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Life span of solutions to a nonlocal in time nonlinear fractional Schrödinger equation

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Abstract

In this paper, we study the initial value problem for the nonlocal in time nonlinear Schrödinger equation

$$\begin{aligned} iu_{t}+\Delta u = \lambda J^{\alpha}_{0\vert t} \vert u\vert^p, \quad x \in \mathbb{R}^N, \quad t > 0,\\ u(x,0) = f(x), \quad x \in \mathbb{R}^N.\quad \quad \end{aligned}$$

Using the test function method, we derive a blow-up exponent. Then based on integral inequalities, we estimate the life span of blowing-up solutions.

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Authors and Affiliations

  1. Laboratoire MIA, EA 3165, Pôle sciences et Technologies, Université de La Rochelle, Avenue M. Crépeau, 17000, La Rochelle, France

    M. Kirane & A. Nabti

  2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    M. Kirane

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  1. M. Kirane
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  2. A. Nabti
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Correspondence to M. Kirane.

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Kirane, M., Nabti, A. Life span of solutions to a nonlocal in time nonlinear fractional Schrödinger equation. Z. Angew. Math. Phys. 66, 1473–1482 (2015). https://doi.org/10.1007/s00033-014-0473-y

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  • DOI: https://doi.org/10.1007/s00033-014-0473-y

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