Search

This paper investigates the complete regularity of the weak solutions, the existence of the strong (X,X2α)$$ \left(X,{X}_{2\alpha}\right) $$‐global and exponential attractors, and their stability on dissipative index α$$ \alpha $$ for the structurally damped Kirchhoff wave equation: utt−M(‖∇u‖2)Δu+(−Δ)αut+f(u)=g(x)$$ {u}_{tt}-M\left({\left\Vert \nabla u\right\Vert}^2\right)\Delta u+{\left(-\Delta \right)}^{\alpha }{u}_t+f(u)=g(x) $$, together with the Dirichlet boundary condition, where the perturbed parameter α∈(1/2,1)$$ \alpha \in \left(1/2,1\right) $$ is called a dissipative index, X$$ X $$ is energy space, and X2α$$ {X}_{2\alpha } $$ is strong solution space. We show that when the nonlinearity f(u)$$ f(u) $$ is of supercritical growth: p∗=N+2N−2≤p

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]

This paper investigates the stability of strong exponential attractors with respect to dissipative index θ ∈ [ 1 / 2 , 1) for the Kirchhoff wave model with structural nonlinear damping: u t t − ϕ (‖ ∇ u ‖ 2) Δ u + σ (‖ ∇ u ‖ 2) (− Δ) θ u t + f (u) = g (x). It proves that for each θ 0 ∈ [ 1 / 2 , 1) , there exists a family of strong bi-space exponential attractors, which are also the standard exponential attractors of optimal regularity and are Hölder continuous at the point θ 0 in the topology of the strong solution space provided that the nonlinearity f (u) is of the optimal subcritical growth. The method used here allows to overcome the difficulties in literatures before to get the Lipschitz continuity and quasi-stability of the evolution semigroup S θ (t) to obtain the desired result. [ABSTRACT FROM AUTHOR]