A Gradient Bound for the Allen-Cahn Equation Under Almost Ricci Solitons

In this paper, we consider positive solutions for the Allen-Cahn equation\begin{equation*}\Delta u+\left(۱-u^{۲}\right)u=۰,\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^{۲}. 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.

کلیدواژه ها:

Gradient estimate ، Almost Ricci soliton ، Allen-Cahn equation