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

Ivonne Rivas; Muhammad Usman; Bing-Yu Zhang

doi:10.3934/mcrf.2011.1.61

Article Contents

2011, Volume 1, Issue 1: 61-81. Doi: 10.3934/mcrf.2011.1.61

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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
2.
Department of Mathematics, University of Dayton, Dayton, OH 45431

Received: September 30, 2010

Revised: December 31, 2010

Published: February 2011

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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.

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Mathematics Subject Classification: Primary: 35Q53, 35Q35; Secondary: 35S15.

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