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