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The paper investigates the well-posedness and the complete regularity of the weak solutions, and the existence of strong global attractor for the strongly damped wave equation with critical nonlinearities on R 3 : u tt - Δ u - Δ u t + h (x , u t) + g (x , u) = f (x) . We show that when both nonlinearities h (x , u t) and g(x, u) are of at most critical growth, (1) the model is well-posed and its weak solution is of higher complete regularity as t > 0 , which ensures that the weak solution is exactly the strong one; (2) the related dynamical system (S (t) , H) possesses a strong (H , H 2) -global attractor of optimal topological property, which is also the standard global attractor of optimal regularity of S(t) in H . The method developed here allows breaking through the long-standing restriction for this model on R 3 : the partial regularity of the weak solutions and almost linearity of h (x , u t) , and helps obtaining the optimal complete regularity of the weak solutions and the existence of strong global attractor. [ABSTRACT FROM AUTHOR]

The paper investigates the well-posedness and the regularity of the solutions, the existence and the continuity of the strong attractors for the structurally damped Kirchhoff wave models with subcritical-critical nonlinearities: u tt - (1 + ϵ ‖ ∇ u ‖ 2 ) Δ u + (- Δ) α u t + h (u t) + g (u) = f (x) , where ϵ ∈ [ 0 , 1 ] is a perturbed extensibility parameter, α ∈ [ 1 / 2 , 1) is a dissipative index. We show that when the nonlinearity g(u) is of either critical growth as α ∈ (1 / 2 , 1) or subcritical growth as α = 1 / 2 , while h (u t) is of critical growth depending on α , the model is well-posed and its weak solution is exactly the strong one; the related solution semigroup S ϵ (t) has a strong (X, Y)-global attractor and a strong (X, Y)-exponential attractor, which are also the standard global and exponential attractor of optimal regularity of S ϵ (t) in X, respectively, where X is the energy space and Y is the strong solution space; these global attractors are upper semicontinuous and these exponential attractors are Hölder continuous with respect to perturbed parameter ϵ in the sense of Y-topology, respectively. [ABSTRACT FROM AUTHOR]

The paper investigates the robustness of pullback attractors and pullback exponential attractors of the non-autonomous Kirchhoff wave models with strong nonlinear damping: u tt - (1 + ϵ ‖ ∇ u ‖ 2 ) Δ u - σ (‖ ∇ u ‖ 2) Δ u t + f (u) = g (x , t) , where ϵ ∈ [ 0 , 1 ] is an extensibility parameter. It shows that when the growth exponent p of the nonlinearity f(u) is up to the supercritical range: 1 ≤ p < p ∗ ∗ ≡ N + 4 (N - 4) + , (i) the related process has a pullback attractor A ϵ in natural energy space H = (H 0 1 ∩ L p + 1) × L 2 for each ϵ , and it is upper semicontinuous on the perturbation ϵ ; (ii) the related process has a partially strong pullback exponential attractor M ϵ for each ϵ , and it is Hölder continuous on ϵ ∈ [ 0 , 1 ] . These results deepen and extend those in recent literatures (Chueshov in J Diff Equ 252:1229–1262, 2012; Ding et al. in Appl Math Lett 76:40–45, 2018; Wang and Zhong in Discrete Contin Dyn Syst 7:3189–3209, 2013). The main novelty of this paper is that it provides a new method to investigate the upper semicontinuity of pullback attractors and the stability of pullback exponential attractors in supercritical nonlinearity case. [ABSTRACT FROM AUTHOR]