We study nonlinear stability of pulled fronts in scalar parabolic equations on the real line of arbitrary order, under conceptual assumptions on existence and spectral stability of fronts. In this general setting, we establish sharp algebraic decay rates and temporal asymptotics of perturbations to the front. Some of these results are known for the specific example of the Fisher-KPP equation, and our results can thus be viewed as establishing universality of some aspects of this simple model. We also give a precise description of how the spatial localization of perturbations to the front affects the temporal decay rate, across the full range of localizations for which asymptotic stability holds. Technically, our approach is based on a detailed study of the resolvent operator for the linearized problem, through which we obtain sharp linear time decay estimates that allow for a direct nonlinear analysis.
We establish selection of critical pulled fronts in invasion processes. Our result shows convergence to a pulled front with a logarithmic shift for open sets of steep initial data, including one-sided compactly supported initial conditions. We rely on robust, conceptual assumptions, namely existence and marginal spectral stability of a front traveling at the linear spreading speed. We demonstrate that the assumptions hold for open classes of spatially extended systems. Previous results relied on comparison principles or probabilistic tools with implied non-open conditions on initial data and s…
We prove that the critical pulled front of Lotka-Volterra competition systems is nonlinearly asymptotically stable. More precisely, we show that perturbations of the critical front decay algebraically with rate $t^{-3/2}$ in a weighted $L^\infty$ space. Our proof relies on pointwise semigroup methods and utilizes in a crucial way that the faster decay rate $t^{-3/2}$ is a consequence of the lack of an embedded zero of the Evans function at the origin for the linearized problem around the critical front.
We analyze the transition between pulled and pushed fronts both analytically and numerically from a model-independent perspective. Based on minimal conceptual assumptions, we show that pushed fronts bifurcate from a branch of pulled fronts with an effective speed correction that scales quadratically in the bifurcation parameter. Strikingly, we find that in this general context without assumptions on comparison principles, the pulled front loses stability and gives way to a pushed front when monotonicity in the leading edge is lost. Our methods rely on far-field core decompositions that identif…
We investigate the inside structure of one-dimensional reaction-diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front conver…
We investigate a specific reaction-diffusion system that admits a monostable pulled front propagating at constant critical speed. When a small parameter changes sign, the stable equilibrium behind the front destabilizes, due to essential spectrum crossing the imaginary axis, causing a Turing bifurcation. Despite both equilibrium states are unstable, the front continues to exist, and is shown to be asymptotically stable, against suitably-localized perturbations, with algebraic temporal decay rate $t^{-3/2}$. To obtain such decay, we rely on point-wise semigroup estimates, and show that the Turi…
In this work we investigate the issue of logarithmic drifts in the position of the level sets of solutions of monostable reaction-diusion equations, with respect to the traveling front with minimal speed. On the one hand, it is a celebrated result of Bramson that such a logarithmic drift occurs when the reaction is of the KPP (or sublinear) type. On the other hand, it is also known that this drift phenomenon disappears when the minimal front speed is nonlinearly determined. However, some monostable reaction-diusion equations fall in neither of those cases and our aim is to fill that gap. We pr…
We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength $β\in\mathbb{R}$. This equation exhibits a transition from pulled to pushed front behavior at $β_c=2$. We prove convergence of the solutions to a traveling wave in a reference frame centered at a position $m_β(t)$ and study the asymptotics of the front location $m_β(t)$. When $β< 2$, it has the same form as for the standard Fisher-KPP equation established by Bramson \cite{Bramson1,Bramson2}: $m_β(t) = 2t - (3/2)\log(t) + x_\infty + o(1)$ as $t\to+\infty$. This form is typical of pulled…
We uncover a seemingly previously unnoticed algebraic structure of a large class of reaction-diffusion equations and use it, in particular, to study the long time behavior of the solutions and their convergence to traveling waves in the pulled and pushed regimes, as well as at the pushmi-pullyu boundary. One such new object introduced in this paper is the shape defect function, which, indirectly, measures the difference between the profiles of the solution and the traveling wave. While one can recast the classical notion of `steepness' in terms of the positivity of the shape defect function, i…
We establish spreading properties of the Lotka-Volterra competition-diffusion system. When the initial data vanish on a right half-line, we derive the exact spreading speeds and prove the convergence to homogeneous equilibrium states between successive invasion fronts. Our method is inspired by the geometric optics approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. Our main result settles an open question raised by Shigesada et al. in 1997, and shows that one of the species spreads to the right with a nonlocally pulled front.
We study the effect of a cut-off on the speed of pulled fronts of the one dimensional reaction diffusion equation. We prove rigorous upper and lower bounds on the speed in terms of the cut-off parameter epsilon. From these bounds we estimate the range of validity of the Brunet--Derrida formula for a general class of reaction terms.
We study invasion fronts in the FitzHugh--Nagumo equation in the oscillatory regime using singular perturbation techniques. Phenomenologically, localized perturbations of the unstable steady-state grow and spread, creating temporal oscillations whose phase is modulated spatially. The phase modulation appears to be selected by an invasion front that describes the behavior in the leading edge of the spreading process. We construct these invasion fronts for large regions of parameter space using singular perturbation techniques. Key ingredients are the construction of periodic orbits, their unsta…
We revisit the nonlinear stability of the critical invasion front in the Ginzburg-Landau equation. Our main result shows that the amplitude of localized perturbations decays with rate $t^{-3/2}$, while the phase decays diffusively. We thereby refine earlier work of Bricmont and Kupiainen as well as Eckmann and Wayne, who separately established nonlinear stability but with slower decay rates. On a technical level, we rely on sharp linear estimates obtained through analysis of the resolvent near the essential spectrum via a far-field/core decomposition which is well suited to accurately describi…
Traveling oscillating fronts (TOFs) are specific waves of the form $U_\star (x,t) = e^{-i ωt} V_\star(x - ct)$ with a profile $V_{\star}$ which decays at $- \infty$ but approaches a nonzero limit at $+\infty$. TOFs usually appear in complex Ginzburg Landau equations of the type $U_t = αU_{xx} + G(|U|^2)U$. In this paper we prove a theorem on the asymptotic stability of TOFs, where we allow the initial perturbation to be the sum of an exponentially localized part and a front-like part which approaches a small but nonzero limit at $+ \infty$. The underlying assumptions guarantee that the operato…
Subdiffusive motion takes place at a much slower timescale than diffusive motion. As a preliminary step to studying reaction-subdiffusion pulled fronts, we consider here the hyperbolic limit $(t,x) \to (t/\varepsilon, x/\varepsilon)$ of an age-structured equation describing the subdiffusive motion of, e.g., some protein inside a biological cell. Solutions of the rescaled equations are known to satisfy a Hamilton-Jacobi equation in the formal limit $\varepsilon \to 0$. In this work we derive uniform Lipschitz estimates, and establish the convergence towards the viscosity solution of the limitin…
