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

Parabolic problems in generalized Sobolev spaces