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This paper contains new results about the well-posedness and the asymptotic dynamics of solutions for a general abstract coupled system that arises in connection with thermoelastic Balakrishnan–Taylor beam models with fractional operators. [ABSTRACT FROM AUTHOR]

The paper is concerned with the exponential attractors for the viscoelastic wave model in Ω ⊂ R 3 : u tt - h t (0) Δ u - ∫ 0 ∞ ∂ s h t (s) Δ u (t - s) d s + f (u) = g , with time-dependent memory kernel h t (·) which is used to model aging phenomena of the material. Conti et al. (Am J Math 140(2):349–389, 2018a; Am J Math 140(6):1687–1729, 2018b) recently provided the correct mathematical setting for the model and a well-posedness result within the novel theory of dynamical systems acting on time-dependent spaces, recently established by Conti et al. (J Differ Equ 255:1254–1277, 2013), and proved the existence and the regularity of the time-dependent global attractor. In this work, we further study the existence of the time-dependent exponential attractors as well as their regularity. We establish an abstract existence criterion via quasi-stability method introduced originally by Chueshov and Lasiecka (J Dyn Differ Equ 16:469–512, 2004), and on the basis of the theory and technique developed in Conti et al. (2018a, b) we further provide a new method to overcome the difficulty of the lack of further regularity to show the existence of the time-dependent exponential attractor. And these techniques can be used to tackle other hyperbolic models. [ABSTRACT FROM AUTHOR]

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 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]