American Institute of Mathematical Sciences

June  2017, 22(4): 1645-1671. doi: 10.3934/dcdsb.2017079

Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces

 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

Received  December 2015 Revised  November 2016 Published  February 2017

Fund Project: The author is supported by NSF of China under Grant 11501289.

In this paper, we first prove the well-posedness for the nonautonomous reaction-diffusion equations with fractional diffusion in the locally uniform spaces framework. Under very minimal assumptions, then we study the asymptotic behavior of solutions of such equation and show the existence of $(H^{2(\alpha -ε),q}_U(\mathbb{R}^N),H^{2(\alpha -ε),q}_φ(\mathbb{R}^N))(0<ε<\alpha <1)$-uniform(w.r.t.$g∈\mathcal{H}_{L^q_U(\mathbb{R}^N)}(g_0)$) attractor $\mathcal{A}_{\mathcal{H}_{L^q_U(\mathbb{R}^N)}(g_0)}$ with locally uniform external forces being translation uniform bounded but not translation compact in $L_b^p(\mathbb{R};L^q_U(\mathbb{R}^N)).$ The key to that extensions is a new the space-time estimates in locally uniform spaces for the linear fractional power dissipative equation.

Citation: Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079
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