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Preprint 05-2013

Global attractors of sixth order PDEs describing the faceting of growing surfaces

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Author(s) : Maciek Korzec , Piotr Nayar , Piotr Rybka

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 05-2013

MSC 2000

35G25 Initial value problems for nonlinear higher-order PDE, nonlinear evolution equations

Abstract :
A spatially two-dimensional sixth order PDE describing the evolution of a growing crystalline surface h(x,y,t) that undergoes faceting is considered with periodic boundary conditions, such as its reduced one-dimensional version. These equation are expressed in terms of the slopes $u_1=h_{x}$ and $u_2=h_y$ to establish the existence of global, connected attractors for both of the equations. Since unique solutions are guaranteed for initial conditions in $\dot H^2_{per}$, we consider the solution operator $S(t): \dot H^2_{per} \rightarrow \dot H^2_{per}$, to gain the results. We prove the necessary continuity, dissipation and compactness properties.

Keywords : Global attractor, long-time dynamics, Cahn-Hilliard type equation, high order PDE, faceting

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