Geodesic
The Showalter–Sidorov problem as a phenomena of the Sobolev-type equations
The Bulletin of Irkutsk State University. Series Mathematics, Tome 3 (2010) no. 1, pp. 104-125 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This article has a survey character. It contains: 1. The Sowalter approach, 2. The Sidorov approach, 3. Applications of the Showalter–Sidorov problem, 4. A generalization of the Showalter–Sidorov problem.
Keywords: the Cauchy problem, the Showalter–Sidorov problem, initial-finish value problem, the Sobolev-type equations. 
@article{IIGUM_2010_3_1_a10,
     author = {G. A. Sviridyuk and S. A. Zagrebina},
     title = {The {Showalter{\textendash}Sidorov} problem as a phenomena of the {Sobolev-type} equations},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {104--125},
     year = {2010},
     volume = {3},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2010_3_1_a10/}
}
TY  - JOUR
AU  - G. A. Sviridyuk
AU  - S. A. Zagrebina
TI  - The Showalter–Sidorov problem as a phenomena of the Sobolev-type equations
JO  - The Bulletin of Irkutsk State University. Series Mathematics
PY  - 2010
SP  - 104
EP  - 125
VL  - 3
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/IIGUM_2010_3_1_a10/
LA  - ru
ID  - IIGUM_2010_3_1_a10
ER  - 
%0 Journal Article
%A G. A. Sviridyuk
%A S. A. Zagrebina
%T The Showalter–Sidorov problem as a phenomena of the Sobolev-type equations
%J The Bulletin of Irkutsk State University. Series Mathematics
%D 2010
%P 104-125
%V 3
%N 1
%U http://geodesic.mathdoc.fr/item/IIGUM_2010_3_1_a10/
%G ru
%F IIGUM_2010_3_1_a10
G. A. Sviridyuk; S. A. Zagrebina. The Showalter–Sidorov problem as a phenomena of the Sobolev-type equations. The Bulletin of Irkutsk State University. Series Mathematics, Tome 3 (2010) no. 1, pp. 104-125. http://geodesic.mathdoc.fr/item/IIGUM_2010_3_1_a10/

[1] G. I. Barenblatt, Yu. P. Zheltov, I. N. Kochina, “Ob osnovnykh predstavleniyakh teorii filtratsii v treschinnovatykh sredakh”, Priklad. matematika i mekhanika, 24:5 (1960), 58–73 | MR

[2] N. N. Verigin, “Ob odnom klasse gidromekhanicheskikh zadach dlya oblastei s podvizhnymi granitsami”, Dinamika zhidkosti so svobodnoi granitsei, 46, Novosibirsk, 1980, 23–32 | MR | Zbl

[3] V. N. Vragov, Kraevye zadachi dlya neklassicheskikh uravnenii matematicheskoi fiziki, NGU, Novosibirsk, 1983

[4] A. F. Gilmutdinova, Issledovanie matematicheskikh modelei s fenomenom needinstvennosti, dis. ... kand. fiz.-mat. nauk, Chelyabinsk, 2009

[5] A. F. Gilmutdinova, “O needinstvennosti reshenii zadachi Shouoltera–Sidorova dlya odnoi modeli Plotnikova”, Vestn. SamGU, 2007, no. 9/1, 85–90 | MR | Zbl

[6] S. A. Zagrebina, “Zadacha Shouoltera–Sidorova–Verigina dlya lineinykh uravnenii sobolevskogo tipa”, Neklassicheskie uravneniya matematicheskoi fiziki, Differentsialnye uravneniya, teoriya funktsii i prilozheniya, sb. tr. mezhdunar. konf., posvyasch. 100-letiyu so dnya rozhdeniya akad. I. N. Vekua (Novosibirsk, 2007), 150–157

[7] S. A. Zagrebina, N. P. Soloveva, “Nachalno-konechnaya zadacha dlya lineinykh evolyutsionnykh uravnenii sobolevskogo tipa na grafe”, Obozrenie priklad. i prom. matematiki, 16:2 (2009), 329–330

[8] S. A. Zagrebina, “O zadache Shouoltera–Sidorova”, Izv. vuzov. Matematika, 2007, no. 3, 22–28 | MR | Zbl

[9] S. A. Zagrebina, M. A. Sagadeeva, “Obobschennaya zadacha Shouoltera–Sidorova dlya uravnenii sobolevskogo tipa s silno (L,p)-radialnym operatorom”, Vestn. MaGU. Ser.: Matematika, 2006, no. 9, 17–27 | Zbl

[10] A. V. Keller, “Chislennoe reshenie zadachi startovogo upravleniya dlya sistemy uravnenii leontevskogo tipa”, Obozrenie priklad. i prom. matematiki, 16:2 (2009), 345–346

[11] A. V. Keller, “Chislennoe reshenie zadachi zhestkogo upravleniya dlya sistemy uravnenii leontevskogo tipa”, Obozrenie priklad. i prom. matematiki, 16:4 (2009), 666–667

[12] M. O. Korpusov, Yu. D. Pletner, A. G. Sveshnikov, “O kvazistatsionarnykh protsessakh v provodyaschikh sredakh bez dispersii”, Zhurn. vychislit. matematiki i mat. fiziki, 40:8 (2000), 1237–1249 | MR | Zbl

[13] O. A. Ladyzhenskaya, Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, 2-e izd., Nauka, M., 1970 | MR

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

[15] Zh.-L. Lions, Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[16] N. A. Manakova, “Zadacha optimalnogo upravleniya dlya uravneniya Oskolkova nelineinoi filtratsii”, Differents. uravneniya, 43:9 (2007), 1185–1192 | MR | Zbl

[17] N. A. Manakova, Issledovanie zadach optimalnogo upravleniya dlya neklassicheskikh uravnenii matematicheskoi fiziki, dis. ... kand. fiz.-mat. nauk, Chelyabinsk, 2005

[18] A. P. Oskolkov, “Nelokalnye problemy dlya odnogo klassa nelineinykh operatornykh uravnenii, voznikayuschikh v teorii uravnenii tipa S. L. Soboleva”, Zap. nauch. seminarov LOMI, 198, 1991, 31–48 | Zbl

[19] I. G. Petrovskii, Lektsii ob uravneniyakh s chastnymi proizvodnymi, Fizmatgiz, M., 1961 | MR

[20] O. A. Romanova, Zhordanovy nabory i psevdoobratnye operatory v teorii nekotorykh klassov vyrozhdennykh differentsialnykh uravnenii, dis. ... kand. fiz.-mat. nauk, Irkutsk, 1984

[21] G. A. Sviridyuk, S. A. Zagrebina, “Zadacha Verigina dlya lineinykh uravnenii sobolevskogo tipa s otnositelno $p$-sektorialnymi operatorami”, Differents. uravneniya, 38:12 (2002), 1646–1652 | MR | Zbl

[22] G. A. Sviridyuk, A. A. Efremov, “Zadacha optimalnogo upravleniya dlya odnogo klassa lineinykh uravnenii tipa Soboleva”, Izv. vuzov. Matematika, 1996, no. 12, 75–83 | MR | Zbl

[23] G. A. Sviridyuk, N. A. Manakova, “Zadacha optimalnogo upravleniya dlya uravneniya Khoffa”, Sib. zhurn. industr. matematiki, 8:2 (2005), 144–151 | MR

[24] G. A. Sviridyuk, “Mnogoobraziya reshenii odnogo klassa evolyutsionnykh i dinamicheskikh uravnenii”, DAN SSSR, 304:2 (1989), 301–304 | Zbl

[25] G. A. Sviridyuk, A. F. Karamova, “O skladke fazovogo prostranstva odnogo neklassicheskogo uravneniya”, Differents. uravneniya, 41:10 (2005), 1476–1581 | MR

[26] G. A. Cviridyuk, “Ob odnoi zadache Showalter”, Differents. uravneniya, 25:2 (1989), 338–339 | MR

[27] G. A. Sviridyuk, “Ob odnoi modeli slaboszhimaemoi vyazkouprugoi zhidkosti”, Izv. vuzov. Matematika, 1994, no. 1, 62–70 | MR | Zbl

[28] G. A. Sviridyuk, “Odna zadacha dlya obobschennogo filtratsionnogo uravneniya Bussineska”, Izv. vuzov. Matematika, 1989, no. 2, 55–61 | MR | Zbl

[29] G. A. Sviridyuk, M. V. Klimentev, “Fazovye prostranstva uravnenii tipa Soboleva s $s$-monotonnymi i silno koertsitivnymi operatorami”, Izv. vuzov. Matematika, 1994, no. 11, 75–82 | MR | Zbl

[30] G. A. Sviridyuk, S. V. Brychev, “Chislennoe reshenie sistem uravnenii leontevskogo tipa”, Izv. vuzov. Matematika, 2003, no. 8, 46–52 | MR | Zbl

[31] N. A. Sidorov, O. A. Romanova, “O primenenii nekotorykh rezultatov teorii vetvleniya pri reshenii differentsialnykh uravnenii”, Differents. uravneniya, 19:9 (1983), 1516–1526 | MR | Zbl

[32] N. A. Sidorov, “Ob odnom klasse vyrozhdennykh differentsialnykh uravnenii s konvergentsiei”, Mat. zametki, 25:4 (1984), 569–578 | MR

[33] N. A. Sidorov, M. V. Falaleev, “Obobschennye resheniya differentsialnykh uravnenii s fredgolmovym operatorom pri starshikh proizvodnykh”, Differents. uravneniya, 23:4 (1987), 726–728 | MR | Zbl

[34] N. A. Sidorov, Obschie voprosy regulyarizatsii v zadachakh teorii vetvleniya, IrGTU, Irkutsk, 1982 | MR

[35] R. Temam, Uravneniya Nave–Stoksa. Teoriya i chislennyi analiz, 2-e izd., Mir, M., 1981 | MR | Zbl

[36] M. V. Falaleev, Elementy teorii obobschennykh reshenii nekotorykh klassov vyrozhdennykh differentsialnykh i integralnykh uravnenii v banakhovykh prostranstvakh, avtoref. ... kand. fiz.-mat. nauk, Sverdlovsk, 1988

[37] V. E. Fedorov, M. V. Plekhanova, “Optimalnoe upravlenie lineinymi uravneniyami sobolevskogo tipa”, Differents. uravneniya, 40:11 (2004), 1548–1556 | MR | Zbl

[38] P. J. Chen, M. E. Gurtin, “On a theory of heat conduction involving two temperatures”, Z. Angew. Math. Phys., 19 (1968), 614–627 | DOI | Zbl

[39] P. L. Davis, “A quasilinear parabolic and a related third order problem”, J. Math. Anal. and Appl., 40:2 (1972), 327–335 | MR | Zbl

[40] G. V. Demidenko, S. V. Uspenskii, Partial differential equations and systems not solvable with respect to the highest-order derivative, Marcel Dekker, Inc., N.Y.–Basel–Hong Kong, 2003 | MR | Zbl

[41] A. Favini, A. Yagi, Degenerate differential equations in Banach spaces, Marcel Dekker, Inc., N.Y., 1999 | MR | Zbl

[42] M. Hallaire, “On a theory of moisture-transfer”, Inst. Rech. Agronom., 1964, no. 3, 60–72

[43] N. J. Hoff, “Creep buckling”, Aeron. Quarterly, 7:1 (1956), 1–20

[44] N. Sidorov, B. Loginov, A. Sinithyn, M. Falaleev, Lyapunov–Shmidt method in nonlinear analysis and applications, Kluwer Academic Publishers, Dordrecht–Boston–London, 2002 | MR | Zbl

[45] S. G. Pyatkov, Operator theory. Nonclassical problems, VSP, Utrecht–Boston–Koln–Tokyo, 2002 | MR | Zbl

[46] R. E. Showalter, Hilbert Space Methods for Partial Differential Equations, Pitman, London–San Francisco–Melbourne, 1977 | MR | Zbl

[47] R. E. Showalter, Monotone operators in Banach Space and and nonlinear partial differential equations, AMS, Providence, 1997 | MR

[48] R. E. Showalter, “Nonlinear degenerate evolution equations and partial differential equations of mixed type”, SIAM J. Math. Anal., 6:1 (1975), 25–42 | DOI | MR | Zbl

[49] R. E. Showalter, “The Sobolev equations. I; II”, Appl. Anal., 5:1 (1975), 15–22 ; 2, 81–99 | DOI | MR | Zbl | MR | Zbl

[50] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht–Boston–Köln–Tokyo, 2003 | MR | Zbl

[51] T. W. Ting, “Parabolic and pseudoparabolic partial differential equations”, J. Math. Soc. Jap., 21:3 (1969), 440–453 | DOI | MR | Zbl