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

Abstract

AbstractWe 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 S1. We establish existence, uniqueness, and regularity of solutions, locally in time, for positive initial data in H1(S1), 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 limtT* w(t,·) exists in C1(S1) but vanishes at some point. We show further that if T* > (1 + √3)/16π2 √3 then T* = ∞ and limt∞ w(t,.) exists and is constant. We discuss also some explicit solutions and propose a generalization to higher dimensions. © 1994 John Wiley & Sons, Inc.