| Contents |
| We consider the Neumann problem for
u_t=-\Delta^2 u - \mu\Delta u - \lambda \Delta |\nabla u|^2 + f(x)
with $\mu\ge 0$ and $\lambda>0$ and a given source function $f$,
which has been proposed as a model for the evolution of a thin surface
when exposed to molecular beam epitaxy.
Numerical simulations suggest that even in the spatially
one-dimensional setting, this problem may admit
solutions for which the negative part $u_-$ blows up with respect to the norm in $L^\infty$,
whereas $u_+$ apparently enjoys comparatively strong boundedness properties.
The goal of the presentation is to point out some mathematical
challenges stemming from this, and to describe
how nevertheless a basic theory on global existence of appropriately
defined weak solutions can be established.
Moreover, some additional integral estimates are derived for these
solutions which indicate that indeed $u_+$
enjoys quite favorable regularity properties, whereas $u_-$ possibly might not.
Finally, despite the lack of a genuine energy functional a statement
on the large time behavior of solutions
is presented under a suitable smallness condition on $\mu$ and $\|f\|_{L^\infty(\Omega)}$. |
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