Existence of a weak solution for a nonlinear parabolic problem with fractional derivates

The primary objective of this study was to demonstrate the existence and uniqueness of a weak solution for a nonlinear parabolic problem with fractional derivatives for the spatial and temporal variables on a one-dimensional domain. Using the Nehari manifold method and its relationship with the Fibering maps, the existence of a weak solution for the stationary case was demonstrated. Finally, using the Arzela-Ascoli theorem and Banach’s fixed point theorem, the existence and uniqueness of a weak solution for the nonlinear parabolic problem were shown.