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 asymptotic properties using analytical methods like Moser iterations and attractor theory.

Contribution

It provides new results on the existence, stability, and asymptotic behavior of solutions to a nonlocal diffusion equation with a parameter, including $L^ abla$ estimates and attractor analysis.

Findings

Existence of a unique branch of stable stationary solutions.

Establishment of $L^ abla$ estimates for solutions.

Analysis of asymptotic behavior as time approaches infinity.

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 $t\to \infty$.