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Existence and positivity of solutions of a fourth‐order nonlinear PDE describing interface fluctuations
Author(s) -
Bleher Pavel M.,
Lebowitz Joel L.,
Speer Eugene R.
Publication year - 1994
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160470702
Subject(s) - mathematics , uniqueness , interval (graph theory) , limit (mathematics) , generalization , connection (principal bundle) , mathematical analysis , constant (computer programming) , partial differential equation , nonlinear system , order (exchange) , pure mathematics , scaling , mathematical physics , combinatorics , geometry , physics , quantum mechanics , finance , economics , computer science , programming language
We study the partial differential equationwhich 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.