Global existence and decay to equilibrium for some crystal surface models

Abstract

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.

First replace: corrected typo pag. 4; reference [14] added; acknowledgements added; results unchanged. Second replace: added changes following the comments of the referees; added reference [1]; results unchanged

References in corpus (5)

• On The Muskat Problem With Viscosity Jump: Global In Time Results
• On the thin film Muskat and the thin film Stokes equations
• Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime
• Gradient flow approach to an exponential thin film equation: global existence and latent singularity
• A vicinal surface model for epitaxial growth with logarithmic free energy

Cited by in corpus (2)

• Global mild solutions of the Landau and non-cutoff Boltzmann equations
• A Proximal-Gradient Algorithm for Crystal Surface Evolution