Adan J. Corcho, Lindolfo P. Mallqui
Abstract:
In this work, we consider the Cauchy problem for the cubic Schrodinger
equation posed on cylinder \(\mathbb{R}\times\mathbb{T}\) with fractional
derivatives \((-\partial_y^2)^{\alpha}\), \(\alpha >0\), in the periodic direction.
The spatial operator includes elliptic and hyperbolic regimes.
We prove \(L^2\) global well-posedness results when \(\alpha \ge 1\) by
proving a \(L^4\)-\(L^2\) Strichartz inequality for the linear equation,
following the ideas in [19],
where it was considered the elliptical case with \(\alpha=1\).
Further, these results remain valid on the Euclidean environment
\(\mathbb{R}^2\), so well-posedness in \(L^2\) are also achieved in this case.
Our proof in the elliptic (hyperbolic) case does not work in the small
directional dispersion case \(0<\alpha <1\) (\(0<\alpha \leq 1\)), respectively.
Submitted August 7, 2024. Published February 21, 2025.
Math Subject Classifications: 35Q55, 35Q35, 35Q60.
Key Words: Elliptic/hyperbolic cubic nonlinear Schrodinger equation; Cauchy problem; well-poseddness.
DOI: 10.58997/ejde.2025.16
Show me the PDF file (590 KB), TEX file for this article.
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Adán J. Corcho Universidad de Córdoba - UCO Departamento de Matemáticas Campus de Rabanales 14071, Córdoba, Spain email: a.corcho@uco.es |
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Lindolfo P. Mallqui Universidade Federal do Rio de Janeiro - UFRJ Instituto de Matemática 21941-909, Rio de Janeiro - RJ, Brazil email: lindolfciencias@gmail.com |
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