Holomorphic Degenerate Operator Semigroups and Evolutionary Sobolev Type Equations in Quasi-Sobolev Spaces of Sequences

Abstract

The interest to Sobolev type equations has significantly  increased recently, moreover, the need occured to consider them in quasi-Banach spaces.  This need is explained not by the desire to enrich the theory but rather by the aspiration to comprehend non-classical models of mathematical physics in quasi-Banach spaces.

It should be noted that Sobolev type equations are called evolutionary, provided their solutions exist only on R₊. The theory of holomorphic degenerate semigroups of operators constructed earlier in Banach and Frechet spaces is transferred to quasi-Sobolev spaces of sequences.

Besides the introduction and references the paper contains four paragraphs. In the first, quasi-Banach spaces and linear bounded and closed operators defined on them are considered. Quasi-Sobolev spaces and powers of the Laplace quasi-operator are also taken into consideration. In the second paragraph polynomials of the Laplace quasi-operator are considered for operators L and M and conditions for the existence of degenerate holomorphic operator semigroups in quasi-Banach spaces of sequences are obtained. In other words, the first part of the generalization of the Solomyak–Iosida theorem to quasi-Banach spaces of sequences is stated. In the third paragraph the phase space of the homogeneous equation is constructed. The last paragraph investigates the "quasi-Banach" analogue of the homogeneous Dirichlet problem in a bounded domain with a smooth boundary for the linear Dzektser equation