On connection between the order of a stationary one-dimensional dispersive equation and the growth of its convective term
Abstract
A boundary value problem for a stationary nonlinear dispersive equation of 2l+1 order with a convective term in the form u^ku_x, k\in N was considered on an interval (0,L). The existence, uniqueness and continuous dependence of a regular solution as well as a relation between the order l and critical values of k of the equation have been established.
Downloads
References
Adams, R., Sobolev Spaces, Second Ed., Academic Press, Elsevier Science, (2003).
Araruna F. D., Capistriano-Filho R. A. and Doronin G. G., Energy decay for the modified Kawahara equation posed in a bounded domain, J. Math. Anal. Appl. 385, 743-756, (2012).
https://doi.org/10.1016/j.jmaa.2011.07.003
Biagioni, H. A. and Linares, F., On the Benney - Lin and Kawahara equations, J. Math. Anal. Appl. 211, 131-152, (1997).
https://doi.org/10.1006/jmaa.1997.5438
Bona, J. L., Sun, S. M. and Zhang, B. Y., Nonhomogeneous problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane, Ann. Inst. H. Poincar'e Anal. Non Lin'eaire 25, 1145-1185, (2008).
https://doi.org/10.1016/j.anihpc.2007.07.006
Bubnov, B. A., Solvability in the large of nonlinear boundary-value problems for the Korteweg-de Vries equation in a bounded domain, (Russian) Differentsial'nye uravneniya 16, No 1, 34-41, (1980), Engl. transl. in: Differ. Equations 16, 24-30, (1980).
Ceballos, J., Sepulveda, M. and Villagran, O., The Korteweg-de Vries- Kawahara equation in a bounded domain and some numerical results, Appl. Math. Comput. 190, 912-936, (2007).
https://doi.org/10.1016/j.amc.2007.01.107
Colin, T. and Ghidaglia, J. M., An initial-boundary-value problem for the Korteweg-de Vries Equation posed on a finite interval, Adv. Differential Equations 6, 1463-1492, (2001).
Cui, S. B., Deng, D. G. and Tao, S. P.,Global existence of solutions for the Cauchy problem of the Kawahara equation with L2 initial data, Acta Math. Sin. (Engl. Ser.) 22, 1457-1466, (2006).
https://doi.org/10.1007/s10114-005-0710-6
Doronin, G. G. and Larkin, N. A., Boundary value problems for the stationary Kawahara equation, Nonlinear Analysis. Series A: Theory, Methods & Applications, 1655-1665, (2007).
https://doi.org/10.1016/j.na.2007.07.005
Evans, L. C., Partial Differential Equations, American Mathematical Society, (1998).
Faminskii, A. V. and Larkin, N. A., Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval, Electron. J. Differ. Equations, 1-20, (2010).
Farah, L. G., Linares, F. and Pastor, A., The supercritical generalized KDV equation: global well-posedness in the energy space and below, Math. Res. Lett. 18, no. 02, 357-377, (2011).
https://doi.org/10.4310/MRL.2011.v18.n2.a13
Hasimoto, H., Water waves, Kagaku 40, 401-408, (1970 (Japanese)).
Isaza, P., Linares, F. and Ponce, G., Decay properties for solutions of fifth order nonlinear dispersive equations, J. Differ. Equats. 258, 764-795, (2015).
https://doi.org/10.1016/j.jde.2014.10.004
Jeffrey, A. and Kakutani, T., Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries equation, SIAM Review, vol 14 no 4, 582-643, (1972).
https://doi.org/10.1137/1014101
Jia, Y. and Huo, Z., Well-posedness for the fifth-order shallow water equations, Journal of Differential Equations 246, 2448-2467, (2009).
https://doi.org/10.1016/j.jde.2008.10.027
Kakutani, T. and Ono, H., Weak non linear hydromagnetic waves in a cold collision free plasma, J. Phys. Soc. Japan 26, 1305-1318, (1969).
https://doi.org/10.1143/JPSJ.26.1305
Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equations, Advances in Mathematics Suplementary Studies, Stud. Appl. Math. 8, 93-128, (1983).
Kawahara, T., Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan 33, 260-264, (1972).
https://doi.org/10.1143/JPSJ.33.260
Kenig, C. E., Ponce, G. and Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation and the contraction principle, Commun. Pure Appl. Math. 46 No 4, 527-620, (1993).
https://doi.org/10.1002/cpa.3160460405
Kenig, C. E., Ponce, G. and Vega, L., Higher -order nonlinear dispersive equations, Proc. Amer. Math. Soc. 122 (1), 157-166, (1994).
https://doi.org/10.1090/S0002-9939-1994-1195480-8
Khanal, N., Wu J. and Yuan, J-M., The Kawahara equation in weighted Sobolev spaces, Nonlinearity 21, 1489-1505, (2008).
https://doi.org/10.1088/0951-7715/21/7/007
Kuvshinov, R. V. and Faminskii, A. V., A mixed problem in a half-strip for the Kawahara equation, (Russian) Differ. Uravn. 45, N. 3, 391-402, (2009), translation in Differ. Equ. 45 N. 3, 404-415, (2009).
https://doi.org/10.1134/S0012266109030100
Ladyzhenskaya, O. A., Solonnikov, V. A. and Uraltseva, N. N., Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, Rhode Island, (1968).
Larkin, N. A.,Korteweg-de Vries and Kuramoto-Sivashinsky equations in bounded domains, J. Math. Anal. Appl. 297, 169-185, (2004).
https://doi.org/10.1016/j.jmaa.2004.04.053
Larkin, N. A., Correct initial boundary value problems for dispersive equations, J. Math. Anal. Appl. 344, 1079-1092, (2008).
https://doi.org/10.1016/j.jmaa.2008.03.055
Larkin, N. A. and Luchesi, J., Higher-order stationary dispersive equations on bounded intervals, Advances in Mathematical Physics, vol. 2018, Article ID 7874305, (2018). doi:10.1155/2018/7874305
https://doi.org/10.1155/2018/7874305
Larkin, N. A. and Sim˜oes, M. H., The Kawahara equation on bounded intervals and on a half-line, Nonlinear Analysis 127, 397-412, (2015).
https://doi.org/10.1016/j.na.2015.07.008
Linares, F. and Pazoto, A., On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping, Proc. Amer. Math. Soc. 135, 1515-1522, (2007).
https://doi.org/10.1090/S0002-9939-07-08810-7
Martel, Y. and Merle, F., Instability of solutions for the critical generalized Korteweg-de Vries equation, Geometrical and Funct. Analysis 11, 74-123, (2001).
https://doi.org/10.1007/PL00001673
Merle, F., Existence of blow up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14, 555-578, (2001).
https://doi.org/10.1090/S0894-0347-01-00369-1
Nirenberg, L., An extended interpolation inequality, Annali della Scuola Nomale Superiore di Pisa, Classe di Scienze 3a serie, tome 20, no 4, 733-737, (1966).
Nirenberg, L., On elliptic partial differential equations, Annali della Scuola Nomale Superiore di Pisa, Classe di Scienze 3a serie, tome 13, no 2, 115-162, (1959).
Pilod, D., On the Cauchy problem for higher-order nonlinear dispersive equations, Journal of Differential Equations 245, 2055-2077, (2008).
https://doi.org/10.1016/j.jde.2008.07.017
Saut, J. C., Sur quelques generalizations de l'equation de Korteweg- de Vries, J. Math. Pures Appl. 58, 21-61, (1979).
Tao, S. P. and Cui, S.B., The local and global existence of the solution of the Cauchy problem for the seven-order nonlinear equation, Acta Matematica Sinica 25 A , (4) 451-460, (2005).
Temam, R., Navier-Stokes Equations. Theory and Numerical Analysis, Noth-Holland, Amsterdam, (1979).
Copyright (c) 2020 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
Funding data
