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Amorphous molecular beam epitaxy: global solutions and absorbing sets

Published online by Cambridge University Press:  21 October 2005

O. STEIN
Affiliation:
Department of Mathematics – C, RWTH Aachen University, Germany email: stein@mathC.rwth-aachen.de
M. WINKLER
Affiliation:
Department of Mathematics – I, RWTH Aachen University, Germany email: winkler@math1.rwth-aachen.de

Abstract

The parabolic equation \[u_t + u_{xxxx} + u_{xx} = - (|u_x|^\alpha)_{xx}, \qquad \alpha>1\], is studied under the boundary conditions $u_x|_{\partial\Omega}=u_{xxx}|_{\partial\Omega}=0$ in a bounded real interval $\Omega$. Solutions from two different regularity classes are considered: It is shown that unique mild solutions exist locally in time for any $\alpha>1$ and initial data $u_0 \in W^{1,q}(\Omega)$ ($q>\alpha$), and that they are global if $\alpha \le \frac{5}{3}$. Furthermore, from a semidiscrete approximation scheme global weak solutions are constructed for $\alpha < \frac{10}{3}$, and for suitable transforms of such solutions the existence of a bounded absorbing set in $L^1(\Omega)$ is proved for $\alpha \in [2,\frac{10}{3})$. The article closes with some numerical examples which do not only document the roughening and coarsening phenomena expected for thin film growth, but also illustrate our results about absorbing sets.

Type
Papers
Copyright
2005 Cambridge University Press

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