Asymptotic behavior of the Korteweg–de Vries equation posed in a quarter plane

Abstract

The purpose of this work is to study the exponential stabilization of the Korteweg–de Vries equation in the right half-line under the effect of a localized damping term. We follow the methods in [G.P. Menzala, C.F. Vasconcellos, E. Zuazua, Stabilization of the Korteweg–de Vries equation with localized damping, Quart. Appl. Math. LX (1) (2002) 111–129] which combine multiplier techniques and compactness arguments and reduce the problem to prove the unique continuation property of weak solutions. Here, the unique continuation is obtained in two steps: we first prove that solutions vanishing on the support of the damping function are necessarily smooth and then we apply the unique continuation results proved in [J.C. Saut, B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations 66 (1987) 118–139]. In particular, we show that the exponential rate of decay is uniform in bounded sets of initial data.