The initial-value problem for
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\begin{document}$$u_t=-\Delta^2 u - \mu\Delta u - \lambda \Delta |\nabla u|^2 + f(x)\qquad \qquad (\star)$$\end{document}is studied under the conditions \documentclass[12pt]{minimal}
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\begin{document}$${{\frac{\partial}{\partial\nu}} u={\frac{\partial}{\partial\nu}} \Delta u=0}$$\end{document} on the boundary of a bounded convex domain \documentclass[12pt]{minimal}
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\begin{document}$${\Omega \subset {\mathbb{R}}^n}$$\end{document} 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 \documentclass[12pt]{minimal}
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\begin{document}$${f \in L^\infty(\Omega)}$$\end{document} satisfies \documentclass[12pt]{minimal}
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\begin{document}$${{\int_\Omega} f \ge 0}$$\end{document}, for each prescribed initial distribution \documentclass[12pt]{minimal}
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\begin{document}$${u_0 \in L^\infty(\Omega)}$$\end{document} fulfilling \documentclass[12pt]{minimal}
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\begin{document}$${{\int_\Omega} u_0 \ge 0}$$\end{document}, there exists at least one global weak solution \documentclass[12pt]{minimal}
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\begin{document}$${u \in L^2_{loc}([0,\infty); W^{1,2}(\Omega))}$$\end{document} satisfying \documentclass[12pt]{minimal}
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\begin{document}$${{\int_\Omega} u(\cdot,t) \ge 0}$$\end{document} 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 \documentclass[12pt]{minimal}
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\begin{document}$${\|f\|_{L^\infty(\Omega)}}$$\end{document}, it is shown that there exists a bounded set \documentclass[12pt]{minimal}
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\begin{document}$${S\subset L^1(\Omega)}$$\end{document} which is absorbing for \documentclass[12pt]{minimal}
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\begin{document}$${(\star)}$$\end{document} in the sense that for any such solution, we can pick T > 0 such that \documentclass[12pt]{minimal}
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\begin{document}$${e^{2\lambda u(\cdot,t)}\in S}$$\end{document} 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 \documentclass[12pt]{minimal}
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\begin{document}$${{\int_\Omega} e^{2\lambda u}dx}$$\end{document}, but also the use of a maximum principle for second-order elliptic equations.
