Asymptotic behaviour for a diffusion equation governed by nonlocal interactions

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal nonlinear diffusion equation, establishing existence, stability, and attractor properties, with detailed estimates and asymptotic analysis as time approaches infinity.

Contribution

It provides new results on the existence of stable solution branches, $L^$ estimates via Moser iterations, and the asymptotic behavior of solutions for a nonlocal diffusion equation.

Findings

Existence of a unique branch of stable stationary solutions

Establishment of $L^$ estimates for solutions

Analysis of asymptotic behavior as $t o $

Abstract

In this paper we study the asymptotic behaviour of a nonlocal nonlinear parabolic equation governed by a parameter. After giving the existence of unique branch of solutions composed by stable solutions in stationary case, we gives for the parabolic problem $L^\infty $ estimates of solution based on using the Moser iterations and existence of global attractor. We finish our study by the issue of asymptotic behaviour in some cases when $t\to \infty$.