  We investigate a viscoelastic flow model with a generalized memory, in which
a weak-singular component is introduced in the exponential convolution kernel
of classical viscoelastic flow equations that remains untreated in the
literature. We prove the well-posedness and regularity of the solutions, based
on which we prove the exponential convergence of the solutions to the steady
state. The proposed model serves as an extension of classical viscoelastic flow
equations by adding a dimension characterized by the power of the weak-singular
kernel, and the derived results provide theoretical supports for designing
numerical methods for both the considered equation and its steady state.
