Unique continuation and decay for the Korteweg-de Vries equation with localized damping

Abstract

Abstract. This work is devoted to prove the exponential decay for the energy of solutions of the Korteweg-de Vries equation in a bounded interval with a localized damping term. Following the method in Menzala (2002) which combines energy estimates, multipliers and compactness arguments the problem is reduced to prove the unique continuation of weak solutions. In Menzala (2002) the case where solutions vanish on a neighborhood of both extremes of the bounded interval where equation holds was solved combining the smoothing results by T. Kato (1983) and earlier results on unique continuation of smooth solutions by J.C. Saut and B. Scheurer (1987). In this article we address the general case and prove the unique continuation property in two steps. We first prove, using multiplier techniques, that solutions vanishing on any subinterval are necessarily smooth. We then apply the existing results on unique continuation of smooth solutions.Mathematics Subject Classification. 35B40, 35Q53.

Stabilization of the Kawahara equation with localized damping

“…Linares and Pazoto [12] studied the stabilization of the generalized KdV system with critical exponents. Unique continuation and decay for the Korteweg-De Vries equation with localized damping have been studied by Pazoto [16].…”

“…In Section 4, inspired by an argument developed in Pazoto [16] for KdV problem, which was used before for linear, time invariant wave and plate equations respectively by Rauch and Taylor [18] and Zuazua [27], we prove a gain of regularity for solutions of the problem (K) with u xx (0, t) = 0 that vanish on a subset ω contained in (0, L).…”

On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping

“…This allows us to apply the unique continuation property results in [17] on smooth solutions to conclude that u = 0. Later on, performing as in [9], the general case was solved in [11] showing that solutions vanishing on any subinterval are necessarily smooth, which yields enough regularity on u to apply the unique continuation results obtained in [15]. More recently, L. Rosier and B.-Y.…”

Uniform stabilization of a class of coupled systems of KdV equations with localized damping

“…This allows applying the unique continuation property results in [27] on smooth solutions to conclude that u = 0. The general case, that is, the case in which the function a = a(x) is active in any open subset of (0, L) was first solved in [19] for the scalar KdV equation (1.12) and boundary conditions as in (1.2). Combining the multiplier techniques developed in [22] and the so-called "compactness-uniqueness arguments" (see, for instance, [29]) it was shown that the solutions vanishing on any subinterval of (0, L) are necessarily smooth, which yielded enough regularity on u to apply the unique continuation results obtained in [23] by Carleman inequalities.…”