A model of viscoelasticity with time-dependent memory kernels

TL;DR

This paper introduces a mathematical model for viscoelastic materials with time-dependent memory kernels, establishing well-posedness and laying groundwork for analyzing long-term behavior in aging materials.

Contribution

It develops a new framework for analyzing viscoelastic systems with time-dependent memory kernels, including a proper notion of solutions and global well-posedness.

Findings

Established a proper notion of solution for the model

Proved global well-posedness of the system

Framework extends to long-term behavior analysis

Abstract

We consider the model equation arising in the theory of viscoelasticity $$\partial_{tt} u-h_t(0)\Delta u -\int_{0}^\infty h_t'(s)\Delta u(t-s)d s+ f(u) = g.$$ Here, the main feature is that the memory kernel $h_t(\cdot)$ depends on time, allowing for instance to describe the dynamics of aging materials. From the mathematical viewpoint, this translates into the study of dynamical systems acting on time-dependent spaces, according to the newly established theory of Di Plinio et al. In this first work, we give a proper notion of solution, and we provide a global well-posedness result. The techniques naturally extend to the analysis of the longterm behavior of the associated process, and can be exported to cover the case of general systems with memory in presence of time-dependent kernels.