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

Gaocheng Yue

doi:10.3934/dcdsb.2017079

Article Contents

2017, Volume 22, Issue 4: 1645-1671. Doi: 10.3934/dcdsb.2017079

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

Gaocheng Yue

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

Received: November 30, 2015

Revised: October 31, 2016

Published: August 2017

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

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Abstract

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.

Keywords:

Reaction-diffusion equations,
uniform attractors,
locally uniform spaces,
fractional powers of sectorial operator.

Mathematics Subject Classification: Primary:35K57, 35B40;Secondary:35B41.

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