In this paper we study the well-posedness for a fourth-order parabolic equation modeling epitaxial thin film growth. Using Kato's Method ,  and  we establish existence, uniqueness and regularity of the solution to the model, in suitable spaces, namelyC0([0,T];Lp(Ω)) where with 1<α<2, n∈N and n≥2. We also show the global existence solution to the nonlinear parabolic equations for small initial data. Our main tools are Lp–Lq-estimates, regularization property of the linear part of e−tΔ2 and successive approximations. Furthermore, we illustrate the qualitative behavior of the approximate solution through some numerical simulations. The approximate solutions exhibit some favorable absorption properties of the model, which highlight the stabilizing effect of our specific formulation of the source term associated with the upward hopping of atoms. Consequently, the solutions describe well some experimentally observed phenomena, which characterize the growth of thin film such as grain coarsening, island formation and thickness growth.
Sandjo, A. N., Moutari, S., & Gningue, Y. (2015). Solutions of fourth-order parabolic equation modeling thin film growth. Journal of Differential Equations, 259(12), 7260-7283. https://doi.org/10.1016/j.jde.2015.08.022