Solutions for fourth order parabolic equation modeling epitaxial thin film growth

• Schwache Lösung der nichtlinearen parabolischen Differentialgleichung vierter Ordnung, welche das epitaxiale Dünnfilmwachstum beschreibt

Sandjo, Albert N.; Wiegner, Michael (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2011, 2012)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2011

Abstract

In this thesis we study the continuum model for some classes of fourth order nonlinear parabolic equation modeling epitaxial thin film growth. In the literature not so much is known about higher order parabolic differential equations, in particular fourth order parabolic equations which is very important in applications (Epitaxial thin films growth, Cahn-Hilliard equation, Image segmentation, etc.). Most of the work on higher order parabolic equation have been done in the case of whole space. However, only few of them are entirely devoted to the particular case of a bounded domain which is realistic and practical. Following Kato’s work there has been a lot of interest in the last decade in mild solutions of epitaxial thin films growth. All these results rely on variations of Kato’s method which allows to obtain global solutions if the initial data are small by a fixed point argument (which is based on Banach fixed point theorem or, equivalently, on a direct fixed point iteration). In this thesis two forms of f will be considered. In the first case we use Interpolation-Extrapolation method. In the second case we consider an integral form of the equation and employ successive approximation combine with Lp-Lq estimate of analytic semigroup to show existence of mild solutions. The basic idea is similar to Wiegner’s approach. However the estimate of the second derivative of $exp(-tDelta^2)$ is needed, this complicates the proof of the fixed point argument. The most crucial thing in our result is that we can control the solution and its derivatives by the initial data. Indeed, global solutions can only be obtained if initial data are small. Finally we give a unified method to construct local and global mild solutions for a class of nonlinear parabolic equation.