Global well-posedness and exponential decay rates for a KdV–Burgers equation with indefinite damping
Cavalcanti, M.M. ; Domingos Cavalcanti, V.N. ; Komornik, V. ; Rodrigues, J.H.
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 1079-1100 / Harvested from Numdam
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We consider the KdV–Burgers equation u t +u xxx -u xx +λu+uu x =0 and its linearized version u t +u xxx -u xx +λu=0 on the whole real line. We investigate their well-posedness their exponential stability when λ is an indefinite damping.

Publié le : 2014-01-01
DOI : https://doi.org/10.1016/j.anihpc.2013.08.003
Classification:  35Q53,  93D15 
@article{AIHPC_2014__31_5_1079_0,
     author = {Cavalcanti, M.M. and Domingos Cavalcanti, V.N. and Komornik, V. and Rodrigues, J.H.},
     title = {Global well-posedness and exponential decay rates for a KdV--Burgers equation with indefinite damping},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {31},
     year = {2014},
     pages = {1079-1100},
     doi = {10.1016/j.anihpc.2013.08.003},
     zbl = {1302.35332},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2014__31_5_1079_0}
}

Cavalcanti, M.M.; Domingos Cavalcanti, V.N.; Komornik, V.; Rodrigues, J.H. Global well-posedness and exponential decay rates for a KdV–Burgers equation with indefinite damping. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) pp. 1079-1100. doi : 10.1016/j.anihpc.2013.08.003. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_5_1079_0/

[1] C. Amick, J. Bona, M. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differ. Equ. 81 (1989), 1 -49 | MR 1012198 | Zbl 0689.35081

[2] J.L. Bona, S.M. Sun, B.-Y. Zhang, Forced oscillations of a damped Korteweg–de Vries equation in a quarter plane, Commun. Contemp. Math. 5 (2003), 369 -400 | Zbl 1054.35076

[3] B.A. Bubnov, Solvability in the large of nonlinear boundary-value problems for the Korteweg–de Vries equation in a bounded domain, Differ. Uravn. 16 (1980), 34 -41 , Differ. Equ. 16 (1980), 24 -30 | MR 559805 | Zbl 0451.35066

[4] M.M. Cavalcanti, V.N. Domingos Cavalcanti, A. Faminskii, F.M.A. Natali, Decay of solutions to damped Korteweg–de Vries type equation, Appl. Math. Optim. 65 (2012), 221 -251 | MR 2891222 | Zbl 1252.35242

[5] T. Dlotko, The generalized Korteweg–de Vries–Burgers equation in H 2 (ℝ) , Nonlinear Anal. 74 (2011), 721 -732 | MR 2738624 | Zbl 1205.35248

[6] A.V. Faminskii, N.A. Larkin, Initial–boundary value problems for quasilinear dispersive equations posed on a bounded interval, Electron. J. Differ. Equ. (2010) | MR 2591986 | Zbl 1184.35229

[7] A.V. Faminskii, N.A. Larkin, Odd order evolution equations posed on a bounded interval, Bol. Soc. Parana. Mat. 28 (2010), 67 -77 | MR 2679035 | Zbl 1213.35186

[8] O. Goubet, J. Shen, On the dual Petrov–Galerkin formulation of the KdV equation, Adv. Differ. Equ. 12 (2007), 817 -830 | MR 2294504

[9] N. Hyayashi, E. Kaikina, J.G. Zavala, On the boundary-value problem for the Korteweg–de Vries equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), 2861 -2884 | MR 2015994 | Zbl 1050.35099

[10] V. Komornik, D.L. Russel, B.-Y. Zhang, Stabilization de l'équation de Korteweg–de Vries, C. R. Acad. Sci. Paris, Sér. I Math. 312 (1991), 841 -843 | MR 1108503 | Zbl 0742.35058

[11] V. Komornik, On the stabilization of the Korteweg–de Vries equation, Bol. Soc. Parana. Mat. (3) 28 no. 2 (2010), 33 -48 | MR 2727429 | Zbl 1209.35118

[12] N.A. Larkin, Korteweg–de Vries and Kuramoto–Sivashinsky equations in bounded domains, J. Math. Anal. Appl. 297 (2004), 169 -185 | MR 2080374 | Zbl 1075.35070

[13] N.A. Larkin, Modified KdV equation with a source term in a bounded domain, Math. Methods Appl. Sci. 29 (2006), 751 -765 | MR 2215992 | Zbl 1099.35114

[14] C. Laurent, L. Rosier, B.-Y. Zhang, Control and stabilization of the Korteweg–de Vries equation on a periodic domain, Commun. Partial Differ. Equ. 35 (2010), 707 -744 | MR 2753618 | Zbl 1213.93015

[15] F. Linares, A.F. Pazoto, On the exponential decay of the critical generalized Korteweg–de Vries equation with localized damping, Proc. Am. Math. Soc. 135 (2007), 1515 -1522 | MR 2276662 | Zbl 1107.93030

[16] F. Linares, A.F. Pazoto, Asymptotic behavior of the Korteweg–de Vries equation posed in a quarter plane, J. Differ. Equ. 833 (2009), 1342 -1353 | MR 2488687 | Zbl 1157.93031

[17] G.P. Menzala, C.F. Vasconcellos, E. Zuazua, Stabilization of the Korteweg–de Vries equation with localized damping, J. Q. Appl. Math. 60 (2002), 111 -129 | MR 1878262 | Zbl 1039.35107

[18] A.F. Pazoto, Unique continuation and decay for the Korteweg–de Vries equation with localized damping, ESAIM Control Optim. Calc. Var. 11 (2005), 473 -486 | Numdam | MR 2148854 | Zbl 1148.35348

[19] A.F. Pazoto, L. Rosier, Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line, Discrete Contin. Dyn. Syst., Ser. B 14 (2010), 1511 -1535 | Zbl 1219.93101

[20] L. Rosier, Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var. 2 (1997), 33 -55 | Numdam | MR 1440078 | Zbl 0873.93008

[21] L. Rosier, A.F. Pazoto, Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line, Discrete Contin. Dyn. Syst., Ser. B 14 (2010), 1511 -1535 | MR 2679653 | Zbl 1219.93101

[22] L. Rosier, B.-Y. Zhang, Global stabilization of the generalized Korteweg–de Vries equation posed on a finite domain, SIAM J. Control Optim. 3 (2006), 927 -956 | MR 2247720 | Zbl 1116.35108

[23] L. Rosier, B.-Y. Zhang, Control and stabilization of the Korteweg–de Vries equation: recent progresses, J. Syst. Sci. Complex. 22 (2009), 647 -682 | MR 2565262 | Zbl 1300.93091

[24] D.L. Russell, B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg–de Vries equation, Trans. Am. Math. Soc. 348 (1996), 3643 -3672 | MR 1360229 | Zbl 0862.93035

[25] L. Tartar, Interpolation non linéaire et régularité, J. Funct. Anal. 9 (1972), 469 -489 | MR 310619 | Zbl 0241.46035

[26] J.L. Bona, L.R. Scott, Solutions of Korteweg–de Vries equation in fractional order Sobolev spaces, Duke Math. J. 43 (1976), 87 -99 | MR 393887 | Zbl 0335.35032

[27] J.-L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Grundlehren Math. Wiss. vol. 181 , Springer-Verlag, New York, Heidelberg (1972) | MR 350177 | Zbl 0223.35039

[28] F. Linares, G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext , Springer-Verlag, New York (2009) | MR 2492151 | Zbl 1178.35004