Instability of solutions in a degenerate reaction diffusion equation

We study the spectral stability of travelling and stationary front and pulse solutions in a class of degenerate reaction-diffusion systems. We characterise the essential spectrum of the linearised operator in full generality and identify conditions under which it lies entirely in the left-half plane. For a number of special cases we obtain analytical results, including explicit Evans functions, and complete spectral descriptions for certain stationary waves. In regimes where analytical methods are not available, we compute the point spectrum numerically using a Riccati-Evans function approach. Our results show that stable travelling fronts can occur, while travelling pulses are typically unstable.

💡 Research Summary

This paper investigates the spectral stability of travelling and stationary front and pulse solutions in a class of degenerate reaction‑diffusion systems of the form
 uₜ = uₓₓ + g(u,v), vₜ = D vₓₓ – g(u,v), D ≥ 0.
The authors first recast the equations in travelling‑wave coordinates z = x – ct, yielding a coupled ODE system for the wave profiles ˆu(z), ˆv(z). Linearising about a given wave leads to a fourth‑order non‑autonomous system (2.3) with spectral parameter λ. The spectrum σ(L) of the linearised operator L is split into essential spectrum and point spectrum.

To characterize the essential spectrum, the authors examine the asymptotic constant‑coefficient system as z → ±∞, obtaining the characteristic quartic equation (2.4) for ν. By substituting ν = i k (k ∈ ℝ) they derive a quadratic relation (2.6) for λ – i c k, whose discriminant Δ determines whether λ is real or forms an ellipse in the complex plane. The analysis shows that if the far‑field derivatives of the reaction term satisfy ĝᵥ > ĝᵤ and ĝᵥ > D ĝᵤ, the essential spectrum lies entirely in the left half‑plane, possibly touching the origin with a quadratic tangency. When both ĝᵤ and ĝᵥ are positive, Δ can become negative for a band of wave numbers, producing a bounded elliptical region of oscillatory modes; this signals a possible Turing‑type instability for stationary waves.

The point spectrum is governed by the Fredholm index. When the index is zero, isolated eigenvalues may appear; otherwise they belong to the essential spectrum. The authors employ two complementary tools to locate point eigenvalues: (i) an explicit Evans function, derived analytically for special parameter choices, and (ii) a Riccati‑Evans function, which is numerically robust for the general case.

Key analytical results:

• For equal diffusion (D = 1) and stationary waves (c = 0) the Evans function can be written in closed form, allowing a complete spectral description. All eigenvalues are shown to have negative real parts, establishing linear and nonlinear stability of these fronts.
• When D = 0 (the v‑component does not diffuse) the stationary solutions consist of piecewise‑constant states. The eigenvalue problem reduces to matching conditions at the discontinuities, yielding explicit formulas for the point spectrum. Stability is determined by simple algebraic inequalities on the reaction derivatives.

Numerical investigations using the Riccati‑Evans approach cover the remaining regimes:

• Non‑stationary fronts with D = 0, c > 0 (relevant to SIS models with immobile infected individuals) exhibit parameter regions where the point spectrum lies strictly in the left half‑plane, confirming that stable travelling fronts can exist even in the fully degenerate case.
• Stationary fronts with D > 0, c = 0 (e.g., ion‑transport models where both intra‑ and extracellular ions diffuse) are also found to be spectrally stable for a wide range of reaction parameters.
• The most general case (D > 0, c ≠ 0) typically displays a leading eigenvalue with positive real part that is complex, indicating a Hopf‑type instability of travelling waves.

Across all examined parameter sets, travelling pulses are found to be linearly unstable. In every case studied, at least one eigenvalue with positive real part appears, often accompanied by a complex conjugate pair, confirming the conjecture that the Tuckwell‑Miura model, while capable of supporting travelling pulses, is not the minimal model for stable travelling pulses.

The paper concludes with a synthesis of the findings: stable travelling fronts are possible in degenerate reaction‑diffusion systems, especially when the diffusion of the v‑component is absent or when diffusion coefficients are equal. In contrast, travelling pulses are generically unstable, regardless of diffusion settings. The combined use of explicit Evans functions and Riccati‑Evans numerics provides a powerful framework for spectral analysis of degenerate systems, opening the way for future studies of more intricate reaction terms and higher‑dimensional extensions.