Complete regularity and strong attractor for the strongly damped wave equation with critical nonlinearities on R3

The paper investigates the well-posedness and the complete regularity of the weak solutions, and the existence of strong global attractor for the strongly damped wave equation with critical nonlinearities on R-3 : u(tt) - delta(u) - delta(ut) + h(x, u(t)) + g(x, u) = f (x). We show that when both nonlinearities h(x, ut) and g(x, u) are of at most critical growth, (1) the model is well-posed and its weak solution is of higher complete regularity as t > 0, which ensures that the weak solution is exactly the strong one; (2) the related dynamical system (S(t), H) possesses a strong (H, H-2)-global attractor of optimal topological property, which is also the standard global attractor of optimal regularity of S(t) in H. The method developed here allows breaking through the long-standing restriction for this model on R-3: the partial regularity of the weak solutions and almost linearity of h(x, ut), and helps obtaining the optimal complete regularity of the weak solutions and the existence of strong global attractor.