Note on a Allen-Cahn equation with Caputo-Fabrizio derivative

In this short note, we investigate the Allen-Cahn equation with appearance of Caputo-Fabizzio derivative. We obtain a local solution when initial value is small enough. This an that has many practical applications. The power term in nonlinear component source function and operator combine to make finding space more difficult than classical problem. discovered new technique, connecting Hilbert scale $L^p$ spaces, overcome these difficulties. Evaluation smoothness was also performed. research ideas paper can be used for other models.