On Some Properties of Solutions to Dzektser Mathematical Model in Quasi-Sobolev Spaces

Abstract

The theory of holomorphic degenerate semigroups of operators constructed earlier in Banach spaces and Frechet spaces is transferred to quasi-Sobolev spaces of sequences. This article contains results about existence of the exponential dichotomies of solutions to evolution Sobolev type equation in quasi-Sobolev spaces. To obtain this result we proved the relatively spectral theorem and the existence of invariant spaces of solutions. All abstract results are applied to investigation of properties of solutions to Dzektser mathematical model in quasi-Sobolev spaces. The article besides the introduction and references contains three paragraphs. In the first one, quasi-Banach spaces, quasi-Sobolev spaces and polynomials of Laplace quasi-operator are defined. Moreover the conditions for existence of degenerate holomorphic operator semigroups in quasi-Banach spaces of sequences are obtained. In other words, we state the first part of generalization of the Solomyak -- Iosida theorem to quasi-Sobolev spaces of sequences. In the second paragraph the phase space of the homogeneous equation is constructed. Here we show the existence of invariant spaces of equation and get the conditions for exponential dichotomies of solutions. The last paragraph presents results on properties of solutions to Dzektser equation.