Strong attractors and their stability for the structurally damped Kirchhoff wave equation with supercritical nonlinearity

Abstract

This paper investigates the complete regularity of the weak solutions, the existence of the strong (X, X 2𝛼 )-global and exponential attractors, and their stability on dissipative index 𝛼 for the structurally damped Kirchhoff wave equation:, together with the Dirichlet boundary condition, where the perturbed parameter 𝛼 ∈ (1∕2, 1) is called a dissipative index, X is energy space, and X 2𝛼 is strong solution space. We show that when the nonlinearity 𝑓 (u) is of supercritical growth:≤ p < p𝛼 , (i) the weak solutions of the model are just the strong ones; (ii) the global and exponential attractors of the related dynamical system (S 𝛼 (t), X) obtained in literature before are exactly the strong (X, X 2𝛼 )-ones, and the family of strong (X, X 2𝛼 )-global attractors { 𝛼 } 𝛼∈(1∕2,1) is upper semicontinuous on 𝛼 in X 2𝛼 -topology; (iii) for each 𝛼 0 ∈ (1∕2, 1), S 𝛼 (t) has a family of strong (X, X 2𝛼 )-exponential attractors {𝔄 𝛼 exp }, which is Hölder continuous at 𝛼 0 in X 2𝛼 0 -topology. The method developed here allows establishing the above-mentioned results, which breakthrough the restriction 1 ≤ p < p * on this topic in literature before.