SUMMARYThe aim of this work is to consider the Korteweg-de Vries equation in a finite interval with a very weak localized dissipation namely the H −1 -norm. Our main result says that the total energy decays locally uniform at an exponential rate. Our analysis improves earlier works on the subject (Q.
Abstract. We study the stabilization of solutions of a coupled system of Kortewegde Vries (KdV) equations in a bounded interval under the effect of a localized damping term. We use multiplier techniques combined with the so-called "compactness-uniqueness argument". The problem is then reduced to proving a unique continuation property (UCP) for weak solutions. The exponential decay of solutions was previously obtained in Bisognin, Bisognin, and Menzala (2003) when the damping was effective simultaneously in neighborhoods of both extremes of the bounded interval. In this work we address the general case using a different approach to obtain the UCP and stabilize the system.
We consider a coupled system of Kuramoto-Sivashinsky equations depending on a suitable parameter ν > 0 and study its asymptotic behavior for t large, as ν → 0. Introducing appropriate boundary conditions we show that the energy of the solutions decays exponentially uniformly with respect to the parameter ν. In the limit, as ν → 0, we obtain a coupled system of Korteweg-de Vries equations known to describe strong interactions of two long internal gravity waves in a stratified fluid for which the energy tends to zero exponentially as well. The decay fails when the length of the space interval L lies in a set of critical lengths.
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