Well-posedness of the Basset problem in spaces of smooth functions

In the present paper we consider the initial value problem for the fractional differential equation d u ( t ) d t + D t 1 2 u ( t ) + A u ( t ) = f ( t ) , 0 < t < 1 , u ( 0 ) = 0 in a Banach space E with the strongly positive operator A . The well-posedness of this problem in spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of problems for 2 m th order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions in space variable are obtained.