Global attractors of sixth order PDEs describing the faceting of growing surfaces

A spatially two-dimensional sixth order PDE describing the evolution of a growing crystalline surface h(x,y,t) that undergoes faceting is considered with periodic boundary conditions, such as its reduced one-dimensional version. These equation are expressed in terms of the slopes $u_1=h_{x}$ and $u_2=h_y$ to establish the existence of global, connected attractors for both of the equations. Since unique solutions are guaranteed for initial conditions in $\dot H^2_{per}$, we consider the solution operator $S(t): \dot H^2_{per} \rightarrow \dot H^2_{per}$, to gain the results. We prove the necessary continuity, dissipation and compactness properties