Global existence and decay to equilibrium for some crystal surface models

In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equations $$\partial_t u=\Delta e^{-\Delta u},$$ $$\partial_t u=-u^2\Delta^2(u^3).$$ These two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97, 281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771-1784, 2011), respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev spaces.