Document Type : Research Paper
Authors
Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.
Abstract
In this paper, we consider positive solutions for the Allen-Cahn equation
\begin{equation*}
\Delta u+\left(1-u^{2}\right)u=0,
\end{equation*}
on an almost Ricci soliton without a boundary. Firstly, using volume comparison Theorem and Sobolev inequality, we estimate the upper bound of $\vert \nabla u\vert^{2}$. As one of the applications, we extend this result to a gradient Ricci almost soliton. Finally, we obtain a Liouville-type theorem for almost Ricci solitons.
Keywords
Main Subjects
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