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.