University of MaraghehSahand Communications in Mathematical Analysis2322-5807Articles in Press20230814A Gradient Bound for the Allen-Cahn Equation Under Almost Ricci Solitons70680610.22130/scma.2023.1972568.1199ENSakinehHajiaghasiDepartment of Pure Mathematics, Faculty of Science,
Imam Khomeini International University,
Qazvin, Iran.ShahroudAzamiDepartment of Pure Mathematics, Faculty of Science,
Imam Khomeini International University,
Qazvin, Iran.0000-0002-8976-2014Journal Article20221112In this paper, we consider positive solutions for the Allen-Cahn equation<br />\begin{equation*}<br />\Delta u+\left(1-u^{2}\right)u=0,<br />\end{equation*}<br />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.