Weak Solutions of a Mixed Problem in a Half-Strip for a Generalized Kawahara Equation
Matematičeskie zametki, Tome 85 (2009) no. 1, pp. 98-109.

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

We consider a mixed problem in a half-strip for a generalized Kawahara equation describing long nonlinear waves in inhomogeneous media with weak dispersion. We study the existence and uniqueness of generalized solutions in Sobolev weight spaces.
Keywords: Kawahara equation, long nonlinear wave, inhomogeneous medium, weak dispersion, Korteweg–de Vries (KdV) equation, Parseval's equality.
@article{MZM_2009_85_1_a8,
     author = {K. Sangare and A. V. Faminskii},
     title = {Weak {Solutions} of a {Mixed} {Problem} in a {Half-Strip} for a {Generalized} {Kawahara} {Equation}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {98--109},
     publisher = {mathdoc},
     volume = {85},
     number = {1},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_85_1_a8/}
}
TY  - JOUR
AU  - K. Sangare
AU  - A. V. Faminskii
TI  - Weak Solutions of a Mixed Problem in a Half-Strip for a Generalized Kawahara Equation
JO  - Matematičeskie zametki
PY  - 2009
SP  - 98
EP  - 109
VL  - 85
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2009_85_1_a8/
LA  - ru
ID  - MZM_2009_85_1_a8
ER  - 
%0 Journal Article
%A K. Sangare
%A A. V. Faminskii
%T Weak Solutions of a Mixed Problem in a Half-Strip for a Generalized Kawahara Equation
%J Matematičeskie zametki
%D 2009
%P 98-109
%V 85
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2009_85_1_a8/
%G ru
%F MZM_2009_85_1_a8
K. Sangare; A. V. Faminskii. Weak Solutions of a Mixed Problem in a Half-Strip for a Generalized Kawahara Equation. Matematičeskie zametki, Tome 85 (2009) no. 1, pp. 98-109. http://geodesic.mathdoc.fr/item/MZM_2009_85_1_a8/

[1] T. Kawahara, “Oscillatory solitary waves in dispersive media”, J. Phys. Soc. Japan, 33:1 (1972), 260–264 | DOI | MR

[2] A. B. Marchenko, “O dlinnykh volnakh v melkoi zhidkosti pod ledyanym pokrovom”, PMM, 52:2 (1988), 230–234 | Zbl

[3] A. T. Ilichev, “O svoistvakh odnogo nelineinogo evolyutsionnogo uravneniya pyatogo poryadka, opisyvayuschego volnovye protsessy v sredakh so slaboi dispersiei”, Sovremennye matematicheskie problemy mekhaniki i ikh prilozheniya, Tr. MIAN, 186, Nauka, M., 1989, 222–226 | MR | Zbl

[4] Y. Pomeau, A. Ramani, B. Grammaticos, “Structural stability of the Korteweg–de Vries solitons under a singular perturbation”, Phys. D, 31:1 (1988), 127–134 | DOI | MR | Zbl

[5] J. P. Boyd, “Weakly non-local solitons for capillary-gravity waves: fifth-degree Korteweg–de Vries equation”, Phys. D, 48:1 (1991), 129–146 | DOI | Zbl

[6] J.-C. Saut, “Sur quelques generalisations de l'equation de Korteweg–de Vries”, J. Math. Pures Appl. (9), 58:1 (1979), 21–61 | MR | Zbl

[7] A. B. Faminskii, “Zadacha Koshi dlya kvazilineinykh uravnenii nechetnogo poryadka”, Matem. sb., 180:9 (1989), 1183–1210 | MR | Zbl

[8] S. Cui, S. Tao, “Stricharts estimates for dispersive equations and solvability of Cauchy problems of the Kawahara equation”, J. Math. Anal. Appl., 304:2 (2005), 683–702 | DOI | MR | Zbl

[9] S. Cui, D. Deng, S. Tao, “Global existence of solutions for the Cauchy problem of the Kawahara equation with $L^2$ initial data”, Acta Math. Sin. (Engl. Ser.), 22:5 (2006), 1457–1466 | DOI | MR | Zbl

[10] K. Sangare, “Smeshannaya zadacha v polupolose dlya obobschennogo uravneniya Kavakhary v prostranstve beskonechno differentsiruemykh eksponentsialno ubyvayuschikh funktsii”, Vestn. RUDN. Ser. matem., 10:1 (2003), 91–107

[11] N. A. Larkin, G. G. Doronin, “Kawahara equation in a quarter-plane and in a finite domain”, Bol. Soc. Parana. Mat. (3), 25:1–2 (2007), 9–16 | MR | Zbl

[12] G. G. Doronin, N. A. Larkin, “Kawahara equation in a bounded domain”, Discrete Contin. Dyn. Syst. Ser. B, 10:4 (2008), 783–799 | MR | Zbl

[13] N. A. Larkin, “Correct initial boundary value problems for dispersive equations”, J. Math. Anal. Appl., 344:2 (2008), 1079–1092 | DOI | MR | Zbl

[14] A. B. Faminskii, “Smeshannaya zadacha v polupolose dlya uravneniya Kortevega–de Friza i ego obobschenii”, Tr. MMO, 51 (1988), 54–94 | MR | Zbl

[15] A. V. Faminskii, “An initial boundary-value problem in a half-strip for the Korteweg–de Vries equation in fractional-order Sobolev spaces”, Comm. Partial Differential Equations, 29:11–12 (2004), 1653–1695 | MR | Zbl

[16] A. V. Faminskii, “Global well-posedness of two initial-boundary-value problems for the Korteweg–de Vries equation”, Differential Integral Equations, 20:6 (2007), 601–642 | MR

[17] C. E. Kenig, G. Ponce, L. Vega, “Well-posedness of the initial value problem for the Korteweg–de Vries equation”, J. Amer. Math. Soc., 4:2 (1991), 323–347 | DOI | MR | Zbl

[18] X. Gaevskii, K. Greger, K. Zakharias, Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978 | MR | Zbl

[19] J. L. Bona, S. Sun, B.-Y. Zhang, “A non-homogeneous boundary-value problem for the Korteweg–de Vries equation in a quarter plane”, Trans. Amer. Math. Soc., 354:2 (2002), 427–490 | DOI | MR | Zbl