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 ‐global and exponential attractors, and their stability on dissipative index for the structurally damped Kirchhoff wave equation: , together with the Dirichlet boundary condition, where the perturbed parameter is called a dissipative index, is energy space, and is strong solution space. We show that when the nonlinearity is of supercritical growth: , (i) the weak solutions of the model are just the strong ones; (ii) the global and exponential attractors of the related dynamical system obtained in literature before are exactly the strong ‐ones, and the family of strong ‐global attractors is upper semicontinuous on in ‐topology; (iii) for each , has a family of strong ‐exponential attractors , which is Hölder continuous at in ‐topology. The method developed here allows establishing the above‐mentioned results, which breakthrough the restriction on this topic in literature before.