Linear convective stability of a front superposition with unstable connecting state

TL;DR

The work addresses convective stability of a two-front invasion in a reaction-diffusion system with an unstable connecting state, revealing long-range semi-strong interactions between fronts. By linearizing around the moving two-front terrace and employing time-weighted spaces, the authors derive time-uniform resolvent bounds via numerical-range estimates for the evolving linear operator ${\mathcal L}(t)$. The main contribution is a constructive stability result: for small interspecific coupling $\alpha_1,\alpha_2$ and an appropriately chosen weight $\omega$, perturbations decay exponentially in the weighted $L^2$ norm, with explicit decay rate $\eta$. This work clarifies how the instability of the intermediate state shapes inter-front interactions and provides a framework for analyzing multi-front patterns in systems with semi-strong interactions, potentially extendable to periodic equilibria and residual-term modulation.

Abstract

We study convective stability of a two-front superposition in a reaction-diffusion system. Due to the instability of the connecting equilibrium, long-range semi-strong interaction is expected between the two waves. When restricting to the linear dynamic, we indeed identify that convective stability of superposed waves occurs for fewer propagation speeds than for the corresponding single waves. It reflects the interaction that monostable waves have over long distances. Our method relies on numerical range estimates, that imply time-uniform resolvent bounds.

Figures (2)

• Figure 1: Left: Numerical simulation showcasing $p_2$ instability. Parameter values are $(d,r,\alpha_1,\alpha_2) = (4,2,0.75,0.75)$, and the initial datum is a step function of $e_3$ for $x \in (-50,50)$ and $e_4$ everywhere else, with an additional small perturbation in $u_2$ in a neighborhood of $x = 0$. Initially the solution forms a front with a profile close to $p_2$. However, the perturbation in the wake then grows into a secondary slower front with a profile close to $p_1$. Right: All known single fronts (edges) between pairs of equilibria (nodes).
• Figure 2: Numerical simulations showcasing $p_3$ (in)stability. Left: same parameters as in Figure \ref{['fig:numerics1']} and initial step function with $e_1$ for $x \in (-50,50)$ and $e_4$ everywhere else. The solution initially forms a front connecting $e_1$ and $e_4$ directly, which breaks up quickly into a superposition of $p_1$ and $p_2$. Right: parameter values are $(d,r,\alpha_1,\alpha_2) = (0.2,2,0.75,0.75)$, and initial step function with $e_1$ for $x \in (-10,10)$, $e_3$ for $x \in (-40,40)\setminus (-10,10)$ and $e_4$ everywhere else. Initially a superposition front of $p_1$ and $p_2$ forms with $c_1 > c_2$. This front then collapses to a single interface $p_3$.

Theorems & Definitions (26)

• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8
• proof
• Lemma 9