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Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise

  • * Corresponding author: Wenqiang Zhao

    * Corresponding author: Wenqiang Zhao
This work was supported by CTBU Grant 1751041, Chongqing NSF Grant of China cstc2016jcyjA0262 and China NSF Grant 11601046.
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  • In this paper, we study the dynamics of a non-autonomous semi-linear degenerate parabolic equation on $\mathbb{R}^N$ driven by an unbounded additive noise. The nonlinearity has $(p,q)$ -exponent growth and the degeneracy means that the diffusion coefficient $σ$ is unbounded and allowed to vanish at some points. Firstly we prove the existence of pullback attractor in $L^2(\mathbb{R}^N)$ by using a compact embedding of the weighted Sobolev space. Secondly we establish the higher-attraction of the pullback attractor in $L^δ(\mathbb{R}^N)$ , which implies that the cocycle is absorbing in $L^δ(\mathbb{R}^N)$ after a translation by the complete orbit, for arbitrary $δ∈[2,∞)$ . Thirdly we verify that the derived $L^2$ -pullback attractor is in fact a compact attractor in $L^p(\mathbb{R}^N)\cap L^q(\mathbb{R}^N)\cap D_0^{1,2}(\mathbb{R}^N,σ)$ , mainly by means of the estimate of difference of solutions instead of the usual truncation method.

    Mathematics Subject Classification: Primary: 60H15, 35B40, 35B41; Secondary: 37H10.

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