Existence and positivity of solutions of a fourth‐order nonlinear PDE describing interface fluctuations
Existence and positivity of solutions of a fourth‐order nonlinear PDE describing interface fluctuations
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DOI：
10.1002/cpa.3160470702
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发表时间：
1994-07
影响因子：
3
通讯作者：
P. Bleher;J. Lebowitz;E. Speer
中科院分区：
文献类型：
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作者：
P. Bleher;J. Lebowitz;E. Speer
我们研究偏微分方程，该方程最初是在研究特定自旋系统中的界面涨落时作为标度极限而出现的。在该应用中，x 位于 R 中，但在这里我们主要研究周期性情况 x E S'。我们针对 H' (S' ) 中的正初始数据在局部时间上建立了解的存在性、唯一性和规律性，并证明了演化的几个 Lyapunov 函数族的存在性。从后者，我们在时间上的全局存在性和正性保持之间建立了尖锐的联系：如果 [O,T*) 是方程正解的最大半开区间存在性，其中 T' (1 + &)/16r2& 则 T* = 03 且 lim,-m w(r, .) 存在且为常数。我们还讨论了一些显式解决方案，并提出了更高维度的推广。 0 1994 约翰·威利 &
We study the partial differential equation 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 x E S'. We establish existence, uniqueness, and regularity of solutions, locally in time, for positive initial data in H' (S' ), 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 [O,T*) is a maximal half open interval of existence for a positive solution of the equation, with T' (1 + &)/16r2& then T* = 03 and lim,-m w(r, .) exists and is constant. We discuss also some explicit solutions and propose a generalization to higher dimensions. 0 1994 John Wiley &