Year: 2015
Type: preprint
Abstract: In this paper we consider the Allen-Cahn equation $$ -Δu = u-u^3 \ \mbox{in} \ {\mathbb R}^3 $$ We prove that for each $k\in\left( \sqrt{2},+\infty\right),$ there exists a solution to the equation which has growth rate $k$, i.e. $$ \| u-H(\cdot -k \ln r + c_k) \|_{L^\infty} \to 0$$ The main ingredients of our proof consist: (1) compactness of solutions with growth $k$, (2) moduli space theory of analytical variety of formal dimension one.
Source: arXiv (Cornell University)
Authors: Gui, Changfeng, Liu, Yong, Wei, Juncheng
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1815639022