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**1 - 6**of**6**### Decoupled and unidirectional asymptotic models for the propagation of internal waves

, 2012

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### ftp ejde.math.txstate.edu PERSISTENCE OF SOLUTIONS TO NONLINEAR EVOLUTION EQUATIONS IN WEIGHTED SOBOLEV SPACES

"... Abstract. In this article, we prove that the initial value problem associated with the Korteweg-de Vries equation is well-posed in weighted Sobolev spaces X s,θ, for s ≥ 2θ ≥ 2 and the initial value problem associated with the nonlinear Schrödinger equation is well-posed in weighted Sobolev spaces X ..."

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Abstract. In this article, we prove that the initial value problem associated with the Korteweg-de Vries equation is well-posed in weighted Sobolev spaces X s,θ, for s ≥ 2θ ≥ 2 and the initial value problem associated with the nonlinear Schrödinger equation is well-posed in weighted Sobolev spaces X s,θ, for s ≥ θ ≥ 1. Persistence property has been proved by approximation of the solutions and using a priori estimates.

### ON THE PROPAGATION OF REGULARITY AND DECAY OF SOLUTIONS TO THE k-GENERALIZED KORTEWEG-DE VRIES EQUATION

"... ABSTRACT. We study special regularity and decay properties of solu-tions to the IVP associated to the k-generalized KdV equations. In par-ticular, for datum u0 ∈H3/4+(R) whose restriction belongs to H l((b,∞)) for some l ∈ Z+ and b ∈ R we prove that the restriction of the corre-sponding solution u(· ..."

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ABSTRACT. We study special regularity and decay properties of solu-tions to the IVP associated to the k-generalized KdV equations. In par-ticular, for datum u0 ∈H3/4+(R) whose restriction belongs to H l((b,∞)) for some l ∈ Z+ and b ∈ R we prove that the restriction of the corre-sponding solution u(·, t) belongs to H l((β,∞)) for any β ∈ R and any t ∈ (0,T). Thus, this type of regularity propagates with infinite speed to its left as time evolves. 1.