Classical Solutions of Parabolic Initial-Boundary-Value Problems and HӧRmander Spaces

A condition for generalized solutions of a parabolic problem for a Petrovskii system to be classical

“…The use of Hörmander spaces allows us to attain the minimal admissible values of the number parameters in conditions (10) and (11). If we formulate an analog of this theorem using anisotropic Sobolev spaces (i.e.…”

“…The use of the function parameter ϕ allows us to achieve the minimal admissible value of the number parameter s, which is not possible in the framework of Sobolev spaces or Hölder spaces [3,4,22]. As to scalar parabolic problems, conditions of this type are obtained in [10,11].…”

Some problems for Petrovskii parabolic systems in generalized Sobolev spaces

“…In recent years, other classes of function spaces are applied more and more actively to parabolic problems; namely, spaces with mixed norms, Triebel-Lizorkin spaces, weighted spaces, spaces of generalized smoothness (see, e.g., [5,6,11,15,23,33] and reference therein). Our paper continues the series of works [18][19][20][23][24][25] aimed to build a theory of parabolic initial-boundary value problems in function spaces of generalized anisotropic smoothness. This smoothness is given by a pair of real numbers and by a radial function which varies slowly at infinity and characterizes supplementary smoothness with respect to that given by the numbers.…”

“…This theorem also allows us to obtain new sufficient conditions for the continuity of generalized solutions, specifically to find conditions under which the solutions are classical (c.f. [19][20][21][22][23][24]). These applications will be given in another paper.…”

Parabolic problems in generalized Sobolev spaces

“…This claim is stronger than the condition of Theorem 4.3 due to the left-hand embeddings in (3.9) and (3.10). This theorem can be used to obtain sufficient conditions under which the generalized solution u to the parabolic problem is classical (see [25]).…”