Today differential equations involving the nonlocal operators have become an efficient tool for modelling the constitutive relation of the viscoelastic fluid.
We investigated the Cauchy-Dirichlet problem for the fractional partial differential equation with variable coefficients which models the Couette flow of the fractional viscoelastic fluid.
In the reference [Math. Meth. Appl. Sci. 46 (2023) 12960–12978], the authors proved the well-posedness of the initial-boundary value problem of the fractional partial differential equation with constant coefficients for modelling the Couette flow and established the result for the long-time behavior of the solution.
We used the asymptotic behavior of the multivariate Mittag-Leffler function to prove the unique existence of the mild solution of the fractional partial differential equations with variable coefficients. Also, employing the boundary regularity estimate of the elliptic equation, we obtained the result for the long-time behavior of the mild solution. Our result extends that of the equation with constant coefficients to the case of the equation with variable coefficients.
Our research results were published in "Rend. Circ. Mat. Palermo, II. Ser, 75 (2026) 9" under the title of "Initial-boundary value problems for equations of motion of viscoelastic fluid with fractional derivatives" (https://doi.org/10.1007/s12215-025-01325-2).
