Random dynamics of non-autonomous semi-linear degenerate parabolic equations on <inline-formula><tex-math id="M1">\begin{document} $\mathbb{R}^N$ \end{document}</tex-math></inline-formula> driven by an unbounded additive noise

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In this paper, we study the dynamics of a non-autonomous semi-linear degenerate parabolic equation on \begin{document} $\mathbb{R}^N$ \end{document} driven by an unbounded additive noise. The nonlinearity has \begin{document} $(p,q)$ \end{document} -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 \begin{document} $L^2(\mathbb{R}^N)$ \end{document} by using a compact embedding of the weighted Sobolev space. Secondly we establish the higher-attraction of the pullback attractor in \begin{document} $L^δ(\mathbb{R}^N)$ \end{document} , which implies that the cocycle is absorbing in \begin{document} $L^δ(\mathbb{R}^N)$ \end{document} after a translation by the complete orbit, for arbitrary \begin{document} $δ∈[2,∞)$ \end{document} . Thirdly we verify that the derived \begin{document} $L^2$ \end{document} -pullback attractor is in fact a compact attractor in \begin{document} $L^p(\mathbb{R}^N)\cap L^q(\mathbb{R}^N)\cap D_0^{1,2}(\mathbb{R}^N,σ)$ \end{document} , mainly by means of the estimate of difference of solutions instead of the usual truncation method.