Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion

Wen Tan; Chunyou Sun

doi:10.3934/dcds.2017260

Article Contents

2017, Volume 37, Issue 12: 6035-6067. Doi: 10.3934/dcds.2017260

This issue Previous Article Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system Next Article Impulsive motion on synchronized spatial temporal grids

Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion

Wen Tan^1,2, , and
Chunyou Sun^3,

1.
School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, Guangdong, China
2.
Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education, and Guangdong Province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, Guangdong, China
3.
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, China

^* Corresponding author: Wen Tan
^* Corresponding author: Wen Tan

Received: July 2016

Revised: July 2017

Published: October 2017

The authors were supported by NSF of China (Grant No. 11471148, Grant No. 11522109).

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Abstract

In this paper, we study the dynamics of a non-autonomous reaction diffusion model with the fractional diffusion on the whole space. We firstly prove the existence of a $(L^2,L^2)$ pullback $\mathscr{D}_μ$ -attractor of this model. Then we show that the pullback $\mathscr{D}_μ$ -attractor attract the $\mathscr{D}_μ$ class (especially all $L^2$ -bounded set) in $L^{2+δ}$-norm for any $δ∈[0,∞)$. Moreover, the solution of the model is shown to be continuous in $H^s$ with respect to initial data under a slightly stronger condition on external forcing term. As an application, we prove that the $(L^2,L^2)$ pullback $\mathscr{D}_{μ}$-attractor indeed attract the class of $\mathscr{D}_{μ}$ in $H^s$ -norm, and thus the existence of a $(L^2, H^s)$ pullback $\mathscr{D}_μ$ -attractor is obtained.

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

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