  We consider a general inhomogeneous parabolic initial-boundary value problem
for a $2b$-parabolic differential equation given in a finite multidimensional
cylinder. We investigate the solvability of this problem in some generalized
anisotropic Sobolev spaces. They are parametrized with a pair of positive
numbers $s$ and $s/(2b)$ and with a function $\varphi:[1,\infty)\to(0,\infty)$
that varies slowly at infinity. The function parameter $\varphi$ characterizes
subordinate regularity of distributions with respect to the power regularity
given by the number parameters. We prove that the operator corresponding to
this problem is an isomorphism on appropriate pairs of these spaces. As an
application, we give a theorem on the local regularity of the generalized
solution to the problem. We also obtain sharp sufficient conditions under which
chosen generalized derivatives of the solution are continuous on a given set.
