*preprint*

**Inserted:** 19 jan 2024

**Year:** 2018

**Abstract:**

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.