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 $ \Omega\subset{\mathbb R}^N $: $ u_{tt}+\Delta^2 u-\kappa\phi(\|\nabla u\|^2)\Delta u-M(\|\Delta u\|^2+\|u_t\|^2)\Delta u_t+f(u) = h $, where $ \kappa \in \Lambda $ (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 $ {\mathcal H}_2 $ where the attractors lie in. Under the assumptions that either the nonlinearity $ f(u) $ is of optimal subcritical growth or even $ f(u) $ is a true source term, we show that (ⅰ) the semi-flow originating from any point in the natural energy space $ {\mathcal H} $ lies in the stronger strong solution space $ {\mathcal H}_2 $ when $ t>0 $; (ⅱ) the related solution semigroup $ S^\kappa(t) $ has a strong $ ({\mathcal H},{\mathcal H}_2) $-global attractor $ {\mathscr A}^\kappa $ for each $ \kappa $ and the family of $ {\mathscr A}^\kappa, \kappa\in \Lambda $ is upper semicontinuous on $ \kappa $ in the topology of stronger space $ {\mathcal H}_2 $; (ⅲ) $ S^\kappa(t) $ has a strong $ ({\mathcal H},{\mathcal H}_2) $-exponential attractor $ \mathfrak {A}^\kappa_{exp} $ for each $ \kappa $ and it is Hölder continuous on $ \kappa $ in the topology of $ {\mathcal H}_2 $. 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.