  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.
