Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local diffusion

TL;DR

This paper investigates the long-term behavior of a semilinear heat equation with memory and non-local diffusion, providing new insights into the attracting sets without using the Dafermos transformation.

Contribution

It introduces a novel approach to analyze the asymptotic behavior of heat equations with memory, avoiding the Dafermos transformation and offering comprehensive results on attracting sets.

Findings

Complete characterization of attracting sets for the original problem.

Generalization of previous results to non-local diffusion cases.

Enhanced understanding of long-term dynamics in heat conduction with memory.

Abstract

In this paper, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is different from the previous published literature on the long time behavior of heat equations with memory which is carried out by the Dafermos transformation. As a consequence, the obtained results provide complete information about the attracting sets for the original problem, instead of the transformed one. In particular, the proved results also generalize and complete previous literature in the local case.