Abstract
In this paper we study the global (in time) existence of small data solutions to semi-linear fractional \(\sigma \)-evolution equations with mass or power non-linearity. Our main goal is to explain on the one hand the influence of the mass term and on the other hand the influence of higher regularity of the data on qualitative properties of solutions. Using modified Bessel functions we prove some polynomial decay in \(L^p-L^q\) estimates for solutions to the corresponding linear fractional \(\sigma \)-evolution equations with vanishing right-hand sides. By a fixed point argument the existence of small data solutions is proved for some admissible range of powers p.
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The research of this article is supported by the DAAD, Erasmus+ Project between the Hassiba Benbouali University of Chlef (Algeria) and TU Bergakademie Freiberg, 2015-1-DE01-KA107-002026, during the stay of the first author at Technical University Bergakademie Freiberg within the periods April 2016 to June 2016, April to July 2017. The first author expresses a sincere thankfulness to Prof. Michael Reissig for proposing the interesting topic, for numerous discussions and the staff of the Institute of Applied Analysis for their hospitality. Both authors thank the referee for valuable proposals to improve the readability of the paper. Moreover, the authors thank the referee for pointing out that the property of continuity of solutions with respect to the time variable requires a special treatment. Finally, the authors thank Marcello D'Abbicco for very useful discussion on the topics of this paper.
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For the first author the financial support was provided by Erasmus\(+\) project KA 107 - collaboration with Algeria.
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Kainane Mezadek, A., Reissig, M. Semi-linear fractional \(\varvec{\sigma }\)-evolution equations with mass or power non-linearity. Nonlinear Differ. Equ. Appl. 25, 42 (2018). https://doi.org/10.1007/s00030-018-0530-x
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DOI: https://doi.org/10.1007/s00030-018-0530-x