Periodic Boundary Value Problem for Nonlinear Sobolev-Type Equations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 3, pp. 14-26.

Voir la notice de l'article provenant de la source Math-Net.Ru

The large-time asymptotic behavior of solutions to the periodic boundary value problem for a nonlinear Sobolev-type equation is studied. In particular, the case where the initial perturbations are not small is considered. In this case, the large-time behavior of solutions is dichotomous.
Keywords: periodic boundary value problem, asymptotic behavior.
Mots-clés : Sobolev-type equation
@article{FAA_2010_44_3_a1,
     author = {E. I. Kaikina and P. I. Naumkin and I. A. Shishmarev},
     title = {Periodic {Boundary} {Value} {Problem} for {Nonlinear} {Sobolev-Type} {Equations}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {14--26},
     publisher = {mathdoc},
     volume = {44},
     number = {3},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2010_44_3_a1/}
}
TY  - JOUR
AU  - E. I. Kaikina
AU  - P. I. Naumkin
AU  - I. A. Shishmarev
TI  - Periodic Boundary Value Problem for Nonlinear Sobolev-Type Equations
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2010
SP  - 14
EP  - 26
VL  - 44
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2010_44_3_a1/
LA  - ru
ID  - FAA_2010_44_3_a1
ER  - 
%0 Journal Article
%A E. I. Kaikina
%A P. I. Naumkin
%A I. A. Shishmarev
%T Periodic Boundary Value Problem for Nonlinear Sobolev-Type Equations
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2010
%P 14-26
%V 44
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2010_44_3_a1/
%G ru
%F FAA_2010_44_3_a1
E. I. Kaikina; P. I. Naumkin; I. A. Shishmarev. Periodic Boundary Value Problem for Nonlinear Sobolev-Type Equations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 3, pp. 14-26. http://geodesic.mathdoc.fr/item/FAA_2010_44_3_a1/

[1] G. Gaevskii, K. Greger, K. Zakharias, Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978 | MR

[2] A. L. Gladkov, “Edinstvennost resheniya zadachi Koshi dlya nekotorykh kvazilineinykh psevdoparabolicheskikh uravnenii”, Matem. zametki, 60:3 (1996), 356–362 | DOI | MR | Zbl

[3] G. V. Demidenko, S. V. Uspenskii, Uravneniya i sistemy, ne razreshennye otnositelno starshei proizvodnoi, Nauchnaya kniga, Novosibirsk, 1998 | MR | Zbl

[4] I. E. Egorov, S. G. Pyatkov, S. V. Popov, Neklassicheskie differentsialno-operatornye uravneniya, Nauka, Novosibirsk, 2000 | MR

[5] E. I. Kaikina, P. I. Naumkin, I. A. Shishmarëv, “Asimptotika reshenii pri bolshikh vremenakh dlya nelineinykh uravnenii tipa Soboleva”, UMN, 64:3(387) (2009), 3–72 | DOI | MR | Zbl

[6] A. I. Kozhanov, “Nachalno-kraevaya zadacha dlya uravneniya tipa obobschennogo uravneniya Bussineska s nelineinym istochnikom”, Matem. zametki, 65:1 (1999), 70–75 | DOI | MR | Zbl

[7] M. O. Korpusov, A. G. Sveshnikov, “O razrushenii reshenii klassa silno nelineinykh volnovykh dissipativnykh uravnenii tipa Soboleva s istochnikami”, Izv. RAN, ser. matem., 69:4 (2005), 89–128 | DOI | MR | Zbl

[8] S. I. Lyashko, Obobschennoe upravlenie lineinymi sistemami, Naukova dumka, Kiev, 1998 | Zbl

[9] D. A. Nomirovskii, “O gomeomorfizmakh, osuschestvlyaemykh nekotorymi differentsialnymi operatorami s chastnymi proizvodnymi”, Ukr. matem. zh., 12 (2004), 1243–1252

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

[11] G. A. Sviridyuk, V. E. Fedorov, “Analiticheskie polugruppy s yadrami i lineinye uravneniya tipa Soboleva”, Sib. matem. zh., 36:5 (1995), 1130–1145 | MR | Zbl

[12] S. L. Sobolev, “Ob odnoi novoi zadache matematicheskoi fiziki”, Izv. AN SSSR, ser. matem., 18:1 (1954), 3–50 | MR | Zbl

[13] I. Stein, G. Veis, Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[14] H. Begehr, D. Q. Dai, “Initial boundary value problem for nonlinear pseudoparabolic equations”, Complex Variables, Theory Appl., 18:1–2 (1992), 33–47 | DOI | MR | Zbl

[15] P. Biler, “Large-time behavior of periodic solutions to dissipative equations of Korteweg–de Vries–Burgers type”, Bull. Polish Acad. Sci., Math., 32:7–8 (1984), 401–405 | MR | Zbl

[16] C. Bu, R. Shull, K. Zhao, “A periodic boundary value problem for a generalized 2D Ginzburg–Landau equation”, Hokkaido Math. J., 27:1 (1998), 197–211 | MR | Zbl

[17] E. Di Benedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993 | MR | Zbl

[18] A. Constantin, J. Escher, “Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation”, Commun. Pure Appl. Math., 51:5 (1998), 475–504 | 3.0.CO;2-5 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[19] C. M. Dafermos, “Large time behavior of periodic solutions of hyperbolic systems of conservation laws”, J. Differential Equations, 121:1 (1995), 183–202 | DOI | MR | Zbl

[20] A. Favini, A. Yagi, Degenerate differential equations in Banach spaces, Marcel Dekker, New York, 1999 | MR | Zbl

[21] P. C. Fife, “Asymptotic states for equations of reaction and diffusion”, Bull. Amer. Math. Soc., 84:5 (1978), 693–726 | DOI | MR | Zbl

[22] B. Guo, X. M. Xiang, “The large time convergence of spectral method for generalized Kuramoto–Sivashinsky equations”, J. Comput. Math., 15:1 (1997), 1–13 | MR | Zbl

[23] N. Hayashi, E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, Asymptotics for Dissipative Nonlinear Equations, Lecture Notes in Math., 1884, Springer-Verlag, Berlin, 2006 | MR | Zbl

[24] B. T. Hayes, “Stability of solutions to a destabilized Hopf equation”, Comm. Pure Appl. Math., 48:2 (1995), 157–166 | DOI | MR | Zbl

[25] E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, “Periodic problem for a model nonlinear evolution equation”, Adv. Differential Equations, 7:5 (2002), 581–616 | DOI | MR | Zbl

[26] W. Kirsch, A. Kutzelnigg, “Time asymptotics for solutions of the Burgers equation with a periodic force”, Math. Z., 232:4 (1999), 691–705 | DOI | MR | Zbl

[27] D. Lu, L. Tian, Z. Liu, “Wavelet basis analysis in perturbed periodic KdV equation”, Appl. Math. Mech., Engl. Ed., 19:11 (1998), 1053–1058 | DOI | MR | Zbl

[28] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997 | MR | Zbl

[29] V. V. Varlamov, “On spatially periodic solutions of the damped Boussinesq equation”, Differential Integral Equations, 10:6 (1997), 1197–1211 | DOI | MR | Zbl

[30] J. Xing, “Global strong solution for a class of Burgers–BBM type equation”, Appl. Math., J. Chin. Univ., 6:1 (1991), 31–37 | MR | Zbl