The Attractors of Camassa–Holm Equation in Unbounded Domains

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 $$L_b^2({\mathbb {R}};L^2_U({\mathbb {R}}^1))$$ .