Flow decomposition for heat equations with memory

TL;DR

This paper introduces a novel flow decomposition for heat equations with memory, revealing their hybrid parabolic-hyperbolic nature and the influence of memory on system dynamics.

Contribution

It provides a new decomposition method for heat equations with memory kernels, highlighting the interplay between parabolic and hyperbolic components.

Findings

Decomposition separates flow into parabolic, hyperbolic, and smoothing parts.

Memory significantly alters the parabolic behavior of the heat equation.

The flow exhibits a hybrid nature due to the memory term.

Abstract

We build up a decomposition for the flow generated by the heat equation with a real analytic memory kernel. It consists of three components: The first one is of parabolic nature; the second one gathers the hyperbolic component of the dynamics, with null velocity of propagation; the last one exhibits a finite smoothing effect. This decomposition reveals the hybrid parabolic-hyperbolic nature of the flow and clearly illustrates the significant impact of the memory term on the parabolic behavior of the system in the absence of memory terms.