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

The main objective of this work is to demostrate the existence and unique of 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 demostrated. Finally, using the Arzela-Ascoli Theorem and Banach’s Fixed Point Theorem, the existence and uniqueness of a weak solution for the non-linear parabolic problem were shown.