Addendum to “Property Leading to Morphological Instability”

Abstract

It was described that the property (P) can result in the morphological instabilities that are nonsimplification such that two peaks of the interface never coalesce into a single peak and the (exponential) increase in the perturbation of the interface, with the aid of a generalized evolution equation in two-dimensional space. (P) The driving force of interface development increases with the increase in curvature. In this addendum, we indicate that the corresponding results hold even in three-dimensional space. First, the following evolution equation in three-dimensional space can be obtained as an extension of that in two-dimensional space. zt 1⁄4 ð1þ zx þ zyÞAIðt; x; y; zÞ fIðt; x; y; z; Kðzx; zy; zxx; zxy; zyyÞÞ ð1þ zx þ zyÞAIIðt; x; y; zÞ fIIðt; x; y; z; Kðzx; zy; zxx; zxy; zyyÞÞ; AIðt; x; y; zÞ 0; AIIðt; x; y; zÞ 0; fIðt; x; y; z; KÞ > 0; fIIðt; x; y; z; KÞ > 0; fIðt; x; y; z; 0Þ 1⁄4 fIIðt; x; y; z; 0Þ 1⁄4 1; ðt; x; yÞ 2 ; ð1Þ where zðt; x; yÞ is the interface profile, is a connected open set, K is the mean curvature, Kðzx; zy; zxx; zxy; zyyÞ 1⁄4 1 2 ð1þ zyÞzxx þ ð1þ zxÞzyy 2zxzyzxy ð1þ zx þ zy2Þ ;