Long-time behavior of solutions of a BBM equation with generalized damping

Jean-Paul Chehab; Pierre  Garnier; Youcef  Mammeri

doi:10.3934/dcdsb.2015.20.1897

Article Contents

2015, Volume 20, Issue 7: 1897-1915. Doi: 10.3934/dcdsb.2015.20.1897

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Long-time behavior of solutions of a BBM equation with generalized damping

1.
LAMFA, UMR 6140, Université de Picardie Jules Verne, Pôle Scientifique, 33, rue Saint Leu, 80039 Amiens
2.
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, Université de Picardie Jules Verne, 80039 Amiens

Received: January 31, 2014

Revised: April 30, 2015

Published: August 2015

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Abstract

We study the long-time behavior of the solution of a damped BBM equation $u_t + u_x - u_{xxt} + uu_x + \mathscr{L}_{\gamma}(u) = 0$. The proposed dampings $\mathscr{L}_{\gamma}$ generalize standards ones, as parabolic ($\mathscr{L}_{\gamma}(u)=-\Delta u$) or weak damping ($\mathscr{L}_{\gamma}(u)=\gamma u$) and allows us to consider a greater range. After establish the local well-posedness in the energy space, we investigate some numerical properties.

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Mathematics Subject Classification: 35B40, 35Q53, 76B03, 76B15.

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