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Mathematical Control and Related Fields (MCRF)
 

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

Pages: 61 - 81, Volume 1, Issue 1, March 2011      doi:10.3934/mcrf.2011.1.61

 
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Ivonne Rivas - Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Oh 45221, United States (email)
Muhammad Usman - Department of Mathematics, University of Dayton, Dayton, OH 45431, United States (email)
Bing-Yu Zhang - Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Oh 45221, United States (email)

Abstract: 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.

Keywords:  Global well-posedness, Korteweg-de Vries equation, asymptotic behavior.
Mathematics Subject Classification:  Primary: 35Q53, 35Q35; Secondary: 35S15.

Received: October 2010;      Revised: January 2011;      Available Online: March 2011.

 References