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The Attractors of Camassa–Holm Equation in Unbounded Domains

  • Gaocheng YueEmail author
Article
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Abstract

In this paper, we deal with the existence of global mild solutions and asymptotic behavior to the viscous Camassa–Holm equation in the locally uniform spaces. First we establish the global well-posedness for the Cauchy problem of viscous Camassa–Holm equation in \({\mathbb {R}}^1\) for any initial data \(u_0\in {\dot{H}}^1_U({\mathbb {R}}^1).\) Then we study the long time dynamical behavior of non-autonomous viscous Camassa–Holm equation on \({\mathbb {R}}^1\) with a new class of external forces and show the existence of \((H^1_U({\mathbb {R}}^1),H^1_\phi ({\mathbb {R}}^1))\)-uniform(w.r.t. \(g\in \mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)\)) attractor \(\mathcal {A}_{\mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)}\) with locally uniform external forces being translation uniform bounded but not translation compact in \(L_b^2({\mathbb {R}};L^2_U({\mathbb {R}}^1))\).

Keywords

Camassa–Holm equation Global solutions Uniform attractors Locally uniform spaces 

Mathematics Subject Classification

35G25 35A05 35B40 

Notes

Acknowledgements

The author would like to thank the anonymous referee for the careful reading of the manuscript.

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Authors and Affiliations

  1. 1.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China

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