Global Attractors for a Class of Kirchhoff Wave Models with a Structural Nonlinear Damping
In this paper we deal with well-posedness and long-time dynamics of a class of Kirchhoff wave models with a nonlinear damping perturbed by a critical source. We first prove the existence and uniqueness of weak solutions and study their properties. Our main result deals with longtime dynamics. We prove that in the natural energy space there exists a global compact attractor of finite fractal dimension. We also establish the existence of a fractal exponential attractor and give conditions that guarantee the existence of a finite number of determining functionals. Our arguments involve a recently developed method based on “compensated” compactness and quasistability estimates.