Strong attractors and their robustness for an extensible beam model with energy damping

This paper investigates the existence of strong global and exponential attractors and their robustness on the perturbed parameter for an extensible beam equation with nonlocal energy damping in <tex-math id="M1">{document}Ω⊂ RN{document}</tex-math>: <tex-math id="M2">{document}uₜₜ+Δ² u-κφ(\|∇ u\|²)Δ u-M(\|Δ u\|²+\|uₜ\|²)Δ uₜ+f(u) = h{document}</tex-math>, where <tex-math id="M3">{document}κ ∈ Λ{document}</tex-math> (index set) is an extensibility parameter, and where the "strong" means that the compactness, the attractiveness and the finiteness of the fractal dimension of the attractors are all in the topology of the stronger space <tex-math id="M4">{document}H₂{document}</tex-math> where the attractors lie in. Under the assumptions that either the nonlinearity <tex-math id="M5">{document}$ f(u) ${document}</tex-math> is of optimal subcritical growth or even <tex-math id="M6">{document}$ f(u) ${document}</tex-math> is a true source term, we show that (ⅰ) the semi-flow originating from any point in the natural energy space <tex-math id="M7">{document}H{document}</tex-math> lies in the stronger strong solution space <tex-math id="M8">{document}H₂{document}</tex-math> when <tex-math id="M9">{document}$ t>0 ${document}</tex-math>; (ⅱ) the related solution semigroup <tex-math id="M10">{document}S^κ(t){document}</tex-math> has a strong <tex-math id="M11">{document}( H, H₂){document}</tex-math>-global attractor <tex-math id="M12">{document}A^κ{document}</tex-math> for each <tex-math id="M13">{document}κ{document}</tex-math> and the family of <tex-math id="M14">{document}A^κ, κ∈ Λ{document}</tex-math> is upper semicontinuous on <tex-math id="M15">{document}κ{document}</tex-math> in the topology of stronger space <tex-math id="M16">{document}H₂{document}</tex-math>; (ⅲ) <tex-math id="M17">{document}S^κ(t){document}</tex-math> has a strong <tex-math id="M18">{document}( H, H₂){document}</tex-math>-exponential attractor <tex-math id="M19">{document}A^κₑₓₚ{document}</tex-math> for each <tex-math id="M20">{document}κ{document}</tex-math> and it is Hölder continuous on <tex-math id="M21">{document}κ{document}</tex-math> in the topology of <tex-math id="M22">{document}H₂{document}</tex-math>. These results break through long-standing existed restriction for the attractors of the extensible beam models in energy space and show the optimal topology properties of them in the stronger phase space.