Long-time dynamics of Kirchhoff wave models with strong nonlinear damping

Abstract: We study well-posedness and long-time dynamics of a class of quasilinear wave equations with a strong damping. We accept the Kirchhoff hypotheses and assume that the stiffness and damping coefficients are $C^1$ functions of the $L_2$-norm of the gradient of the displacement. We first prove the existence and uniqueness of weak solutions and study their properties for a rather wide class of nonlinearities which covers the case of possible degeneration (or even negativity) of the stiffness coefficient and the case of a supercritical source term. Our main results deal with global attractors. In the case of strictly positive stiffness factors we prove that in the natural energy space endowed with a partially strong topology there exists a global attractor whose fractal dimension is finite. In the non-supercritical case the partially strong topology becomes strong and a finite dimensional attractor exists in the strong topology of the energy space. Moreover, in this case 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 quasi-stability estimates.