Saddle-point structure of fixed points in a reaction-diffusion equation with discontinuous nonlinearity

TL;DR

This paper investigates the local dynamics near fixed points in a reaction-diffusion equation with discontinuous nonlinearity, revealing a saddle-point structure and hyperbolicity through linearization involving the Dirac delta.

Contribution

It introduces a novel linearization method using the Dirac delta to analyze stability and saddle-point structure of fixed points in discontinuous reaction-diffusion equations.

Findings

Fixed points exhibit saddle-point structure.

Fixed points are hyperbolic.

Linearization involving Dirac delta is effective.

Abstract

In this paper, we study the local behaviour of solutions near the fixed points of a reaction-diffusion equation with discontinuous nonlinearity. By employing an appropriate linearization around the fixed points, which involves the Dirac delta distribution, we analyze the stability of the stationary solutions and demonstrate that they exhibit a saddle-point structure. As a result, we establish the hyperbolicity of the fixed points.