Stability of asymptotic waves in the Fisher–Stefan equation

We establish spectral, linear, and nonlinear stability of the vanishing and slow-moving travelling waves that arise as time asymptotic solutions to the Fisher–Stefan equation. All stability analysis is in terms of the limiting equations that the asymptotic waves satisfy. • Spectral, linear and nonlinear stability of asymptotic waves in the Fisher–Stefan equation. • Stability analysis is in terms of the limiting equations. • We use a dynamical systems approach and compute the Maslov index of the travelling waves.