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Dynamics of stochastic nonlocal partial differential equations

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Abstract

This paper is concerned with the asymptotic behavior of solutions to nonlocal stochastic partial differential equations with multiplicative and additive noise driven by a standard Brownian motion, respectively. First of all, the stochastic nonlocal differential equations are transformed into their associated conjugated random differential equations, we then construct the dynamical systems to the original problems via the properties of conjugation. Next, in the case of multiplicative noise, we establish the existence of the random attractor when it absorbs every bounded deterministic set. Particularly, it is shown the pullback random attractor, which is also forward attracting, becomes a singleton when the external forcing term vanishes at zero. Eventually, in the case of additive noise, two approaches are applied to prove the existence of pullback random attractors with the help of energy estimations. Actually, these two attractors turn out to be the same one.

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Authors and Affiliations

  1. Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012, Seville, Spain

    Jiaohui Xu & Tomás Caraballo

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  1. Jiaohui Xu
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  2. Tomás Caraballo
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Correspondence to Tomás Caraballo.

Additional information

This research has been partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-I00, and by Junta de Andalucía (Consejería de Economía y Conocimiento) and FEDER under projects US-1254251 and P18-FR-4509.

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Xu, J., Caraballo, T. Dynamics of stochastic nonlocal partial differential equations. Eur. Phys. J. Plus 136, 849 (2021). https://doi.org/10.1140/epjp/s13360-021-01818-w

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