# American Institute of Mathematical Sciences

March  2011, 1(1): 61-81. doi: 10.3934/mcrf.2011.1.61

## Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain

 1 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Oh 45221, United States, United States 2 Department of Mathematics, University of Dayton, Dayton, OH 45431, United States

Received  October 2010 Revised  January 2011 Published  March 2011

In this paper, we study a class of initial boundary value problem (IBVP) of the Korteweg-de Vries equation posed on a finite interval with nonhomogeneous boundary conditions. The IBVP is known to be locally well-posed, but its global $L^2$- a priori estimate is not available and therefore it is not clear whether its solutions exist globally or blow up in finite time. It is shown in this paper that the solutions exist globally as long as their initial value and the associated boundary data are small, and moreover, those solutions decay exponentially if their boundary data decay exponentially.
Citation: Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
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##### References:
 [1] Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097 [2] Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45 [3] M. Agrotis, S. Lafortune, P.G. Kevrekidis. On a discrete version of the Korteweg-De Vries equation. Conference Publications, 2005, 2005 (Special) : 22-29. doi: 10.3934/proc.2005.2005.22 [4] Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255 [5] Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509 [6] Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655 [7] Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069 [8] Olivier Goubet. Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 625-644. doi: 10.3934/dcds.2000.6.625 [9] Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353 [10] Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061 [11] Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761 [12] Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857 [13] Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure & Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046 [14] Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control & Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024 [15] John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149 [16] Hongwei Wang, Amin Esfahani. Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation. Evolution Equations & Control Theory, 2019, 8 (4) : 709-735. doi: 10.3934/eect.2019035 [17] Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019229 [18] Belkacem Said-Houari. Long-time behavior of solutions of the generalized Korteweg--de Vries equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 245-252. doi: 10.3934/dcdsb.2016.21.245 [19] Boling Guo, Zhaohui Huo. The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in $L^2$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 121-136. doi: 10.3934/dcds.2006.16.121 [20] Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297

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