In this paper we study the bi-spatial dynamics of non-autonomous parabolic equations with nonlinear Laplacian on n+1\documentclass[12pt]{minimal}
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\begin{document}$$n+1$$\end{document}-dimensional unbounded thin domains, where the nonlinearity has a (p, q)-growth exponents. We first prove the existence and uniqueness of tempered pullback attractors in L2\documentclass[12pt]{minimal}
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\begin{document}$$ L^2$$\end{document} on n+1\documentclass[12pt]{minimal}
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\begin{document}$$n+1$$\end{document}-dimensional unbounded thin domains, and then obtain the upper semi-continuity of these attractors in L2\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document} when the n+1\documentclass[12pt]{minimal}
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\begin{document}$$n+1$$\end{document}-dimensional thin domains degenerates onto a n-dimensional entire space Rn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^n$$\end{document}. Finally by borrowing an inductive method and a bootstrap technique, we show that the difference of solutions near the initial time is higher-order integrable for any space dimension n≥1\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 1$$\end{document}, which further shows the existence of pullback attractors in Lp∩Lq\documentclass[12pt]{minimal}
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\begin{document}$$ L^p\cap L^q$$\end{document}. Based on the higher-order integrability, we find that the obtained pullback attractor is attracting under the topology of Lδ\documentclass[12pt]{minimal}
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\begin{document}$$L^\delta $$\end{document} with δ∈[2,∞)\documentclass[12pt]{minimal}
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\begin{document}$$\delta \in [2,\infty )$$\end{document}.
