This paper can be considered as an introductory review of scale invariance theories illustrated by the study of the equation ${\partial }_{t}h=-{\partial }_x\left[{\left({\partial }_{x}h\right)}^{1-2\nu }+{\partial }_{xxx}h\right],$ where $\nu >1/2.$ The $d-$dimensionals version of this equation is proposed for $\nu \ge 1$ to discuss the coarsening of growing interfaces that induce a mound-type structure without slope selection (Golubović, 1997). Firstly, the above equation is investigated in detail by using a dynamic scaling approach, thus allowing for obtaining a wide range of dynamic scaling functions (or pseudosimilarity solutions) which lend themselves to similarity properties. In addition, it is shown that these similarity solutions are spatial periodic solutions for any $\nu >1/2,$ confirming that the interfacial equation undergoes a perpetual coarsening process. The exponents $\beta$ and $\alpha$ describing, respectively, the growth laws of the interfacial width and the mound lateral size are found to be exactly $\beta =(1+\nu )/4\nu$ and $\alpha =1/4,$ for any $\nu >\frac{1}{2}.$ Our analytical contribution examines the scaling analysis in detail and exhibits the geometrical properties of the profile or scaling functions. Our finding coincides with the result previously presented by Golubović for $0<\nu \le 3/2$.

本文可视为对通过方程研究说明的尺度不变性理论的介绍性回顾${\partial }_{吨}H=-{\partial }_X\left[{\left({\partial }_{X}H\right)}^{1-2\nu }+{\partial }_{XXX}H\right],$在哪里$\nu >1/2.$这$d-$这个方程的维度版本被提议用于$\nu \ge 1$讨论生长界面的粗化，导致没有坡度选择的土丘型结构（Golubović，1997）。首先，通过使用动态缩放方法对上述方程进行详细研究，从而允许获得范围广泛的动态缩放函数（或伪相似性解），这些函数本身具有相似性。此外，这些相似性解是任何空间周期解$\nu >1/2,$确认界面方程经历了一个永久的粗化过程。指数$\beta$和$\alpha$分别描述了界面宽度和丘横向尺寸的增长规律，发现准确$\beta =(1+\nu )/4\nu$和$\alpha =1/4,$对于任何$\nu >\frac{1}{2}.$我们的分析贡献详细检查了缩放分析，并展示了轮廓或缩放函数的几何特性。我们的发现与 Golubović 先前提出的结果一致$0<\nu \le 3/2$.