On the two dimensional fast phase transition equation: well-posedness and long-time dynamics

We consider the initial boundary value problem for the two dimensional equation describing the evolution of the systems with the fast phase transition. Under critical growth and dissipativity conditions on the nonlinearities, we prove the existence of the global attractor. In particular, we establish the existence of the global attractor for the strong solutions of the weakly damped 2D Kirchhoff plate equation involving p-Laplacian operator with the critical exponent p= 5 , and thereby improve the previously obtained results. We also show that the approach of this paper can be applied to the hyperbolic relaxation of the 2D Cahn-Hilliard equation with the critical quartic nonlinearity.