On the decay of solutions to a class of defocusing NLS

2008/11/12 by Nicola Visciglia, Visciglia, Nicola · 1 citation
Mathematics · Physics and Astronomy · #35B40 #35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP #msc:35B40 #msc:35Q55

paper · pdf · doi:10.48550/arxiv.0811.1849

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arxiv created 2008/11/12 · arxiv updated 2009/12/01

Abstract

We consider the following family of Cauchy problems: equation* i∂_t u= Δu - u|u|^α, (t,x) ∈ \R × \R^d equation* u(0)=φ∈ H¹(\R^d) where 0<α<\frac 4d-2 for d≥ 3 and 0<α<∞ for d=1,2. We prove that the L^r-norms of the solutions decay as t→ ± ∞, provided that 2<r<(2d)/(d-2) when d≥ 3 and 2<r<∞ when d=1,2. In particular we extend previous results obtained by Ginibre and Velo for d≥ 3 and by Nakanishi for d=1,2, where the same decay results are proved under the extra assumption α>\frac 4d.

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• Well-posedness and scattering of odd solutions for the defocusing INLS in one dimension

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