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Global existence and decay to equilibrium for some crystal surface models

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  • In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equations

    $ \ \ \ \ \ {\partial _t} u = \Delta e^{-\Delta u}, \\ {\partial _t} u = -u^2\Delta ^2(u^3). $

    These two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97,281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771-1784, 2011), respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev spaces.

    Mathematics Subject Classification: 35A01, 35B40, 35G25, 35K30, 35K55.


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