In this paper we prove local well-posedness of a space-time fractional generalization of the nonlinear Schrodinger equation with a power-type nonlinearity. The linear equation coincides with a model proposed by Naber, and displays a nonlocal behavior both in space and time which accounts for long-range interactions and a so-called memory effect. Because of a loss of derivatives produced by the latter and the lack of a semigroup structure of the solution operator, we employ a strategy of proof based on exploiting some smoothing effect similar to that used by Kenig, Ponce, and Vega for the KdV equation. Finally, we prove analytic ill-posedness of the data-to-solution map in the supercritical case.

Space-Time Fractional Nonlinear Schrödinger Equation / Grande Izquierdo, R.. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 51:5(2019), pp. 4172-4212. [10.1137/19m1247140]

Space-Time Fractional Nonlinear Schrödinger Equation

Grande Izquierdo, R.
2019-01-01

Abstract

In this paper we prove local well-posedness of a space-time fractional generalization of the nonlinear Schrodinger equation with a power-type nonlinearity. The linear equation coincides with a model proposed by Naber, and displays a nonlocal behavior both in space and time which accounts for long-range interactions and a so-called memory effect. Because of a loss of derivatives produced by the latter and the lack of a semigroup structure of the solution operator, we employ a strategy of proof based on exploiting some smoothing effect similar to that used by Kenig, Ponce, and Vega for the KdV equation. Finally, we prove analytic ill-posedness of the data-to-solution map in the supercritical case.
2019
51
5
4172
4212
https://epubs.siam.org/doi/10.1137/19M1247140
https://arxiv.org/abs/1604.05072
Grande Izquierdo, R.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/135450
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