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, as well as its reduced one-dimensional version. These equations are expressed in terms of the slopes (Formula presented.) and (Formula presented.) to establish the existence of global, connected attractors for both equations. Since unique solutions are guaranteed for initial conditions in (Formula presented.), we consider the solution operator (Formula presented.), to gain our results. We prove the necessary continuity, dissipation and compactness properties.
Bibliographical noteFunding Information:
MK would like to acknowledge the financial support by the DFG Research Center Matheon . Furthermore MK thanks the University of Warsaw for the hospitality during two visits in 2012. The work of PR was supported in part by NCN through 2011/01/B/ST1/01197 Grant. The authors thank the referee for his/her extensive comments which helped to improve the text.
© 2015, The Author(s).
- Anisotropic surface energy
- Cahn-Hilliard type equation
- Global attractor