Global existence and decay to equilibrium for some crystal surface models
Rafael Granero-Belinchón, Martina Magliocca
TL;DR
This work analyzes global existence and exponential decay to equilibrium for two fourth-order nonlinear crystal-surface models, recast as $\partial_t u = \Delta e^{-\Delta u}$ and $\partial_t u = -u^2 \Delta^2(u^3)$. By reformulating with $v = \Delta u$ (and related variables) and employing a regularized Galerkin scheme in the Wiener algebra $\mathscr{A}^0$, the authors derive explicit smallness thresholds $\delta(|v_0|_0)>0$ (or $|v_0|_0<1$ for the second model) that guarantee global weak solutions and exponential decay in $\mathscr{A}^0$ and in Sobolev spaces $\mathscr{H}^r$ for $0\le r<2$. Central to the analysis are Lyapunov functionals $\mathcal{L}_1(v)=\int e^{-v}$ and $\mathcal{L}_2(v)=\int \frac{1}{(1+v)^2}$, together with energy estimates and compactness to pass to the limit in regularized problems, yielding sharp, computable criteria on initial data. The results place the evolution in a critical Wiener-space framework and extend to higher regularity, providing exponential decay and uniform bounds that hold on the torus and reflect the models’ physics, with potential applicability to related thin-film and crystal-surface PDEs.
Abstract
In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equations $$ \partial_t u=Δe^{-Δu}, $$ $$ \partial_t u=-u^2Δ^2(u^3). $$ 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.