  We study a new exactly solvable model of coupling of the Dark Energy and Dark
Matter, in the framework of which the kernel of non-gravitational interaction
is presented by the integral Volterra-type operator well-known in the classical
theory of fading memory. Exact solutions of this isotropic homogeneous
cosmological model were classified with respect to the sign of the discriminant
of the cubic characteristic polynomial associated with the key equation of the
model. Energy-density scalars of the Dark Energy and Dark Matter, the Hubble
function and acceleration parameter are presented explicitly; the scale factor
is found in quadratures. Asymptotic analysis of the exact solutions has shown
that the Big Rip, Little Rip, Pseudo Rip regimes can be realized with the
specific choice of guiding parameters of the model. We show that the
Coincidence problem can be solved if we consider the memory effect associated
with the interactions in the Dark Sector of the Universe.
