Weak solution for nonlinear fractional P(.)-Laplacian Problem with variable order via Rothe's Time-discretization method

In this paper, we prove the existence and uniqueness results of weak solutions to a class of nonlinear fractional parabolic p(.)-Laplacian problem with variable order. The main tool used here is the Rothe’s method combined with the theory of variable-order fractional Sobolev spaces with variable exponent.