We consider the KdV–Burgers equation and its linearized version on the whole real line. We investigate their well-posedness their exponential stability when λ is an indefinite damping.
Keywords: KdV–Burgers equation, Well-posedness, Stabilization by feedback, Decay rate
@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{\textendash}Burgers} equation with indefinite damping}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1079--1100}, publisher = {Elsevier}, volume = {31}, number = {5}, year = {2014}, doi = {10.1016/j.anihpc.2013.08.003}, zbl = {1302.35332}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.003/} }
TY - JOUR AU - Cavalcanti, M.M. AU - Domingos Cavalcanti, V.N. AU - Komornik, V. AU - Rodrigues, J.H. TI - Global well-posedness and exponential decay rates for a KdV–Burgers equation with indefinite damping JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 1079 EP - 1100 VL - 31 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.003/ DO - 10.1016/j.anihpc.2013.08.003 LA - en ID - AIHPC_2014__31_5_1079_0 ER -
%0 Journal Article %A Cavalcanti, M.M. %A Domingos Cavalcanti, V.N. %A Komornik, V. %A Rodrigues, J.H. %T Global well-posedness and exponential decay rates for a KdV–Burgers equation with indefinite damping %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 1079-1100 %V 31 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.003/ %R 10.1016/j.anihpc.2013.08.003 %G en %F 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, Volume 31 (2014) no. 5, pp. 1079-1100. doi : 10.1016/j.anihpc.2013.08.003. http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.003/
[1] Decay of solutions of some nonlinear wave equations, J. Differ. Equ. 81 (1989), 1 -49 | MR | Zbl
, , ,[2] Forced oscillations of a damped Korteweg–de Vries equation in a quarter plane, Commun. Contemp. Math. 5 (2003), 369 -400 | Zbl
, , ,[3] 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 | Zbl
,[4] Decay of solutions to damped Korteweg–de Vries type equation, Appl. Math. Optim. 65 (2012), 221 -251 | MR | Zbl
, , , ,[5] The generalized Korteweg–de Vries–Burgers equation in , Nonlinear Anal. 74 (2011), 721 -732 | MR | Zbl
,[6] Initial–boundary value problems for quasilinear dispersive equations posed on a bounded interval, Electron. J. Differ. Equ. (2010) | EuDML | MR | Zbl
, ,[7] Odd order evolution equations posed on a bounded interval, Bol. Soc. Parana. Mat. 28 (2010), 67 -77 | MR | Zbl
, ,[8] On the dual Petrov–Galerkin formulation of the KdV equation, Adv. Differ. Equ. 12 (2007), 817 -830 | MR
, ,[9] 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 | Zbl
, , ,[10] Stabilization de l'équation de Korteweg–de Vries, C. R. Acad. Sci. Paris, Sér. I Math. 312 (1991), 841 -843 | MR | Zbl
, , ,[11] On the stabilization of the Korteweg–de Vries equation, Bol. Soc. Parana. Mat. (3) 28 no. 2 (2010), 33 -48 | MR | Zbl
,[12] Korteweg–de Vries and Kuramoto–Sivashinsky equations in bounded domains, J. Math. Anal. Appl. 297 (2004), 169 -185 | MR | Zbl
,[13] Modified KdV equation with a source term in a bounded domain, Math. Methods Appl. Sci. 29 (2006), 751 -765 | MR | Zbl
,[14] Control and stabilization of the Korteweg–de Vries equation on a periodic domain, Commun. Partial Differ. Equ. 35 (2010), 707 -744 | MR | Zbl
, , ,[15] On the exponential decay of the critical generalized Korteweg–de Vries equation with localized damping, Proc. Am. Math. Soc. 135 (2007), 1515 -1522 | MR | Zbl
, ,[16] Asymptotic behavior of the Korteweg–de Vries equation posed in a quarter plane, J. Differ. Equ. 833 (2009), 1342 -1353 | MR | Zbl
, ,[17] Stabilization of the Korteweg–de Vries equation with localized damping, J. Q. Appl. Math. 60 (2002), 111 -129 | MR | Zbl
, , ,[18] Unique continuation and decay for the Korteweg–de Vries equation with localized damping, ESAIM Control Optim. Calc. Var. 11 (2005), 473 -486 | EuDML | Numdam | MR | Zbl
,[19] 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
, ,[20] Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var. 2 (1997), 33 -55 | EuDML | Numdam | MR | Zbl
,[21] 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 | Zbl
, ,[22] Global stabilization of the generalized Korteweg–de Vries equation posed on a finite domain, SIAM J. Control Optim. 3 (2006), 927 -956 | MR | Zbl
, ,[23] Control and stabilization of the Korteweg–de Vries equation: recent progresses, J. Syst. Sci. Complex. 22 (2009), 647 -682 | MR | Zbl
, ,[24] Exact controllability and stabilizability of the Korteweg–de Vries equation, Trans. Am. Math. Soc. 348 (1996), 3643 -3672 | MR | Zbl
, ,[25] Interpolation non linéaire et régularité, J. Funct. Anal. 9 (1972), 469 -489 | MR | Zbl
,[26] Solutions of Korteweg–de Vries equation in fractional order Sobolev spaces, Duke Math. J. 43 (1976), 87 -99 | MR | Zbl
, ,[27] Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Grundlehren Math. Wiss. vol. 181 , Springer-Verlag, New York, Heidelberg (1972) | MR | Zbl
, ,[28] Introduction to Nonlinear Dispersive Equations, Universitext , Springer-Verlag, New York (2009) | MR | Zbl
, ,Cited by Sources: