Parabolic problems in generalized Sobolev spaces. (English) Zbl 1479.35516
Summary: 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 this solution are continuous on a given set.
MSC:
| 35K35 | Initial-boundary value problems for higher-order parabolic equations |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
Hörmander space; slowly varying function; isomorphism property; local regularity; quadratic interpolation of spacesReferences:
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