Ademir F. Pazoto

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 JC 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

TL;DR: This paper investigates the boundary stabilization of the Boussinesq system of KdV-KdV type posed on a bounded domain and designs a two-parameter family of feedback laws for which the solutions issuing from small data are globally defined and exponentially decreasing in the energy space.

TL;DR: In this paper, a viscous finite-difference space semi-discretization of a locally damped wave equation in a regular 2D domain is studied, where the damping term is supported in a suitable subset of the domain, so that the energy of solutions of the damped continuous wave equation decays exponentially to zero.

TL;DR: In this paper, a linear dissipative wave equation in ℝN with periodic coefficients is considered and an expansion of solutions as t→∞ is obtained by means of Bloch wave decomposition, and it is shown that the solutions behave as the homogenized heat kernel.

TL;DR: In this article, the control properties of the Kortewegde Vries (KdV) equation posed on a bounded interval (0,L ) with a distributed control were investigated.