Dual-phase-lag one-dimensional thermo-porous-elasticity with microtemperatures

This paper is devoted to studying the linear system of partial differential equations modelling a one-dimensional thermo-porous-elastic problem with microtemperatures in the context of the dual-phase-lag heat conduction. Existence, uniqueness, and exponential decay of solutions are proved. Polynomial stability is also obtained in the case that the relaxation parameters satisfy a certain equality. Our arguments are based on the theory of semigroups of linear operators.

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Using techniques of the theory of semigroups of linear operators, we study the question of approximating solutions to equations governing diffusion in thin layers separated by a semi-permeable membrane. We show that as thickness of the layers converges to 0, the solutions, which by nature are functions of 3 variables, gradually lose dependence on the vertical variable and thus may be regarded as functions of 2 variables. The limit equation describes diffusion on the lower and upper sides of a two-dimensional surface (the membrane) with jumps from one side to the other. The latter possibility is expressed as an additional term in the generator of the limit semigroup, and this term is built from permeability coefficients of the membrane featuring in the transmission conditions of the approximating equations (i.e., in the description of the domains of the generators of the approximating semigroups). We prove this convergence result in the spaces of square integrable and continuous functions, and study the way the choice of transmission conditions influences the limit.

Feynman averaging of semigroups generated by Schrödinger operators

The extension of averaging procedure for operator-valued function is defined by means of the integration of measurable map with respect to complex-valued measure or pseudomeasure. The averaging procedure of one-parametric semigroups of linear operators based on Chernoff equivalence for operator-valued functions is constructed. The initial value problem solutions are investigated for fractional diffusion equation and for Schrödinger equation with relativistic Hamiltonian of free motion. It is established that in these examples the solution of evolutionary equation can be obtained by applying the constructed averaging procedure to the random translation operators in classical coordinate space.

On continuous dependence for the mixed problem of microstretch bodies

We do a qualitative study on the mixed initial-boundary value problem in the elastodynamic theory of microstretch bodies. After we trans- form this problem in a temporally evolutionary equation on a Hilbert space, we will use some results from the theory of semigroups of linear operators in order to prove the continuous dependence of the solutions upon initial data and supply terms.