Existência, Unicidade e Decaimento Exponencial das Soluç ... - UFRJ

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. Combining energy estimatives, multipliers and compactness
arguments the problem is reduced to prove the unique continuation of weak
solutions. The case where solutions vanish on a neighborhood of both extremes
of the bounded interval where equation holds is solved combining a
smoothing result by T. Kato [4] and results of unique continuation of smooth
solutions by J. C. Saut and B. Scheurer [17].
In this work we study the general case and prove the unique continuation
property in two steps: we first prove, using multipler techniques, that solutions
vanishing on any subinterval are necessarilly smooth. We then apply
the existing results on unique continuation of smooth solutions by J. C. Saut
and B. Sheurer.
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