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Problems of the type described in the title were have been studied [4, Ch. 5] for various partial differential equations, not including parabolic equations. The book just cited also contains a survey and bibliography of papers related to nonlocal problems for partial differential equations and operator-differential equations. Among more recent studies relating to this theme we note the papers [3,5]. We shall use the following notation below; f/P is the p-dimensional torus {x E R p : 0 < xr < 27r, r = 1,p}, D = [0.,T] x tiP, k = (kt,.. .,kp) E

Conditions for the existence and uniqueness of a classical solution of a nonlocal boundary-value problem for a differential equation with a regularized Riemann-Liouville fractional time derivative with variable coefficients are investigated.