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

  • * Corresponding author: Wen Tan

    * Corresponding author: Wen Tan 
The authors were supported by NSF of China (Grant No. 11471148, Grant No. 11522109).
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  • 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.

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


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