A Linearized Model of Vibrations in the DNA Molecule in the Quasi-Banach Spaces

Abstract

The system of Boussinesq equations, which simulates the vibrations in the DNA molecule, is researched in this paper. Earlier this model with a nonlinear right side was considered in a Banach space. In this paper, the model with a linear one is considered, but in quasi-Banach spaces of sequences. In the article the method of phase space and the theory of $(L,~p)$-bounded operators, developed by G. A. Sviridyuk and T. G. Sukacheva for first order equations, are used. It is also based on the theory of the Cauchy problem for linear equations of Sobolev type of high order in quasi-Banach spaces. A mathematical model of vibrations in the DNA molecule is reduced to an equation of Sobolev type in a quasi-Sobolev space. We construct the phase space of a system of linear Boussinesq equations. Sufficient conditions for the solvability of the initial boundary value problem for the Boussinesq equations in terms of the theory of $(L,~p)$-bounded operators are obtained.