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

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 functions of the L2-norm of the gradient of the displacement. We prove the existence and uniqueness of weak solutions and study their properties for a 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. For strictly positive stiffness factors we prove that in the natural energy space endowed with a partially strong topology there exists a global finite-dimensional attractor. In the non-supercritical case this attractor is strong. 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.