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

Source abstract

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.