Strong unique continuation properties and a classification of the asymptotic profiles are established for the fractional powers of a Schrödinger operator with a Hardy-type potential, by means of an Almgren monotonicity formula combined with a blow-up analysis.

We consider the equation Δu = Vu in the half-space ${\open R}_ + ^d $, d ⩾ 2 where V has certain periodicity properties. In particular, we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation Δu = Vu is studied as part of a broader class of elliptic evolution equations.

In this paper we prove a unique continuationresult for a cascade system of parabolic equations, in which the solution of the firstequation is (partially) used as a forcing term for the second equation. As aconsequence we prove the existence of ε-insensitizing controls for someparabolic equations when the control region and the observability region do not intersect.

This work is devoted to prove the exponential decay for the energyof solutions of the Korteweg-de Vries equation in a bounded intervalwith a localized damping term. Following the method in Menzala (2002)which combines energy estimates, multipliers and compactnessarguments the problem is reduced to prove the unique continuation ofweak solutions. In Menzala (2002) the case where solutions vanish on aneighborhood of both extremes of the bounded interval where equationholds was solved combining the smoothing results by T. Kato (1983)and earlier results on unique continuation of smooth solutions byJ.C. Saut and B. Scheurer (1987). In this article we address thegeneral case and prove the unique continuation property in twosteps. We first prove, using multiplier techniques, that solutionsvanishing on any subinterval are necessarily smooth. We then applythe existing results on unique continuation of smooth solutions.