In this paper, we prove several results on the exponential decay in $L^{2}$ norm of the KdV equation on the real line with localized dampings. First, for the linear KdV equation, the exponential decay holds if and only if the averages of the damping coefficient on all intervals of a fixed length have a positive lower bound. Moreover, under the same damping condition, the exponential decay holds for the (nonlinear) KdV equation with small initial data. Finally, with the aid of certain properties of propagation of regularity in Bourgain spaces for solutions of the associated linear system and the unique continuation property, the exponential decay for the KdV equation with large data holds if the damping coefficient has a positive lower bound on $E$, where $E$ is equidistributed over the real line and the complement $E^{c}$ has a finite Lebesgue measure.

We consider a damped abstract second order evolution equation with an additionalvanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, wedo not assume the operator defining the main damping to be bounded. First, using aconstructive frequency domain method coupled with a decomposition of frequencies and theintroduction of a new variable, we show that if the limit system is exponentially stable,then this evolutionary system is uniformly − with respect to the calibration parameter −exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decayestimates of the underlying semigroup provided such decay estimates hold for the limitsystem. Finally, we discuss some applications of our results; in particular, the case ofboundary damping mechanisms is accounted for, which was not possible in the earlier workmentioned above.

We study the stabilization of global solutions of theKawahara (K) equation in a bounded interval, under the effect ofa localized damping mechanism. The Kawahara equation is a modelfor small amplitude long waves. Using multiplier techniques andcompactness arguments we prove the exponential decay of the solutions of the (K) model. The proofrequires of a unique continuation theorem and the smoothing effectof the (K) equation on the real line, which are proved in this work.

First, we consider a semilinear hyperbolic equation with a locally distributed damping in a boundeddomain. The damping is located on a neighborhood of a suitable portion of theboundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt.46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman estimate [Ruiz, J. Math. Pures Appl.71 (1992) 455–467], we address the same type of problem in an exterior domain for a locally damped semilinear wave equation. For both problems, our method of proof is constructive, and much simpler than those found in the literature. In particular, we improve in some way on earlier results by Dafermos, Haraux, Nakao, Slemrod and Zuazua.