On a Variable-Order Fractional Parabolic Problems

In this article, we have investigated the existence and uniqueness of a weak solution to a parabolic problem involving the variable-order fractional p-Laplacian. We have demonstrated that this solution exists and is unique under certain assumptions on the second member. The problem studied is a parabolic case of the one examined in reference [5] by M. Hsini et al., with the replacement of the fractional Laplacian operator with variable exponent by the variable-order fractional operator. The functional framework involves spaces of variable-order fractional Sobolev spaces.