On solvability of perturbed Sobolev type equations

V. E. Fedorov ; O. A. Ruzakova

Algebra i analiz, Tome 20 (2008) no. 4, pp. 189-217.

Résumé

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 the cases the perturbed operators are integral or differential.

Détail

Keywords: Perturbation theory, semigroup, Cauchy problem, Sobolev type equation.

Bibliographie

[1] Sviridyuk G. A., “K obschei teorii polugrupp operatorov”, Uspekhi mat. nauk, 49:4 (1994), 47–74 | MR | Zbl

[2] Demidenko G. V., Uspenskii C. V., Uravneniya i sistemy, ne razreshennye otnositelno starshei proizvodnoi, Nauch. kn., Novosibirsk, 1998 | MR | Zbl

[3] Sviridyuk G. A., Fedorov V. E., Linear Sobolev type equations and degenerate semigroups of operators, VSP, Utrecht, 2003 | MR | Zbl

[4] Sveshnikov A. G., Alshin A. B., Korpusov M. O., Pletner Yu. D., Lineinye i nelineinye uravneniya sobolevskogo tipa, Fizmatlit, M., 2007 | Zbl

[5] Favini A., Yagi A., Degenerate differential equations in Banach spaces, Monogr. Textbooks Pure Appl. Math., 215, Marcel Dekker, Inc., New York, 1999 | MR | Zbl

[6] Fedorov V. E., “Vyrozhdennye silno nepreryvnye polugruppy operatorov”, Algebra i analiz, 12:3 (2000), 173–200 | MR

[7] Fedorov V. E., “Golomorfnye razreshayuschie polugruppy uravnenii sobolevskogo tipa v lokalno vypuklykh prostranstvakh”, Mat. sb., 195:8 (2004), 131–160 | MR | Zbl

[8] Fedorov V. E., “Obobschenie teoremy Khille–Iosidy na sluchai vyrozhdennykh polugrupp v lokalno vypuklykh prostranstvakh”, Sib. mat. zh., 46:2 (2005), 426–448 | MR | Zbl

[9] Phillips R. S., “Perturbation theory for semi-groups of linear operators”, Trans. Amer. Math. Soc., 74 (1953), 199–221 | DOI | MR

[10] Khille E., Fillips R. S., Funktsionalnyi analiz i polugruppy, IL, M., 1962

[11] Krein S. G., Khazan M. I., “Differentsialnye uravneniya v banakhovom prostranstve”, Itogi nauki i tekhn. Ser. Mat. analiz, 21, VINITI, M., 1983, 130–264 | MR

[12] Ivanov V. V., “Vozmuschenie ravnomerno summiruemykh polugrupp operatorov”, Dokl. AN SSSR, 250:2 (1980), 269–273 | MR | Zbl

[13] Ivanov V. V., “Ravnomerno summiruemye polugruppy operatorov. II. Vozmuschenie polugrupp”, Issledovaniya po geometrii “v tselom” i matematicheskomu analizu, Tr. In-ta mat. SO, 9, Nauka, Novosibirsk, 1987, 159–182 | MR

[14] Danford N., Shvarts Dzh. T., Lineinye operatory. Obschaya teoriya, Editorial URSS, M., 2004

[15] Sviridyuk G. A., Manakova N. A., “Regulyarnye vozmuscheniya odnogo klassa lineinykh uravnenii sobolevskogo tipa”, Differents. uravneniya, 38:3 (2002), 423–425 | MR | Zbl

[16] Tribel X., Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980 | MR

[17] Showalter R. E., “Existence and representation theorems for a semilinear Sobolev equation in Banach space”, SIAM J. Math. Anal., 3:3 (1972), 527–543 | DOI | MR | Zbl

[18] Khenri D., Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii, Mir, M., 1985 | MR

[19] Plotnikov P. I., Starovoitov V. N., “Zadacha Stefana s poverkhnostnym natyazheniem kak predel modeli fazovogo polya”, Differents. uravneniya, 29:3 (1993), 461–471 | MR | Zbl

[20] Plotnikov P. I., Klepacheva A. V., “Uravneniya fazovogo polya i gradientnye potoki marginalnykh funktsii”, Sib. mat. zh., 42:3 (2001), 651–669 | MR | Zbl

[21] Fedorov V. E., Urazaeva A. V., “Obratnaya zadacha dlya odnogo klassa singulyarnykh lineinykh operatorno-differentsialnykh uravnenii”, Tr. Voronezhsk. zimn. mat. shk., VGU, Voronezh, 2004, 161–172