On iterating concentration and periodic regimes at the anomalous diffusion in polymers

TL;DR

This paper models anomalous diffusion in polymers considering viscoelastic effects, proving the existence of global weak solutions and time-periodic solutions under certain conditions, advancing understanding of complex diffusion behaviors.

Contribution

It introduces a mathematical framework for anomalous diffusion in polymers, demonstrating existence of solutions and periodic regimes without restrictions on period length.

Findings

Existence of global in time weak solutions for the boundary value problem.

Existence of time-periodic weak solutions under additional coefficient assumptions.

Solutions maintain concentration values at segment endpoints, indicating stability.

Abstract

Diffusion of a penetrating liquid in a polymeric material does not often satisfy the classical diffusion equations and requires taking relaxational (viscoelastic) properties of the polymer into account. We investigate a boundary value problem on a bounded domain in space for the set of equations modelling this abnormal diffusion. It is proved that, for any sufficiently short time segment and any stress prescribed at the beginning of this segment, there exists a global in time weak solution of the boundary value problem (a pair: concentration - stress) such that the concentrations at the beginning and at the end of the segment are the same. Under an additional assumption on coefficients, existence of time-periodic weak solutions (without any restrictions of the period length) is shown.