Global solutions in higher dimensions to a fourth-order parabolic equation modeling epitaxial thin-film growth

Abstract

The initial-value problem for $$u_t=-\Delta^2 u - \mu\Delta u - \lambda \Delta |\nabla u|^2 + f(x)\qquad \qquad (\star)$$ is studied under the conditions \({{\frac{\partial}{\partial\nu}} u={\frac{\partial}{\partial\nu}} \Delta u=0}\) on the boundary of a bounded convex domain \({\Omega \subset {\mathbb{R}}^n}\) with smooth boundary. This problem arises in the modeling of the evolution of a thin surface when exposed to molecular beam epitaxy. Correspondingly the physically most relevant spatial setting is obtained when n = 2, but previous mathematical results appear to concentrate on the case n = 1. In this work, it is proved that when n ≤ 3, μ ≥ 0, λ > 0 and \({f \in L^\infty(\Omega)}\) satisfies \({{\int_\Omega} f \ge 0}\), for each prescribed initial distribution \({u_0 \in L^\infty(\Omega)}\) fulfilling \({{\int_\Omega} u_0 \ge 0}\), there exists at least one global weak solution \({u \in L^2_{loc}([0,\infty); W^{1,2}(\Omega))}\) satisfying \({{\int_\Omega} u(\cdot,t) \ge 0}\) for a.e. t > 0, and moreover, it is shown that this solution can be obtained through a Rothe-type approximation scheme. Furthermore, under an additional smallness condition on μ and \({\|f\|_{L^\infty(\Omega)}}\), it is shown that there exists a bounded set \({S\subset L^1(\Omega)}\) which is absorbing for \({(\star)}\) in the sense that for any such solution, we can pick T > 0 such that \({e^{2\lambda u(\cdot,t)}\in S}\) for all t > T, provided that Ω is a ball and u0 and f are radially symmetric with respect to x = 0. This partially extends similar absorption results known in the spatially one-dimensional case. The techniques applied to derive appropriate compactness properties via a priori estimates include straightforward testing procedures which lead to integral inequalities involving, for instance, the functional \({{\int_\Omega} e^{2\lambda u}dx}\), but also the use of a maximum principle for second-order elliptic equations.