Stability and hyperbolicity of equilibria for a scalar nonlocal one-dimensional quasilinear parabolic problem

TL;DR

This paper investigates the stability and hyperbolicity of equilibria in a nonlocal scalar quasilinear parabolic equation, revealing spectral properties and saddle point behavior similar to local models but requiring complex spectral analysis.

Contribution

It provides a detailed spectral analysis of nonlocal quasilinear parabolic equations and establishes the saddle point property of equilibria, extending known results from local equations.

Findings

Spectral analysis of the associated linear operators

Equilibria exhibit saddle point properties

Nonlocal equations share similarities with local counterparts

Abstract

In this work, we present results on stability and hyperbolicity of equilibria for a scalar nonlocal one-dimensional quasilinear parabolic problem. We show that this nonlocal version of the well-known Chafee-Infante equation bares some resemblance with the local version. However, its nonlocal characteristc requires a fine analysis of the spectrum of the associated linear operators, a lot more ellaborated than the local case. The saddle point property of equilibria is shown to hold for this quasilinear model.