A magneto-viscoelasticity problem with a singular memory kernel

TL;DR

This paper proves the existence of solutions for a one-dimensional magneto-viscoelasticity model involving a non-linear integro-differential system with an unbounded relaxation kernel, extending previous models by allowing more general kernels.

Contribution

It generalizes prior work by allowing the relaxation function to be unbounded at zero, broadening the class of kernels for magneto-viscoelastic models.

Findings

Existence of solutions for the coupled integro-differential PDE system.

Extension of previous models to include unbounded relaxation kernels.

Application of the penalized Ginzburg-Landau magnetic evolution equation.

Abstract

The existence of solutions to a one-dimensional problem arising in magneto-viscoelasticity is here considered. Specifically, a non-linear system of integro-differential equations is analyzed, it is obtained coupling an integro-differential equation modeling the viscoelastic behaviour, in which the kernel represents the relaxation function, with the non-linear partial differential equations modeling the presence of a magnetic field. The case under investigation generalizes a previous study since the relaxation function is allowed to be unbounded at the origin, provided it belongs to $L^1$; the magnetic model equation adopted, as in the previous results [21,22, 24, 25] is the penalized Ginzburg-Landau magnetic evolution equation.