Stability of equilibrium solutions of a double power reaction diffusion equation with a Dirac interaction

TL;DR

This paper analyzes the stability of equilibrium solutions in a one-dimensional nonlinear reaction-diffusion equation with a Dirac interaction, providing explicit formulas and eigenvalue calculations to determine instability conditions.

Contribution

It introduces explicit formulas for equilibrium solutions and calculates the number of positive eigenvalues to assess stability, advancing understanding of localized reaction-diffusion equations.

Findings

Explicit formulas for equilibrium solutions are derived.

The number of positive eigenvalues is computed to determine instability.

The dimension of the unstable manifold is characterized.

Abstract

In this paper we provide detailed information about the instability of equilibrium solutions of a nonlinear family of localized reaction-difussion equations in dimensione one. Beyond we provide explicit formulas to the equilibrium solutions, via perturbation method and we calculate the exact number of positive eigenvalues of the linear operator associated to the stability problem, which allow us to compute the dimension of the unstable manifold.