  In this paper, we investigate the well-posedness and the long-time asymptotic
behavior for the initial-boundary value problem for multi-term time-fractional
diffusion equations, where the time differentiation consists of a finite
summation of Caputo derivatives with decreasing orders in (0,1) and positive
constant coefficients. By exploiting several important properties of
multinomial Mittag-Leffler functions, various estimates follow from the
explicit solutions in form of these special functions. Then the uniqueness and
continuous dependency upon initial value and source term are established, from
which the continuous dependence of solution of Lipschitz type with respect to
various coefficients is also verified. Finally, by a Laplace transform
argument, it turns out that the decay rate of the solution as time tends to
infinity is dominated by the minimum order of the time-fractional derivatives.

Analysis of PDEs