On Integro-Differential Inclusions with Operator-valued Kernels

TL;DR

This paper establishes conditions for the well-posedness of integro-differential inclusions with operator-valued kernels in Hilbert spaces, covering various equations including those in visco-elasticity and phase transition models.

Contribution

It introduces a unified framework for analyzing integro-differential inclusions with operator-valued kernels and provides criteria for their well-posedness, encompassing several classes of equations.

Findings

Criteria for well-posedness of integro-differential inclusions

Application to visco-elasticity equations

Application to phase transition models

Abstract

We study integro-differential inclusions in Hilbert spaces with operator-valued kernels and give sufficient conditions for the well-posedness. We show that several types of integro-differential equations and inclusions are covered by the class of evolutionary inclusions and we therefore give criteria for the well-posedness within this framework. As an example we apply our results to the equations of visco-elasticity and to a class of nonlinear integro-differential inclusions describing Phase Transition phenomena in materials with memory.