# Los’ V. M.

### Systems parabolic in Petrovskii's sense in Hörmander spaces

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 365-380

We study a general parabolic initial-boundary-value problem for systems parabolic in Petrovskii’s sense with zero initial Cauchy data in some anisotropic H¨ormander inner-product spaces.We prove that the operators corresponding to this problem are isomorphisms between the appropriate H¨ormander spaces. As an application of this result, we establish a theorem on the local increase in regularity of solutions of the problem. We also obtain new sufficient conditions of continuity for the generalized partial derivatives of a given order of a chosen component of the solution.

### Sufficient conditions under which the solutions of general parabolic initial-boundaryvalue problems are classical

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1518-1527

We establish new sufficient conditions under which the generalized solutions of initial-boundary-value problems for the linear parabolic differential equations of any order with complex-valued coefficients are classical. These conditions are formulated in the terms of belonging of the right-hand sides of this problem to certain anisotropic H¨ormander spaces. In the definition of classical solution, its continuity on the line connecting the lateral surface with the base of the cylinder (in which the problem is considered) is not required.

### Classical solutions of parabolic initial-boundary value problems and Hormander spaces.

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1229-1239

For the second-order linear parabolic differential equations with complex-valued coefficients, we establish new sufficient conditions under which the generalized solutions of these problems are continuous. The conditions are formulated in the terms of belonging of the right-hand sides of these problems to certain anisotropic Ho¨rmander spaces.

### Theorems on isomorphisms for some parabolic initial-boundary-value problems in Hörmander spaces: limiting case

Ukr. Mat. Zh. - 2016. - 68, № 6. - pp. 786-799

In Hilbert Hörmander spaces, we study the initial-boundary-value problems for arbitrary parabolic differential equations of the second order with Dirichlet boundary conditions or general boundary conditions of the first order in the case where the solutions of these problems belong to the space $H^{2,1,\varphi}$. It is shown that the operators corresponding to these problems are isomorphisms between suitable Hörmander spaces. The regularity of the functions that form these spaces is characterized by a couple of numerical parameters and a functional parameter $\varphi$ slowly varying at infinity in Karamata’s sense. Due to the presence of the parameter $\varphi$, the Hörmander spaces describe the regularity of the functions more precisely than the anisotropic Sobolev spaces.

### Mixed Problems for the Two-Dimensional Heat-Conduction Equation in Anisotropic Hörmander Spaces

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 645-656

For some anisotropic inner-product Hörmander spaces, we prove the theorems on well-posedness of initial-boundary-value problems for the two-dimensional heat-conduction equation with Dirichlet or Neumann boundary conditions. The regularity of the functions from these spaces is characterized by a couple of numerical parameters and a function parameter regularly varying at infinity in Karamata’s sense and characterizing the regularity of functions more precisely than in the Sobolev scale.

### On the Sobolev problem in the complete scale of Banach spaces

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1181–1192

In a bounded domain*G* with boundary ∂*G* that consists of components of different dimensions, we consider an elliptic boundary-value problem in complete scales of Banach spaces. The orders of boundary expressions are arbitrary; they are pseudodifferential along ∂*G*. We prove the theorem on a complete set of isomorphisms and generalize its application. The results obtained are true for elliptic Sobolev problems with a parameter and parabolic Sobolev problems as well as for systems with the Douglis-Nirenberg structure.