Mat. Nauk 49 (1994), no. 4(298), 47–74 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 4, 45–74. MR 1309441,
  • 2. G. V. Demidenko and S. V. Uspenskiĭ, \cyr Uravneniya i sistemy, ne razreshennye otnositel′no starsheĭ proizvodnoĭ, Nauchnaya Kniga (NII MIOONGU), Novosibirsk, 1998 (Russian, with Russian summary). \cyr K 90-letiyu Akademika Sergeya L′vovicha Soboleva. [Dedicated to Academician Sergeĭ L′vovich Sobolev on the 90th anniversary of his birth]; With an appendix containing a reprint of a paper by Sobolev [Izv. Akad. Nauk SSSR Ser. Mat. 18 (1954), 3–50; MR0069382 (16,1029d)]. MR 1831690
    Gennadii V. Demidenko and Stanislav V. Uspenskii, Partial differential equations and systems not solvable with respect to the highest-order derivative, Monographs and Textbooks in Pure and Applied Mathematics, vol. 256, Marcel Dekker, Inc., New York, 2003. Translated from the 1998 Russian original by Tamara Rozhkovskaya; With an appendix containing a translation of a paper by S. L. Sobolev [Izv. Akad. Nauk SSSR Ser. Math. 18 (1954), 3–50; MR0069382]. MR 2064818
  • 3. G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev type equations and degenerate semigroups of operators, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2003. MR 2225515
  • 4. A. G. Sveshnikov, A. B. Al'shin, M. O. Korpusov, and Yu. D. Pletner, Linear and nonlinear equations of Sobolev type, Fizmatlit, Moscow, 2007. (Russian)
  • 5. Angelo Favini and Atsushi Yagi, Degenerate differential equations in Banach spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 215, Marcel Dekker, Inc., New York, 1999. MR 1654663
  • 6. V. E. Fedorov, Degenerate strongly continuous semigroups of operators, Algebra i Analiz 12 (2000), no. 3, 173–200 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 3, 471–489. MR 1778194
  • 7. V. E. Fedorov, Holomorphic resolving semigroups of Sobolev-type equations in locally convex spaces, Mat. Sb. 195 (2004), no. 8, 131–160 (Russian, with Russian summary); English transl., Sb. Math. 195 (2004), no. 7-8, 1205–1234. MR 2101340,
  • 8. V. E. Fedorov, A generalization of the Hille-Yosida theorem to the case of degenerate semigroups in locally convex spaces, Sibirsk. Mat. Zh. 46 (2005), no. 2, 426–448 (Russian, with Russian summary); English transl., Siberian Math. J. 46 (2005), no. 2, 333–350. MR 2141208,
  • 9. R. S. Phillips, Perturbation theory for semi-groups of linear operators, Trans. Amer. Math. Soc. 74 (1953), 199–221. MR 0054167,
  • 10. Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
  • 11. S. G. Kreĭn and M. I. Khazan, Differential equations in Banach space, Itogi Nauki i Tekhniki. Ser. Mat. Anal., vol. 21, VINITI, Moscow, 1983, pp. 130-264; English transl. in J. Soviet Math. 30 (1985), no. 3. MR 0736523 (85f:34116)
  • 12. V. V. Ivanov, Perturbation of uniformly summable operator semigroups, Dokl. Akad. Nauk SSSR 250 (1980), no. 2, 269-273; English transl., Soviet Math. Dokl. 21 (1980), no. 1, 48-52. MR 0557767 (81d:47030)
  • 13. -, Uniformly summable semigroups of operators. II. Perturbation of the semigroups, Trudy Inst. Mat. (Novosibirsk), vol. 9, Nauka, Novosibirsk, 1987, pp. 159-182. (Russian) MR 0929254 (89d:47090)
  • 14. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
  • 15. G. A. Sviridyuk and N. A. Manakova, Regular perturbations of a class of linear equations of Sobolev type, Differ. Uravn. 38 (2002), no. 3, 423–425, 432 (Russian, with Russian summary); English transl., Differ. Equ. 38 (2002), no. 3, 447–450. MR 2005085,
  • 16. Hans Triebel, Interpolation theory, function spaces, differential operators, 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995. MR 1328645
  • 17. R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal. 3 (1972), 527–543. MR 0315239,
  • 18. D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 0610244 (83j:35084)
  • 19. P. I. Plotnikov and V. N. Starovoĭtov, The Stefan problem with surface tension as a limit of the phase field model, Differentsial′nye Uravneniya 29 (1993), no. 3, 461–471, 550 (Russian, with Russian summary); English transl., Differential Equations 29 (1993), no. 3, 395–404. MR 1236334
  • 20. P. I. Plotnikov and A. V. Klepachëva, Phase field equations and gradient flows of marginal functions, Sibirsk. Mat. Zh. 42 (2001), no. 3, 651–669, iii (Russian, with Russian summary); English transl., Siberian Math. J. 42 (2001), no. 3, 551–567. MR 1852242,
  • 21. V. E. Fedorov and A. V. Urazaeva, An inverse problem for a class of singular linear operator-differential equations, Trudy Voronezh. Zimneĭ Mat. Shkoly, Voronezh. Gos. Univ., Voronezh, 2004, pp. 16-172. (Russian)

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Additional Information

V. E. Fedorov
Affiliation: Chelyabinsk State University, Ul. Br. Kashirinykh, 454021 Chelyabinsk, Russia

O. A. Ruzakova
Affiliation: Chelyabinsk State University, Ul. Br. Kashirinykh, 454021 Chelyabinsk, Russia

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

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