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Benjamin–Bona–Mahony Equations with Memory and Rayleigh Friction

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Abstract

This paper is concerned with the integrodifferential Benjamin-Bona-Mahony equation

$$\begin{aligned} u_t - u_{txx} + \alpha u - \int _0^\infty g(s) u_{xx}(t-s) \mathrm{d}s + (f(u))_x = h \end{aligned}$$

complemented with Dirichlet boundary conditions, in the presence of a possibly large external force h. The nonlinearity f is allowed to exhibit a superquadratic growth, and the dissipation is due to the simultaneous interaction between the nonlocal memory term and the Rayleigh friction. The existence of regular global and exponential attractors of finite fractal dimension is shown.

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Notes

  1. In [8] the exponential stability of a more general (nonlinear) version of (6.2) has been proved.

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133, Milano, Italy

    Filippo Dell’Oro

  2. Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, Université de Picardie Jules Verne, 80039, Amiens, France

    Youcef Mammeri

  3. Institut de Génétique, Environnement et Protection des Plantes, INRA UMR 1349, 35650, Le Rheu, France

    Youcef Mammeri

Authors
  1. Filippo Dell’Oro
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  2. Youcef Mammeri
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Correspondence to Filippo Dell’Oro.

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Dell’Oro, F., Mammeri, Y. Benjamin–Bona–Mahony Equations with Memory and Rayleigh Friction. Appl Math Optim 83, 813–831 (2021). https://doi.org/10.1007/s00245-019-09568-z

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