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

In this paper, we consider positive solutions for the Allen-Cahn equation ∆u +(1 − u2) u=0, on an almost Ricci soliton without a boundary. Firstly, using vol-ume comparison Theorem and Sobolev inequality, we estimate the upper bound of |∇u|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.