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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 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]