Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations

✍ Scribed by Pavel M. Bleher; Joel L. Lebowitz; Eugene R. Speer

• Publisher: John Wiley and Sons
• Year: 1994
• Tongue: English
• Weight: 799 KB
• Volume: 47
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160470702

✦ Synopsis

Abstract

We study the partial differential equation magnified image which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system. In that application x lies in R, but here we study primarily the periodic case × R S^1^. We establish existence, uniqueness, and regularity of solutions, locally in time, for positive initial data in H^1^(S^1^), and prove the existence of several families of Lyapunov functions for the evolution. From the latter we establish a sharp connection between existence globally in time and positivity preservation: if [0], T*) is a maximal half open interval of existence for a positive solution of the equation, with T* < ∞, then lim__~t~T* w(t,·)__ exists in C^1^(S^1^) but vanishes at some point. We show further that if T* > (1 + √3)/16π^2^ √3 then T* = ∞ and lim__~t~∞ w(t,.)__ exists and is constant. We discuss also some explicit solutions and propose a generalization to higher dimensions. © 1994 John Wiley & Sons, Inc.