The existence of uniform attractors for non-autonomous reaction-diffusion equations on the whole space

Abstract

In this paper, we introduce a new class of functions satisfying spacial absolutely continuous (see Definition 3.1), denoted by \documentclass[12pt]{minimal}\begin{document}$L^{2}_{sac}(\mathbb {R};\mathbb {R}^{n})$\end{document}Lsac2(R;Rn), which are translation bounded but not normal (see [S. S. Lu, H. Q. Wu, and C. K. Zhong, “Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces,” Discrete Contin. Dyn. Syst. A 13(3), 701−719 (2005)]10.3934/dcds.2005.13.701 and Definition 3.1) in \documentclass[12pt]{minimal}\begin{document}$L^{2}_{loc}(\mathbb {R};\mathbb {R}^{n})$\end{document}Lloc2(R;Rn). Then the asymptotic a priori estimate is applied to some nonlinear reaction-diffusion equations with external forces \documentclass[12pt]{minimal}\begin{document}$g(x,s)\in L^{2}_{sac}(\mathbb {R};\mathbb {R}^{n})$\end{document}g(x,s)∈Lsac2(R;Rn). We obtain the existence of uniform attractor together with its structure in the bi-spaces \documentclass[12pt]{minimal}\begin{document}$(L^{2}(\mathbb {R}^{n}), L^{2}(\mathbb {R}^{n}))$\end{document}(L2(Rn),L2(Rn)) and \documentclass[12pt]{minimal}\begin{document}$(L^{2}(\mathbb {R}^{n}), L^{p}(\mathbb {R}^{n}))(p>2)$\end{document}(L2(Rn),Lp(Rn))(p>2) without any restriction on the growing order of the nonlinear term.