On the longtime behavior of solutions to a model for epitaxial growth

Abstract

We consider a fourth-order nonlinear parabolic type equation on a two-dimensional bounded domain $\Omega$. This equation governs the evolution of the height profile of a thin film in an epitaxial growth process. We show that such equation endowed with no-flux boundary conditions generates a dissipative dynamical system under very general assumptions on $\partial\Omega$ on a phase-space of $L^{2}$-type. This system possesses a global as well as an exponential attractor. In addition, if $\partial\Omega$ is smooth enough, we show that every trajectory converges to a single equilibrium by means of a suitable Łojasiewicz--Simon inequality. An estimate of the convergence rate is also obtained.