Global existence and decay to equilibrium for some crystal surface models

TL;DR

This paper analyzes the long-term behavior of solutions to two nonlinear fourth-order PDEs modeling crystal surface evolution, establishing conditions for global existence and exponential decay to equilibrium.

Contribution

It provides explicit initial data size conditions ensuring global solutions and decay, advancing understanding of crystal surface PDEs.

Findings

Global existence under specific initial data norms

Exponential decay to equilibrium proven

Conditions explicitly computed for initial data size

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.