  In H\"ormander inner product spaces, we investigate initial-boundary value
problems for an arbitrary second order parabolic partial differential equation
and the Dirichlet or a general first-order boundary conditions. We prove that
the operators corresponding to these problems are isomorphisms between
appropriate H\"ormander spaces. The regularity of the functions which form
these spaces is characterized by a pair of number parameters and a function
parameter varying regularly at infinity in the sense of Karamata. Owing to this
function parameter, the H\"ormander spaces describe the regularity of functions
more finely than the anisotropic Sobolev spaces.
