The Allen–Cahn–Nagumo equation is a reaction-diffusion equation with a bistable nonlinearity. This equation appears to be simple, however, it includes a rich behavior of solutions. The Allen–Cahn–Nagumo equation features a solution that constantly maintains a certain profile and moves with a constant speed, which is referred to as a traveling wave solution. In this paper, the entire solution of the Allen–Cahn–Nagumo equation is studied in multi-dimensional space. Here an entire solution is meant by the solution defined for all time including negative time, even though it satisfies a parabolic partial differential equation. Especially, this equation admits traveling wave solutions connecting two stable states. It is known that there is an entire solution which behaves as two traveling wave solutions coming from both sides in one dimensional space and annihilating in a finite time and that this one-dimensional entire solution is unique up to the shift. Namely, this entire solution is symmetric with respect to some point. There is a natural question whether entire solutions coming from all directions in the multi-dimensional space are radially symmetric or not. To answer this question, radially asymmetric entire solutions will be constructed by using super-sub solutions.

中文翻译：

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多维空间中Allen–Cahn–Nagumo方程的整体解

Allen–Cahn–Nagumo方程是具有双稳态非线性的反应扩散方程。这个方程看起来很简单，但是它包含了丰富的解行为。Allen–Cahn–Nagumo方程的特征是不断地保持一定轮廓并以恒定速度运动的解决方案，这被称为行波解决方案。本文在多维空间中研究了Allen–Cahn–Nagumo方程的整体解。这里，整个解决方案是由所有时间定义的解决方案，包括负时间，即使它满足抛物线偏微分方程。特别是，该方程式允许连接两个稳定状态的行波解。众所周知，存在一个整体解，其表现为在一个一维空间中从两侧传来并在有限时间内solutions灭的两个行波解，并且该一维整体解在移位之前是唯一的。即，整个解决方案关于某点是对称的。一个自然的问题是，来自多维空间各个方向的整个解是否都是径向对称的。为了回答这个问题，将使用超子解构造径向非对称的整体解。一个自然的问题是，来自多维空间各个方向的整个解是否都是径向对称的。为了回答这个问题，将使用超子解构造径向非对称的整体解。一个自然的问题是，来自多维空间各个方向的整个解是否都是径向对称的。为了回答这个问题，将使用超子解构造径向非对称的整体解。