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 equationsThese 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(Phys. D, 240, -1784(Phys. D, 240, , 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.

The radius of analyticity for solutions to a problem in epitaxial growth on the torus

“…As discussed in the introduction, the paper [6] proves existence of solutions for all time, on the torus, for Δh 0 A(T n ) 1/10; this corresponds to existence for our parameter α (as defined in Corollary 2) in the interval [3/4, 1]. Thus, our existence result newly allows the values α ∈ (0, 3/4).…”

“…On free space rather than the torus, Liu and Strain have demonstrated global existence of small solutions, decay to equilibrium, and analyticity of solutions [9]. Both the papers [6,9] take initial conditions h 0 such that Δh 0 is small in A, the Wiener algebra. The Wiener algebra on free space is the set of functions with Fourier transform in L 1 , and on the torus is the set of functions with Fourier series in 1 .…”

“…We now discuss the formulation of the problem, which starts the same as the formulation in [6,9], which is by making the Taylor expansion of the exponential in (1). Making this Taylor expansion of the exponential, we can write (1) as…”

On a thin film model with insoluble surfactant

“…In addition we prove uniqueness of the weak solutions provided that they belong to a (slightly) higher regularity class. A similar approach has been employed before for the Muskat problem [14,15,27] (and the references therein) for the doubly parabolic Keller-Segel system [11], PDEs modelling small steepness porous flow [31] and the evolution of crystal surfaces [30]. We consider two cases: The gravity driven film, where surface tension effects are neglected (S = 0), which leads to a coupled system of second order equations; and the capillary driven film, where we take surface tension effects into account (S > 0).…”

Global strong solution with BV derivatives to singular solid-on-solid model with exponential nonlinearity

“…However, the method therein works only for DL regime whose mobility is a constant and the evolution in ADL regime is still an open question for p = 1. More recently, the original exponential equation (1.4) in DL regime is studied in [8,22,16] for p = 2 and in [32] for p ∈ (1,2], where the existence of strong solution with latent singularity and global solution starting from small data are established. For p = 1 in DL regime, [21] constructs some explicit solution to demonstrate the asymmetry of height profile due to exponential effect.…”