Traveling Fronts Of Pyramidal Shapes And General Pyramidal Shapes In The Bisable Allen-Cahn Equation

In this paper,we study the asymptotic stability of pyramid wave and the exis-tence of general pyramid wave in the Allen-Cahn equationThe main result is divided into two chapters.Firstly,we study more about the stability of the three dimensional pyramid wave. By using the squeezing technique combined with the comparison principle, we not only show that such a three-dimensional pyramid wave of Allen-Cahn equation has a global exponential asymptotic stability, but also give their clear asymptotic propagation velocity.Secondly, we further generalize the existing three-dimensional pyramid wave, which obtain a general pyramid wave in the three-dimensional space and prove that this generalized pyramid wave is existence in the general case. It will make the classification of traveling wave solutions, especially in high-dimensional space, to get a new step.