We establish selection of critical pulled fronts in invasion processes as predicted by the marginal stability conjecture. 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 and demonstrate that the assumptions hold for open classes of spatially extended systems. Previous results relied on comparison principles or probabilistic tools with implied nonopen conditions on initial data and structure of the equation. Technically, we describe the invasion process through the interaction of a Gaussian leading edge with the pulled front in the wake. Key ingredients are sharp linear decay estimates to control errors in the nonlinear matching and corrections from initial data.
1. Introduction
The onset of structure formation in spatially extended physical systems is often mediated by an invasion process, in which a pointwise stable state invades a pointwise unstable state. One observes that an initially localized perturbation to the pointwise unstable background state grows in amplitude and spreads spatially. In its wake, this spreading process may select a spatially constant state, a periodic pattern, or more complicated dynamics; see Reference 84 for a thorough review of experimental observations of invasion processes across the sciences. The fundamental objective then is to describe this process, in particular by predicting the spreading speed and the selected state in the wake.
A mathematical study usually focuses first on a one-sided invasion process, describing convergence of solutions to a front that connects the unstable state in the leading edge to the selected state in the wake. Existence of such fronts and stability information is often available through a variety of analytical and computational techniques, ranging from comparison methods and monotonicity Reference 27Reference 87 to integrability Reference 63Reference 85, to perturbative techniques in small-amplitude settings Reference 15Reference 16Reference 20Reference 23Reference 45Reference 72 or in the presence of scale separation Reference 14Reference 32Reference 75, to topological techniques Reference 79Reference 80 and to rigorous computational approaches Reference 6; see also the review Reference 84. The marginal stability conjecture connects such existence and stability information with the physically most interesting question as to which fronts are actually selected, that is, observed when starting from steep, particularly compactly supported initial conditions; see for instance Reference 15Reference 18Reference 84. In essence, the conjecture predicts that out of a family of fronts, the “marginally stable” front is selected and thus reduces, whenever proven, the study of invasion processes to the study of existence and stability of front solutions. Our work here makes this concept of marginal stability precise and establishes the conjecture.
Linear marginal stability. Motivating this conjecture are predictions based on the linearized problem at the trivial unstable state. Spatially localized disturbances in this linearized problem grow exponentially and spread spatially. Analyzing stability in co-moving frames of speed $c$, one finds that perturbations exhibit pointwise exponential decay for all speeds above a critical speed, $c>c_*$. The pointwise decay or growth is typically encoded in the complex linear dispersion relation through pinched double roots$\lambda _\mathrm{dr}(c)$: one finds pointwise exponential behavior $\sim \exp (\lambda _\mathrm{dr}t)$ with $\operatorname {Re}\lambda _\mathrm{dr}(c)<0$ for $c>c_*$ and marginal stability $\operatorname {Re}\lambda _\mathrm{dr}(c_*)=0$ at the “critical” linear spreading speed$c_*$Reference 8Reference 13Reference 49; see Section 1.2 for a brief review. In addition to predictions for the speed, the linear analysis also offers a prediction for the nature of the invasion process: zero versus nonzero frequency $\operatorname {Im}\lambda _\mathrm{dr}(c_*)$ predicts invasion that is
$$\begin{equation*} \text{(S) stationary or (P) time-periodic} \end{equation*}$$
in the co-moving frame.
Nonlinear stability. Nonlinear saturation of this linear growth process leads to formation of fronts. Describing the evolution toward fronts, one invokes subtle stability arguments. Pointwise stability is mostly intractable in nonlinear equations, and one therefore resorts to function spaces with exponential weights, penalizing in particular perturbations in the leading edge of the front that would grow due to the inherent instability of the invaded state. In the leading edge, the linearized evolution is indeed close to the linearization at the unstable state and one finds that fronts with speeds $c<c_*$ are unstable in any weighted space. Fronts with speed $c\geq c_*$ however can often be shown to be stable in exponentially weighted spaces, based on properties of the linearization at the front. One therefore analyzes the spectrum of the linearization, which is composed of essential spectrum related to stability in the leading edge, essential spectrum related to stability in the wake, and point spectrum associated with stability of the front interface. In suitably weighted spaces, one establishes or assumes that essential spectrum associated with the leading edge is stable for $c>c_*$ and marginally stable for $c=c_*$, while point spectrum and essential spectrum associated with the state in the wake are stable or marginally stable; see also Figure 1.
This perspective recovers some notion of stability but does not address the question of speed selection, since all fronts with speeds $c \geq c_*$ are stable in this sense. It turns out that permissible perturbations of the front in such stability arguments are very localized so that initial conditions given by the sum of front and perturbation necessarily preserve the exponential decay rate in the leading edge of the front. The arguments do in particular not describe the behavior of the most relevant steep, for instance step-function like, initial conditions.
The marginal stability conjecture. The marginal stability conjecture states that, nevertheless, information on the existence and stability of fronts does yield a selection criterion:
$$\begin{equation*} \text{\textit{Marginally stable fronts are selected by steep initial conditions.}} \end{equation*}$$
The hypothesis is understood to hold universally across equations for open classes of steep initial conditions, under appropriate notions of marginal stability; see Reference 18Reference 23Reference 24Reference 84 for statements of the conjecture in many specific examples. In this paper, we will define marginal stability through spectral properties of the linearization of the front in norms with exponentially growing weights in the leading edge, encoding marginal stability in the leading edge, absence of unstable or marginally stable spectrum associated with the front interface or the wake; see Hypotheses 1, 2, 3, and 4. Fronts where marginal stability is caused by the leading edge are commonly referred to as pulled fronts. In that regard, our assumptions exclude pushed fronts, with instability in the front interface, and staged invasion with instabilities in the wake. Selection of pushed fronts is easier to establish than that of pulled fronts, since for pushed fronts a spectral gap may be recovered with exponential weights; see Section 8 for a discussion of pushed fronts and other secondary instability mechanisms. We define selection in Definition 1 as allowing for open classes of steep initial conditions. Our main result, precisely stated in Theorem 1, establishes the marginal stability conjecture under these conceptual assumptions.
In addition, we address the universality aspect of the marginal stability conjecture in Theorem 2, which shows that our assumptions hold for open classes of equations. Contrasted with previous work, our result does not rely on the structure of the equation or sign conditions on initial data. We rephrase the front selection problem as a stability problem for fronts similar to Reference 4Reference 24Reference 26Reference 33, but, crucially, with supercritical localization of perturbations to the front. In fact, as we shall see later, dynamics near a critical front profile exhibit diffusive decay in suitable norms for localized perturbations of the front. However, again in suitable norms, steep initial data induces perturbations of fronts that grow linearly in the leading edge and therefore do not decay diffusively. In this respect, our results can be compared to efforts toward establishing diffusive decay in pattern-forming systems, where neutral modes decay diffusively, and where modulation equations attempt to capture dynamics when perturbations are not spatially localized Reference 21Reference 34Reference 52Reference 54Reference 55Reference 56Reference 77.
A brief history and examples. Specific to front selection, the mathematical literature originates with work in the 1930s on the Fisher-KPP equation Reference 31Reference 61,
$$\begin{equation} \partial _t u = \partial _{xx}u + u - u^2, \quad x \in \mathbb{R}, \quad t > 0, \cssId{texmlid1}{\tag{1.1}} \end{equation}$$
where fronts connecting the stable state $u \equiv 1$ to the unstable state $u \equiv 0$ are linearly stable for speeds $c\geq 2$, the linear spreading speed. Kolmogorov, Petrovskii, and Piskunov Reference 61 proved in 1937 that for step function initial data, $u = 1, x<0$ and $u = 0$,$x>0$, the solution to Equation 1.1 indeed converges to shifted pulled fronts,
for some shift $\sigma (t) = 2t + \mathrm{o}(t)$, uniformly in space, where $q(\cdot ; 2)$ is a front solution to Equation 1.1 satisfying $u(x, t) = q(x-2t; 2)$, unique up to spatial translation. In particular, the speed $\sigma '(t)=2+\mathrm{o}(1)$ converges to the linear spreading speed as $t\to \infty$. The basic idea of the proof relies on using comparison principles, with (unstable) fronts at speeds $c\lesssim 2$ as subsolution building blocks, and was subsequently adapted to a plethora of systems that allow comparison principles, sometimes in a more hidden fashion, on the real line and also in higher space dimensions; see e.g. Reference 2Reference 7Reference 40Reference 42Reference 73Reference 87. In a celebrated series of papers, Bramson Reference 11Reference 12 showed that convergence of the speed is quite slow with a universal leading-order correction that induces a $\log$-shift in the position, independent of initial conditions,
The approach there relies on a probabilistic interpretation of Equation 1.1 as an evolution of distributions in a branched random walk. Proofs were greatly simplified later using comparison principles with refined subsolutions in Reference 41Reference 62Reference 66. The new techniques introduced also led to refined asymptotics, allowed adaptations to other systems, and analysis in higher space dimensions; see e.g. Reference 9Reference 10Reference 38Reference 67Reference 74.
Inspecting the comprehensive review of experimental observations and theoretical studies of front propagation into unstable states Reference 84, almost all experimental settings and associated models including for instance fluid instabilities, crystal growth, and phase separation, do not admit a probabilistic interpretation or comparison principles, nor do they preserve positivity of initial data. In fact key examples in Reference 84 are higher-order parabolic equations that do not admit comparison principles. A prototypical case of stationary propagation (S) is the extended Fisher-KPP equation
$$\begin{equation} \partial _t u = - \delta ^2 \partial _{xxxx} u + \partial _{xx} u + f(u), \quad x \in \mathbb{R}, \quad t > 0, \cssId{texmlid2}{\tag{1.4}} \end{equation}$$
for $\delta$ small, where $f$ is a smooth function satisfying $f(0) = f(1) = 0$,$f'(0) > 0, f'(1) < 0$, and (for instance) $f''(u) < 0$ for all $u \in (0,1)$. A basic example for time-periodic propagation, case (P), is the Swift-Hohenberg equation, a prototypical model for pattern formation in contexts such as Rayleigh-Bénard convection,
$$\begin{equation} \partial _t u = -\partial _{xxxx}u - 2\partial _{xx}u + (\mu -1) u - u^3, \quad x \in \mathbb{R}, \quad t > 0, \quad \mu > 0; \cssId{texmlid3}{\tag{1.5}} \end{equation}$$
see Reference 23 for a formulation of the marginal stability conjecture in this particular case. Fourth order (and even sixth order) scalar equations such as Equation 1.4 and Equation 1.5 arise in many physical circumstances, including in phenomenological models for convection rolls Reference 82, in crystal nucleation and growth Reference 25Reference 86, in models for spatial localization of patterns across many physical systems Reference 60, as models for phyllotaxis Reference 70, as amplitude equations derived from reaction-diffusion or fluid systems Reference 65Reference 76, and even in Turing’s late work on morphogenesis Reference 17.
Motivated by these examples, we focus on a setting of higher-order parabolic equations in which we establish selection of pulled fronts and the marginal stability conjecture in the case of stationary invasion (S). We believe that the techniques introduced here will also prove useful in understanding front propagation in the time-periodic case (P), particularly in pattern-forming systems such as Equation 1.5. We further expect the linear theory which we develop here and in Reference 4 to be useful in understanding diffusive decay near coherent structures in other contexts.
1.1. Setup and main results
We consider scalar, spatially homogeneous parabolic equations of arbitrary order, of the form
$$\begin{equation} u_t = \mathcal{P}(\partial _x) u + f(u), \quad u = u (x,t) \in \mathbb{R}, \quad x \in \mathbb{R}, \quad t > 0, \cssId{texmlid8}{\tag{1.6}} \end{equation}$$
with $f$ smooth, $f(0) = f(1) = 0, f'(0) > 0$, and $f'(1) < 0$, and polynomial differential operator
so that $\mathcal{P}(\partial _x)$ is elliptic of order $2m$, although not necessarily symmetric, that is, we allow nonzero coefficients of odd derivatives.
We pass to a co-moving frame of speed $c$ and linearize at the unstable rest state $u \equiv 0$ to find
$$\begin{equation} u_t = \mathcal{P}(\partial _x) u + c \partial _x u + f'(0) u. \cssId{texmlid6}{\tag{1.8}} \end{equation}$$
Informally, the linear spreading speed$c_*$ is a distinguished speed so that solutions to Equation 1.8 with compactly supported initial data grow exponentially pointwise for $c \lesssim c_*$ and decay pointwise for $c \gtrsim c_*$. To characterize $c_*$, we substitute $u = e^{\nu x + \lambda t}$ into Equation 1.8 and find the dispersion relation,
The dispersion relation determines the spectrum of the linearization about the unstable state $\mathcal{P}(\partial _x) + c_* \partial _x + f'(0)$ via the Fourier transform, in the sense that the spectrum of this operator (on, for instance, $L^p (\mathbb{R})$) is given by
$$\begin{equation} \Sigma ^+ = \{ \lambda \in \mathbb{C}: d^+(\lambda , ik) = 0 \text{ for some } k \in \mathbb{R}\}. \cssId{texmlid76}{\tag{1.10}} \end{equation}$$
From now on we fix $c = c_*$ and write $d^+ = d^+_{c_*}$. We also define the left dispersion relation
which determines the spectrum of the linearization $\mathcal{P}(\partial _x) + c_* \partial _x + f'(1)$ at $u=1$ through
$$\begin{equation} \Sigma ^- = \{ \lambda \in \mathbb{C}: d^-(\lambda , ik) = 0 \text{ for some } k \in \mathbb{R}\}. \cssId{texmlid37}{\tag{1.13}} \end{equation}$$
We focus on dynamics in the leading edge and therefore assume (strict) stability in the wake.
According to the marginal stability conjecture, one expects the propagation dynamics to be governed by traveling wave solutions, and we therefore assume existence of such a front.
Possibly translating in space and reflecting $q_*\mapsto -q_*$ if necessary, we assume $b=1$.
Finally, we need to exclude the possibility that the fronts are pushed in the sense that the nonlinearity accelerates the speed of propagation. Typically, in the presence of pushed fronts, the linearization about the critical front has an unstable eigenvalue. This linearization is given by
The assumption $f'(0) > 0$ implies that the essential spectrum of $\mathcal{A}$ on $L^2$ is unstable Reference 30Reference 68Reference 69, but Hypotheses 1 and 2 imply that, with a smooth positive exponential weight $\omega$ satisfying
the essential spectrum of the conjugate operator $\mathcal{L}= \omega \mathcal{A} \omega ^{-1}$ is marginally stable; see Figure 1 for a schematic, and the beginning of Section 3 for further details. To exclude pushed fronts, we assume the following.
In the Fisher-KPP setting, fronts with the linear spreading speed $c=2$ satisfy Hypotheses 1, 2, 3, and 4. Absence of a bounded solution to $\mathcal{L}u = 0$ in the Fisher-KPP equation is a consequence of the weak exponential decay Equation 1.15 of the critical front. Instabilities as excluded in Hypothesis 4 occur for instance when considering an asymmetric cubic nonlinearity and lead to the selection of pushed fronts that propagate at a speed $c_\mathrm{pushed}>c_*$; see Reference 39. Separating pulled fronts with speed $c=2$ from pushed fronts is the case of a bounded solution to $\mathcal{L}u=0$, excluded in Hypothesis 4. In the case that there is such a bounded solution, we say $\lambda = 0$ is a resonance of $\mathcal{L}$.
Before we can state our main results, we address the question: what features should a meaningful front selection result have? First, such a result should establish that, for an open class of initial data, the front interface is located approximately at $x = c_* t$ for large times, so that the asymptotic speed of propagation is the linear spreading speed. Second, the class of initial data should include some data that is compactly supported in the leading edge, commonly the most interesting case. Such a setting rules out selection in this sense of faster-traveling, supercritical frontsReference 22, $c>c_*$, which attract open sets of initial data but only with well-prepared, slowly decaying exponential tails.
Our result includes a specific choice of algebraic weight with smooth, positive weight function $\rho _r$,$r\in \mathbb{R}$, satisfying
$$\begin{equation} \rho _r (x) = \begin{cases} \langle x \rangle ^r, & x \geq 1, \\ 1, & x \leq -1, \end{cases} \cssId{texmlid11}{\tag{1.19}} \end{equation}$$
where $\langle x \rangle = (1+x^2)^{1/2}$.
We emphasize that we do not require any structure of the equation beyond Hypotheses 1 through 4 — in particular, our results apply to equations without comparison principles. The first author together with Garénaux recently proved Reference 3 that the extended Fisher-KPP equation Equation 1.4 satisfies Hypotheses 1 through 4, so that Theorem 1 applies immediately in that setting. Adapting the ideas therein, we show that the class of equations we consider here is open, thereby emphasizing the universality across different equations. We make this precise in Theorem 2.
We note in passing that the assumptions in Theorem 2 on the nonlinearity at $\delta = 0$ imply that $0$ and $1$ perturb smoothly to nearby zeros of $f$ for $\delta$ small. Shifting and rescaling $u$, we may then assume without loss of generality that $f(0; \delta ) = f(1; \delta ) = 0$.
Together, Theorems 1 and 2 establish propagation at the linear spreading speed for open classes of initial data and open classes of equations.
1.2. Overview and preliminaries
Pointwise stability and pinched double roots
We give a brief review of concepts of pointwise decay driving our understanding of invasion processes, and the role of pinched double roots, based on the presentation in Reference 49. To determine pointwise growth or decay for solutions to the linearization about the unstable state, we use the inverse Laplace transform to write the solution to Equation 1.8 as
where $u_0$ is the initial data, $\Gamma$ is a suitable contour to the right of the essential spectrum, and $G_\lambda (\xi )$ is the resolvent kernel, which solves
We restrict to strongly localized, for instance compactly supported, $u_0$. To obtain optimal decay of $u(x, t)$, one aims to shift the integration contour $\Gamma$ as far to the left as possible. Because the initial data is strongly localized and because we are interested in the pointwise growth or decay of the solution, rather than in a fixed norm, the obstruction to shifting the contour $\Gamma$ is not the essential spectrum of $P(\partial _\xi ) + c \partial _\xi + f'(0)$, which depends on the choice of function space and may be moved with exponential weights. Instead, the only obstructions to shifting the contour are singularities of $\lambda \mapsto G_\lambda (\xi )$ for fixed$\xi$.
To track singularities of $G_\lambda (\xi )$, we recast the equation as a first order system, and solve for the matrix Green’s function $T_\lambda (\xi )$, which solves
for some matrix $M(\lambda )$ which is polynomial in $\lambda$. The resolvent kernel $G_\lambda$ may be recovered from $T_\lambda$, and both have precisely the same pointwise singularities Reference 49, Lemma 2.1. The matrix Green’s function $T_\lambda (\xi )$ has the explicit expression
where $P^{\mathrm{s}/\mathrm{u}} (\lambda )$ are, for $\lambda$ sufficiently large, the projections onto the stable and unstable eigenspaces of $M(\lambda )$, which are well-separated for $\operatorname {Re}\lambda \gg 1$ due to the fact that the underlying equation is parabolic. Since $M(\lambda )$ is polynomial in $\lambda$, singularities of $T_\lambda (\xi )$ are precisely the singularities of $P^{\mathrm{s}/\mathrm{u}} (\lambda )$. Using the Dunford integral, these projections may be analytically continued from $\operatorname {Re}\lambda \gg 1$ until an eigenvalue of $M(\lambda )$ which was stable for $\operatorname {Re}\lambda \gg 1$ collides with an eigenvalue of $M(\lambda )$ which was unstable for $\operatorname {Re}\lambda \gg 1$. Eigenvalues $\nu$ of $M(\lambda )$ are precisely roots $\nu$ of the dispersion relation, and such a collision of stable and unstable eigenvalues is a pinched double root of the dispersion relation, by definition; see Reference 49, Definition 4.2. The term “pinched” refers to the fact that the colliding roots come from opposite (that is, stable and unstable) directions for $\operatorname {Re}\lambda \gg 1$.
To summarize, the contour $\Gamma$ can be shifted to the left until we reach a pointwise singularity of $G_\lambda (\xi )$, and all such singularities are pinched double roots of the dispersion relation. The pinched double root with maximal real part therefore gives an upper bound on the pointwise exponential decay rate of $u(x, t)$. It is possible to have a pinched double root that is not a singularity of $G_\lambda (\xi )$, if the eigenvalues collide but have distinct limiting eigenspaces Reference 49, Remark 4.5. We exclude this possibility in Hypothesis 1 by restricting to simple pinched double roots, which are robust (see Lemma 7.1) and always produce pointwise growth modes Reference 49, Lemma 4.4.
The linear spreading speed $c_*$ in case (S) is then characterized by a simple pinched double root at the origin, as in Hypothesis 1(i); compare Reference 49, Section 6. Assumptions (ii)-(iii) of Hypothesis 1 on minimality of critical spectrum guarantee that this is the most unstable pinched double root, so that the linearization precisely exhibits marginal pointwise stability, as required by the marginal stability conjecture. A short calculation shows that the double root $\lambda _\mathrm{dr}$ at the origin moves to the right as $c$ decreases, $\lambda _\mathrm{dr}'(c_*)<0$, so that the origin is pointwise exponentially stable for $c>c_*$ and exponentially unstable for $c<c_*$Reference 49, Remark 6.6. Heuristically, marginal pointwise stability in the leading edge allows solutions evolving from steep initial data to develop a Gaussian tail which does not decay rapidly in time, allowing for matching with the front interface on an intermediate length scale, thus explaining the selection of $c_*$; see Figure 1 and the discussion below.
Note that spreading speeds with marginally stable pinched double roots are quite generally well-defined Reference 49, Corollary 6.5, motivating the conceptual, equation-independent setup here.
Marginal stability as a selection mechanism
Hypothesis 1 guarantees that, in the frame moving with $c_*$, the state $u \equiv 0$ is marginally pointwise stable, and in particular marginally stable in a weighted space with weight $e^{\eta _* x}$. Hypothesis 3 gives us a front to perturb from, although perturbations that cut off the tail of the front, which are the most relevant for front selection, are large perturbations growing linearly in $x$, due to the front asymptotics $xe^{-\eta _*x}$. For initial data that vanish for $x$ sufficiently large, the dynamics for large $x$ are governed by the linearization about the unstable state, which, in the co-moving frame with speed $c_*$ and in a weighted space with weight $e^{\eta _* x}$, is given by
See Section 2 for further details on the precise form of this Gaussian tail. These diffusive dynamics, with no temporal decay, allow for matching with the front $q_*(x) \sim x e^{-\eta _* x}$ on the intermediate length scale $x +\frac{3}{2 \eta _*} \log t \sim t^\mu , 0 < \mu \ll 1$. This is the intuition for the speed selection here: for steep initial data, the dynamics are initially driven by the diffusive repair at $+\infty$, which then pulls the front forward at the natural speed $c_*$ associated to this diffusive repair.
For speeds $c \neq c_*$, the discussion on pointwise growth above implies that the linearization in the leading edge will either grow or decay exponentially in time, precluding matching with the front, which is constant in time in the co-moving frame. Also, instabilities beyond the one associated with the pointwise growth in the leading edge would induce temporal growth in the frame with speed $c_*$, again preventing matching with the front on the intermediate length scale. In this sense, the marginal stability of the front, that is, choosing $c=c_*$ and excluding other instabilities as made precise in Hypotheses 1, 2, 3, and 4, is necessary for matching and selection of the pulled front in the sense that failure of marginal stability would select a different profile. An important boundary case, excluded here by Hypothesis 2, is diffusively stable essential spectrum in the wake, touching the imaginary axis in a parabolic fashion, rather than exponentially stable as assumed in Hypothesis 2, a scenario typical for pattern-forming fronts such as those in Equation 1.5. While this scenario is excluded here, preliminary sharp stability results providing a basis for selection were recently obtained in Reference 5.
Sketch of the main proof
Absent a comparison principle but equipped with assumptions on the linearization at a given front profile, one would like to phrase the selection problem as a stability problem. Initial conditions with vanishing support for large $x>0$ can be thought of as perturbations of size $xe^{-\eta _* x}$, which however are not small perturbations in a suitable function space. Indeed, the weighted front satisfies $\omega (x) q_*(x) \sim x$ as $x \to \infty$ by Hypothesis 3, so that a perturbation which cuts off the front tail is only small in a function space such as $L^\infty _{-r} (\mathbb{R})$ for $r > 1$ (after already including the exponential weight; see below for definitions of weighted spaces). However, by the argument of Reference 4, Proposition 7.6, one can show that the linear evolution to such a perturbation will typically grow like $t^\beta$ for some $\beta > 0$, precluding a nonlinear perturbative argument.
As suggested in the above discussion, we overcome this difficulty by perturbing instead from a refined profile, informed by the formal asymptotics in Reference 22, which resembles the critical front for $x + c_* t - \frac{3}{2\eta _*} \log t \ll \sqrt {t}$ with a Gaussian tail for $x + c_* t - \frac{3}{2 \eta _*} \log t \gg \sqrt {t}$. Such a construction was carried out for the Fisher-KPP equation in Reference 66Reference 67 and used together with the comparison principle to establish a refined description of the asymptotics of the front position. The key insight in this construction is to match on an intermediate length scale $x \sim t^\mu$ for some $\mu > 0$ small.
As a first main ingredient to our result, we construct such an approximate solution in our conceptual setup in a way that guarantees small residuals. Based on this first step, most of our work is concerned with establishing stability in time of such an approximate solution. In order to guarantee small residuals, we let the approximate solution evolve for some large time $T$ to an initial profile, such that small perturbations to the approximate solution include initial conditions which vanish for $x$ sufficiently large; see Figure 1.
In the second step, we establish stability by closing a perturbative argument. The main difficulty here stems from the fact that the logarithmic shift introduces critical terms into the linear dynamics. We therefore need sharp estimates on the linearized evolution which we obtain by refining resolvent estimates originally derived in order to conclude stability of the critical front in Reference 4. In order to close the nonlinear argument, we rely on sharp characterizations of decay and nonlinear contributions in terms of $T$, the characteristic scale of the initial Gaussian tail.
To illustrate the still substantial difficulties in closing this perturbative argument, consider the heavily simplified model problem
$$\begin{equation} \begin{cases} w_t = w_{xx} - \frac{3}{2(t+T)} (w_x - w), & x > 0, t > 0, \\ w = 0, & x = 0, t>0, \end{cases} \tag{1.25} \end{equation}$$
where the diffusive term captures spectral properties in the leading edge and the non-autonomous terms are induced by the logarithmic shift. The autonomous linear evolution, that is, ignoring the $-\frac{3}{2}(t+T)^{-1} (w_x - w)$ terms, allows $t^{-3/2}$ algebraic decay in suitable norms provided the initial data is sufficiently localized. However, it turns out that the term $\frac{3}{2} (t+T)^{-1} w$ is critical, so that in fact $w$ does not decay but instead remains $\mathrm{O}(1)$. This feature is explicit after the simple but insightful change of variables $z = (t+T)^{-3/2} w$, which eliminates the critical term, giving an equation
$$\begin{equation*} \begin{cases} z_t = z_{xx} - \frac{3}{2(t+T)} z_x, & x > 0, t > 0, \\ \ z = 0, & x = 0, t > 0. \end{cases} \end{equation*}$$
The term $-\frac{3}{2} (t+T)^{-1} z_x$ has improved decay properties compared to $\frac{3}{2} (t+T)^{-1} w$ due to the presence of an extra spatial derivative, as spatial derivatives of the heat kernel exhibit faster decay. Using sharp estimates on decay of derivatives and several bootstrap steps, refining in particular the estimates in Reference 4, we find $T$-uniform decay estimates $(t+T)^{-3/2}$ for small initial data. In $z$-variables, the nonlinearity causes additional complications due to the factor $(t+T)^{3/2}$, which we account for using sharp $T$-dependent characterizations of decay. A significant part of our efforts is then concerned with establishing robust decay estimates, equivalent to those in the model problem but based only on our conceptual assumptions, for the linearized evolution near our approximate solution, which in turn we base on estimates on the linearization at the critical front $\mathcal{L}$. Throughout, we cannot and do not rely on comparison principles which are not available for the full problem.
From this perspective, our results can be seen as an extension of stability results for critical fronts to actual selection mechanisms for fronts and invasion speeds, by placing the selection problem in a sufficiently broad perturbative framework. Indeed, our previous work Reference 4 was motivated by stability results for critical Fisher-KPP fronts by Gallay Reference 33 and the more recent approach by Faye and Holzer Reference 26 using more direct pointwise semigroup methods. Our analysis in Reference 4 establishes stability of pulled fronts against localized perturbations that do not alter the exponential tail of the front, thus not sufficiently large in the tail to establish front selection, but also develops the fundamentals of the linear theory we rely on and adapt to our needs here. This linear theory can be viewed as a robust functional analytic alternative to stability problems that have been successfully analyzed using pointwise resolvents, Evans functions, and pointwise semigroup methods Reference 36Reference 51Reference 58. On the level of the resolvent, we indeed replace the pointwise Evans function techniques with an equivalent functional analytic approach to tracking eigenvalues and resonances based on farfield-core decompositions, initially developed in Reference 71.
Outline of the paper
In Section 2, we use a matching procedure to construct a good approximate solution of Equation 1.6 which moves with the expected speed. In Section 3, we use a far-field/core decomposition to prove sharp estimates on the resolvent $(\mathcal{L}- \gamma ^2)^{-1}$. In Section 4, we use carefully chosen Laplace inversion contours as in Reference 4 to translate these resolvent estimates into sharp linear decay estimates. We then carry out a nonlinear stability analysis in Section 5 to prove that certain classes of solutions to Equation 1.6 resemble our approximate solution. We rephrase these results as statements on front propagation in Section 6, thereby proving Theorem 1. In Section 7, we use ideas from Reference 3 to show that our assumptions hold for open classes of equations. We conclude in Section 8 with a discussion of extensions of our results (including to systems of parabolic equations) and some of the challenges therein.
Function spaces
We will need algebraic and exponential weights generalizing those defined in Equation 1.19 and Equation 1.17. For $r_-, r_+ \in \mathbb{R}$, we define a smooth positive algebraic weight
For a non-negative integer $k$ and a real number $1 \leq p \leq \infty$, we define the corresponding algebraically weighted Sobolev space $W^{k, p}_{r_-, r_+} (\mathbb{R})$ through
$$\begin{equation} \| g \|_{W^{k,p}_{r_-, r_+}} = \| \rho _{r_-, r_+} g \|_{W^{k,p}}, \tag{1.27} \end{equation}$$
where $W^{k,p} (\mathbb{R})$ is the standard Sobolev space with differentiability index $k$ and integrability $p$. If $r_- = 0$ and $r_+ = r$, we write $\rho _{0, r_+} = \rho _r$ and $W^{k,p}_{0, r_+}(\mathbb{R})=W^{k, p}_r(\mathbb{R})$ with norm $\| \cdot \|_{W^{k, p}_r}$. For $k = 0$, we write $W^{0, p}_{r_-, r_+} (\mathbb{R}) = L^p_{r_-, r_+} (\mathbb{R})$, and denote the norm by $\| \cdot \|_{L^p_{r_-, r_+}}$, or $\| \cdot \|_{L^p_{r}}$ in the case where $r_- = 0, r_+ = r$. Similarly, for $\eta _-, \eta _+ \in \mathbb{R}$ we let $\omega _{\eta _-, \eta _+}$ be a smooth positive exponential weight satisfying
and define corresponding exponentially weighted Sobolev spaces $W^{k,p}_{\mathrm{exp}, \eta _-, \eta _+} (\mathbb{R})$ through the norms
$$\begin{equation} \| g \|_{W^{k,p}_{\mathrm{exp}, \eta _-, \eta _+}} = \| \omega _{\eta _-, \eta _+} g \|_{W^{k,p}}. \tag{1.29} \end{equation}$$
Again, we write $\omega _{0, \eta _+} = \omega _{\eta }$ when $\eta _- = 0$ and $\eta _+ = \eta$, and $W^{k,p}_{\mathrm{exp}, 0, \eta _+}(\mathbb{R})=W^{k,p}_{\mathrm{exp}, \eta } (\mathbb{R})$, and, for $k = 0$, we write $W^{0,p}_{\mathrm{exp},\eta _-, \eta _+} (\mathbb{R}) = L^p_{\mathrm{exp},\eta _-, \eta _+} (\mathbb{R})$, with corresponding notation for the norms.
Additional notation
For two Banach spaces $X$ and $Y$, we let $\mathcal{B}(X,Y)$ denote the space of bounded linear operators from $X$ to $Y$ equipped with the operator norm topology. For $\delta > 0$, we let $B(0, \delta )$ denote the open ball in the complex plane with radius $\delta$. When the intention is clear, we may abuse notation slightly by writing a function $u(x,t)$ as $u(t) = u(\cdot , t)$, viewing it as an element of some function space for each $t$.
2. Construction of the approximate solution
We cast Equation 1.6 in the co-moving frame with position
$$\begin{equation*} \xi = x - c_* t + \frac{3}{2 \eta _*} \log (t+T) - \frac{3}{2 \eta _*} \log (T), \end{equation*}$$
i.e. in a frame that moves with the linear spreading speed up to the logarithmic delay. Here we write the logarithmic shift as $\frac{3}{2\eta _*} (\log (t+T) - \log (T))$ to capture the phase shift resulting from letting the approximate solution evolve for time $T$. After relabeling $\xi$ as $x$ again, we find
We next use an exponential weight to stabilize the linear part of the equation, defining $v = \omega u$ with $\omega$ from Equation 1.17. The weighted variable $v$ solves $\mathrm{F}_{\mathrm{res}}[v] = 0$, where the nonlinear operator $\mathrm{F}_{\mathrm{res}}$ is
Our goal in this section is to construct an approximate solution $\psi$ such that $\mathrm{F}_{\mathrm{res}}[\psi ] (x,t) = R(x,t)$ with $\| R (\cdot , t) \|_{L^\infty _r}$ small in a suitable sense. We follow the construction in Reference 67, modifying it for the higher order equations considered here and only including the terms which are relevant for our analysis. The basic idea is to use an appropriate shift of the front to construct the “interior” of our approximate solution, and then glue this on the intermediate length scale $x \sim (t+T)^\mu$ to a diffusive tail which we construct in self-similar coordinates. The construction in this section relies only on the dynamics in the leading edge captured by Hypothesis 1(i) and the existence of a pulled front with generic asymptotics assumed in Hypothesis 3. Additional instabilities in the essential spectrum or point spectrum, excluded by Hypothesis 1(ii)-(iii) and Hypothesis 4 respectively, would not prohibit the construction here but instead render the approximate solution constructed in this section unstable.
2.1. Interior of the approximate solution
Fix $\mu > 0$ small. To construct the approximate solution for $x \in (-\infty , (t+T)^\mu )$, we define
where we will choose the shift $\zeta (t+T)$ in Lemma 2.6 to match with a diffusive tail on the length scale $x \sim (t+T)^\mu$. The matching conditions will imply that $\zeta$ is smooth, with $\zeta (t +T) = \mathrm{O}((t+T)^{\mu -1/2})$ and $\dot{\zeta }(t+T) = \mathrm{O}((t+T)^{\mu - 3/2})$, and we therefore assume for the remainder of this section that these conditions hold. Since we choose $T$ large, $\zeta$ will be small uniformly in $t$, and so we expect that $\psi ^-(x,t) \approx \omega (x) q_*(x)$, which we make precise in Lemma 2.1.
2.2. Approximate solution in the leading edge
For $x \geq 1$,$\omega (x) = e^{\eta _* x}$, and so the equation $\mathrm{F}_{\mathrm{res}}[v] = 0$ reduces to
for some constants $\alpha _k \in \mathbb{R}$, with $(-1)^m\alpha _{2m} < 0$. That is, the dynamics near $x = \infty$ are essentially diffusive: since large spatial scales are most relevant for the long time behavior here, the dynamics are governed by the lowest derivative $\alpha \partial _{xx}$. To make this precise, we introduce scaling variables
where $x_0$ is a shift to be chosen later. We introduce $\Psi (\xi , \tau ) = \psi ^+(x,t)$, so that the equation $\mathrm{F}_{\mathrm{res}}[\psi ^+] = R$ for $x \geq 1$ becomes $\mathcal{F}_{\mathrm{res}}[\Psi ] = e^{\tau } \mathcal{R}$ for $\xi \geq (1+x_0)/\sqrt {\alpha (t+T)}$ where $\mathcal{R} (\xi , \tau ) = R(x,t)$ and
provided $e^{-\eta _* x} \psi$ is bounded for $x \geq 1$. We will only use this equation on the length scale $x \geq (t+T)^{\mu }$, which corresponds to $\xi \geq \alpha ^{-1/2} \left( e^{(\mu - 1/2) \tau } + x_0 e^{-\tau /2} \right)$. On this scale, we have the estimate
so that for $\mu > 0$ fixed and for $\tau$ large, this factor dominates $e^{\tau }$, and the nonlinearity is exponentially small in $\tau$. Therefore, to leading order in $e^{\tau /2}$, the equation $\mathcal{F}_{\mathrm{res}}[\Psi ] = 0$ is
revealing in which sense the dynamics are essentially diffusive: this is precisely the heat equation in self-similar variables. The spectrum of the operator
We only need a solution defined for $\xi \geq \alpha ^{-1/2} \left( e^{(\mu - 1/2) \tau } + x_0 e^{-\tau /2} \right) \geq 0$ to match with the interior solution, so we will consider the resulting equations for $\Psi _0$ and $\Psi _1$ on the half-line $\xi \geq 0$. Inserting the ansatz Equation 2.22 into Equation 2.17 and collecting terms in powers of $e^{\tau /2}$ gives
The interior of the front provides a strong absorption effect since the spectrum of $\mathcal{L}^-$ is strictly contained in the left half plane. We reflect this fact in the choice of Dirichlet boundary conditions $\Psi _i (0) = 0, i = 0, 1$. The unique solution $\Psi _0 \in L^2 (\mathbb{R}_+)$ to Equation 2.23 then is Reference 35, Appendix A
for a constant $\beta _0 \in \mathbb{R}$. If we posed these equations on the whole real line, the next eigenvalue for $L_\Delta$ would be at $\lambda = -\frac{1}{2}$, which would present an obstacle to solving Equation 2.24. However, the restriction to the half-line with a Dirichlet boundary condition, equivalent to considering the equation on the real line with odd data, removes this eigenvalue since the corresponding eigenfunction is even; see again Reference 35, Appendix A. One can further obtain Gaussian estimates on the solution to Equation 2.24: conjugating with Gaussian weight $e^{\xi ^2/8}$ transforms $L_\Delta$ into the quantum harmonic oscillator with well-known spectral properties; see e.g. Reference 43. We collect the relevant results in Lemma 2.3.
Reverting to the original variables, the approximate solution $\Psi (\xi , \tau )$ we have constructed has the form
where $\beta _0$ and $x_0$ are parameters to be chosen, and $\alpha$ is the effective diffusivity from Equation 2.15. In particular, we have $|\psi ^+(x,t)| \leq C \langle x \rangle$ uniformly for $t > 0, x > 0$, so that $e^{-\eta _* x} \psi ^+(x,t)$ is uniformly bounded in $t$, and we can use the estimate Equation 2.12 on the nonlinearity to write
The important observation is that every term carries a factor of at least $e^{-\tau /2}$ and has Gaussian localization in $\xi$. More precisely, by Lemma 2.3, the formula Equation 2.25, and the estimate Equation 2.29, there exists a constant $C > 0$ such that
for $T$ large and for $\xi \geq \alpha ^{-1/2} (e^{(\mu - 1/2) \tau } + x_0 e^{-\tau /2})$. In the original variables, one finds $\mathrm{F}_{\mathrm{res}}[\psi ^+] = e^{-\tau } \mathcal{F}_{\mathrm{res}}[\Psi ]$, so that for $x \geq (t+T)^\mu$ and for $T$ large,
where $\mathrm{F}_{\mathrm{res}}[\psi ^+] = R^+$. This pointwise estimate on the residual implies the following estimates in norm.
2.3. Matching the interior to the leading edge
We construct the full approximate solution $\psi$ by matching $\psi ^-$ and $\psi ^+$ at the length scale $x \sim (t+T)^\mu$. Simply matching values at $x = (t+T)^{\mu }$ through choosing the shift $\zeta (t+T)$ would leave us with mismatched derivatives and distributions in the residual. To avoid this, we smoothly blend solutions over the region $x \in [(t+T)^\mu , (t+T)^\mu + 1]$. Therefore, consider the smooth cutoff $\chi _0$ with $0 \leq \chi _0 \leq 1$ and
The remainder of this section is dedicated to proving that $\psi$ satisfies $\mathrm{F}_{\mathrm{res}}[\psi ] = R$ for a small residual $R$, provided we choose the constants $\beta _0$ and $x_0$ appropriately.
where $[\mathrm{F}_{\mathrm{res}}, \chi ]$ is the commutator defined by $[\mathrm{F}_{\mathrm{res}}, \chi ] \psi = \mathrm{F}_{\mathrm{res}}[\chi \psi ] - \chi \mathrm{F}_{\mathrm{res}}[\psi ]$. The terms $\chi \mathrm{F}_{\mathrm{res}}[\psi ^-]$ and $(1-\chi ) \mathrm{F}_{\mathrm{res}}[\psi ^+]$ satisfy Equation 2.38 by Lemmas 2.2 and 2.4, so we only need to prove Equation 2.38 for the term involving the commutator. Since $f(0) = f(1) = 0$, the support of this commutator is contained in $\{ (x,t) : x \in [(t+T)^\mu , (t+T)^\mu + 1]\}$ by construction of $\chi$. We first match the values of $\psi ^-$ and $\psi ^+$ precisely at $x = (t+T)^\mu$, as in Reference 67.
Finally, we collect for later use the following result, which says that, for large times, $\psi$ is well-approximated by $\omega q_*$. The bulk of the work is in proving Lemma 2.1 — the estimates in the leading edge are simple in comparison, so we omit the proof.
2.4. Outlook: Approximating the true solution
The goal of the remainder of the paper is to prove that $\psi$ is a good approximation to the true solution $u$, in an appropriate sense. We let $v$ solve $\mathrm{F}_{\mathrm{res}}[v] = 0$, where $\mathrm{F}_{\mathrm{res}}$ is given by Equation 2.2. This is simply the original equation Equation 1.6, in the co-moving frame with the logarithmic delay, and with the exponential weight $\omega$. We then let $w = v - \psi$, i.e. we view $v$ as a perturbation of $\psi$. We must then control the perturbation $w$, which solves
where $R = \mathrm{F}_{\mathrm{res}}[\psi ]$. Note that we have introduced a term $f'(q_*) w$ so that the principal linear part of this equation has the form $\mathcal{L}w$. We then define
We would like to view $w_t = \mathcal{L}w$ as the principal part of this equation, so that we can control $w$ by estimating nonlinear terms in the variation of constants formula with sharp bounds on the linear semigroup $e^{\mathcal{L}t}$. This is not yet possible at this level, however: as suggested in the discussion of the reduced model problem in Section 1.2, the term $-\frac{3}{2 \eta _*} (t+T)^{-1} \omega (\omega ^{-1})' w$ is critical, and prevents the solution from decaying in time. To unravel this, we introduce $z = (t+T)^{-3/2} w$ and find that $z$ solves
Note that $\omega (\omega ^{-1})' (x) + \eta _* \equiv 0$ for $x > 1$, so that this term is essentially removed, which ultimately allows us to view this equation as a perturbation $z_t = \mathcal{L}z + H(t) [z]$ of the linear equation $z_t = \mathcal{L}z$, with $H(t) [z]$ encoding the nonlinear and nonautonomous terms in Equation 2.53. Our goal is then to obtain sharp temporal decay of $z$, equivalent to boundedness of $w$, by controlling all terms in the variation of constants formula
Closing this argument requires sharp estimates on the decay of the semigroup $e^{\mathcal{L}t}$, which we obtain in the following two sections by a detailed analysis of the resolvent $(\mathcal{L}- \lambda )^{-1}$.
3. Resolvent estimates
We obtain the linear estimates necessary for our analysis through sharp estimates on the resolvent of the weighted linearization $\mathcal{L}$ near its essential spectrum. We start with some preliminary spectral theory. We say the essential spectrum of an operator $A$ is the set of $\lambda \in \mathbb{C}$ such that $A - \lambda$ is not a Fredholm operator of index 0. The following discussion applies in particular on $L^p(\mathbb{R})$,$1 \leq p \leq \infty$ or on any algebraically weighted $L^p$ space. Fredholm properties of the linearization $\mathcal{A} = \mathcal{P}(\partial _x) + c_* \partial _x + f'(q_*)$ are determined by the asymptotic dispersion curves Equation 1.11 and Equation 1.13; see Reference 30Reference 57 for background. Specifically, $\mathcal{A} - \lambda$ is Fredholm if and only if $\lambda \notin \Sigma ^+ \cup \Sigma ^-$ and $\mathcal{A} - \lambda$ has index zero if $\lambda$ is to the right of these curves.
Exponential weights change the asymptotic dispersion relations and thereby move the Fredholm borders. As a result, the Fredholm borders of the weighted linearization $\mathcal{L}$ are given by
$$\begin{equation*} \Sigma ^+_{\eta _*} = \{ d^+ (\lambda , ik - \eta _*) = 0 \text{ for some } ik \in \mathbb{R}\}, \qquad \Sigma ^-_{\eta _*}=\Sigma ^-. \end{equation*}$$
Hypothesis 1 and Hypothesis 2 then imply that the essential spectrum of $\mathcal{L}$ is marginally stable, as depicted in Figure 1.
We start by proving estimates for the resolvent $(\mathcal{L}^+-\lambda )^{-1}$ of the limiting operator at $x = \infty$ near its essential spectrum $\Sigma ^+_{\eta _*}$. Hypothesis 1 implies that the dispersion relation has a branch point at $\lambda = 0$, which we unfold through $\gamma \coloneq \sqrt {\lambda }$ with branch cut along the negative real axis.
3.1. Estimates on the asymptotic resolvent
We analyze the asymptotic resolvent through its integral kernel $G^+_\gamma$, which solves
Since $\mathcal{L}^+$ is a constant coefficient differential operator (defined by Equation 1.24), the solution to $(\mathcal{L}^+ - \gamma ^2) u = g$ is given through convolution with $G^+_\gamma$. In Reference 4, Section 2, we give a detailed description of this resolvent kernel, and use this description to prove that the asymptotic resolvent is Lipschitz in $\gamma$ near the origin in an appropriate sense.
This is essentially Proposition 2.1 of Reference 4, although there we state the result for $L^2$-based spaces only. The proof carries over with minor modifications, so we do not give the full details here. We will use the same general approach to prove further estimates which translate to improved time decay for derivatives, and therefore we outline the overall strategy in the following. We note that since proposition only involves the asymptotic operator $\mathcal{L}^+$, which can be defined by Equation 1.24 independently of the existence of a critical front, Proposition 3.1 relies only on Hypothesis 1. Our approach is based on a decomposition of the resolvent kernel in which the principal piece resembles the resolvent kernel for the heat equation. To arrive at this decomposition, we write $(\mathcal{L}^+ - \gamma ^2)u = g$ as a first order system in $U = (u, \partial _x u, \dots , \partial _x^{2m-1} u)$, which has the form
$$\begin{equation} \partial _x U = M (\gamma ) U + F, \cssId{texmlid39}{\tag{3.4}} \end{equation}$$
where $F = (0, 0, \dots , g)^T$ and $M(\gamma )$ is a $2m$-by-$2m$ matrix which is analytic (in fact a polynomial) in $\gamma$. The structure of this matrix implies
so that eigenvalues of $M(\gamma )$, which we call spatial eigenvalues, correspond with roots of the dispersion relation. Using Fourier transform and asymptotics for large $\gamma >0$, we see that, for $\gamma ^2$ to the right of the essential spectrum, $M(\gamma )$ is a hyperbolic matrix with stable and unstable subspaces $E^\mathrm{s}(\gamma )$ and $E^\mathrm{u}(\gamma )$ satisfying $\dim E^\mathrm{s}(\gamma ) = \dim E^\mathrm{u}(\gamma )$. We let $P^\mathrm{s}(\gamma )$ and $P^\mathrm{u}(\gamma ) = 1 - P^\mathrm{s}(\gamma )$ denote the corresponding spectral projections, which are analytic for $\gamma ^2$ to the right of the essential spectrum. We can use these projections to write the matrix Green’s function for the system Equation 3.4 as in Reference 49
where $P_1$ is the projection onto the first component and $Q_1$ is the embedding into the last component, i.e. $P(u_1, \dots , u_{2m}) = u_1$ and $Q_1 g = (0, \dots , 0, g)^T$; see Reference 4Reference 49 for details.
Following Reference 49, we conclude that singularities of $G_\gamma ^+$ are determined precisely by singularities of the stable projection $P^\mathrm{s}(\gamma )$. Hypothesis 1 implies that the dispersion relation $d^+(\gamma ^2, \nu ^\pm - \eta _*)$ has two roots of the form
for $\gamma$ close to zero, and that all other roots are bounded away from zero for $\gamma$ small. In particular, $\nu ^\pm (\gamma )$ collide as $\gamma \to 0$ and form a Jordan block for $M(0)$ to the eigenvalue 0 Reference 49. Hence, $P^\mathrm{s}(\gamma )$ necessarily has a singularity at $\gamma = 0$Reference 59. We isolate this singularity by splitting
where $P^\mathrm{cs}(\gamma )$ is the spectral projection associated to the eigenvalue $\nu ^-(\gamma )$, and $P^\mathrm{ss}(\gamma )$ is the strong stable projection associated to the rest of the stable eigenvalues. Since the other stable eigenvalues are bounded away from the imaginary axis for $\gamma$ small, $P^\mathrm{ss}(\gamma )$ is analytic in $\gamma$ near the origin, and only $P^\mathrm{cs}(\gamma )$ is singular. Similarly, $P^\mathrm{cu} (\gamma )$ is the spectral projection associated to $\nu ^+(\gamma )$, and the strong unstable projection $P^\mathrm{uu} (\gamma )$ is analytic near $\gamma = 0$.
In Reference 4, Lemma 2.2, we characterize the singularity of $P^\mathrm{cs}(\gamma )$ at $\gamma = 0$ using Lagrange interpolation to write $P^\mathrm{cs}(\gamma )$ as a polynomial in $M(\gamma )$, and we find that the singularity is a simple pole such that $\gamma P^\mathrm{cs} (\gamma ) \big |_{\gamma = 0} = - \gamma P^\mathrm{cu} (\gamma ) \big |_{\gamma =0}$. We let $\tilde{P}^{\mathrm{cs/cu}} (\gamma )$ denote the analytic part of $P^\mathrm{cs/cu}(\gamma )$ which remains after subtracting this pole. We arrive at the following decomposition of the resolvent kernel:
Our goal in the remainder of this section is to use this decomposition of the resolvent kernel to prove the following estimates on derivatives of solutions to $(\mathcal{L}^+ - \gamma ^2) u = g$.
We note again that Proposition 3.2 relies only on Hypothesis 1, since it only involves the linearization about $u \equiv 0$. The assumption that $g$ is odd in Propositions 3.1 and 3.2 models the absorption effect in the wake of the front due to the fact that the spectrum of the linearization at $u \equiv 1$ is strictly contained in the left half plane. Indeed, for a second order equation, $\mathcal{L}^+ = \partial _{xx}$, and so oddness is equivalent to imposing a Dirichlet boundary condition. Although the full operator does not necessarily commute with reflections, we can exploit oddness in Section 3.2 when extending estimates to the full resolvent $(\mathcal{L}- \gamma ^2)^{-1}$.
The main work in proving Proposition 3.2 is to prove the estimate for the piece of the resolvent given by convolution with $G^\mathrm{c}_\gamma$, since this is the worst behaved term from the perspective of $\gamma$ dependence. Indeed, it is clear that $|\tilde{G}^\mathrm{c}_\gamma | \leq C |\gamma | G^\mathrm{c}_\gamma$ for $\gamma$ small, and $G^\mathrm{h}$ is even better behaved, since it is uniformly exponentially localized in space for $\gamma$ small. Hence we only give the proof for the term involving $G^\mathrm{c}_\gamma$. We first need a basic preliminary result on localization of antiderivatives of odd functions, a proof of which can be found for instance in Reference 53, Appendix A.
3.2. Estimates on the full resolvent
As in Reference 4, we use a far-field/core decomposition to transfer estimates from the asymptotic resolvent to the full resolvent $(\mathcal{L}- \gamma ^2)^{-1}$. The main difference from Reference 4 is that here we need to work in both $L^1$- and $L^\infty$-based spaces rather than simply in $L^2$-based spaces, as an interplay of these spaces is needed in order to handle some borderline cases in our estimates. Such a borderline case can be seen already in Proposition 3.2, where only in the space $L^1_{1,1} (\mathbb{R})$ do we get the optimal $|\gamma |^{-1}$ estimate — we cannot close the argument in $L^\infty _{2,2} (\mathbb{R})$ due to the fact that $\langle x \rangle ^{-1}$ is not integrable on $\mathbb{R}$, and for $r > 2$ we find a slightly worse estimate Equation 3.18 in $L^\infty _{r,r} (\mathbb{R})$.
In this section we turn to obtaining estimates on the resolvent $(\mathcal{L}- \gamma ^2)^{-1}$ that translate into optimal decay estimates for the semigroup $e^{\mathcal{L}t}$. We note that since these estimates concern the full linearization about the front $q_*$, the estimates in this section rely on all of Hypotheses 1 through 4. The first estimate corresponds to the $t^{-3/2}$ decay of the semigroup when one gives up sufficient algebraic localization.
The next result contains estimates on derivatives which imply sharp decay estimates on $\partial _x e^{\mathcal{L}t}$ when giving up (essentially) no spatial localization.
In the remainder of this section, we prove Proposition 3.6. The essential ingredients of this proof are used to prove the $L^2$ analogue of Proposition 3.5 in Reference 4, so we do not give the full details of the proof of Proposition 3.5 here. As in Reference 4, we decompose our data $g$ into a left piece, a center piece, and a right piece. We use the left and right pieces as data for the asymptotic operators $\mathcal{L}_\pm$, and then solve the resulting equation on the center piece in exponentially weighted function spaces in which we recover Fredholm properties of $\mathcal{L}$.
To this end, we let $(\chi _-, \chi _c, \chi _+)$ be a partition of unity on $\mathbb{R}$ such that
$$\begin{equation} \chi _+ (x) = \begin{cases} 0, & x \leq 2, \\ 1, & x \geq 3, \end{cases} \tag{3.28} \end{equation}$$
$\chi _- (x) = \chi _+(-x)$, and $\chi _c=1-\chi _+-\chi _-$ is compactly supported. Given $g \in L^1_1 (\mathbb{R})$, we then write
$$\begin{equation} g = \chi _- g + \chi _c g + \chi _+g \eqcolon g_- + g_c + g_+. \tag{3.29} \end{equation}$$
We would like to decompose the solution $u$ to $(\mathcal{L}- \gamma ^2) u = g$ in a similar way, but we need to refine this approach to take advantage of the fact that the spectrum of $\mathcal{L}_-$ is strictly in the left half plane creating a strong absorption effect on the left. For this, let $g^\mathrm{odd}_+ (x) = g_+ (x) - g_+ (-x)$ be the odd extension of $g_+$ and let $u^+$ solve
where $[\mathcal{L}_+, \chi _+] = \mathcal{L}_+ (\chi _+ \cdot ) - \chi _+ \mathcal{L}\cdot$ is the commutator. Note that $\mathcal{L}$ attains its limits exponentially quickly as $x \to \infty$, and the commutator $[\mathcal{L}_+, \chi _+]$ is a differential operator of order $2m-1$ with compactly supported coefficients since $\chi _+$ is constant outside the interval $[2, 3]$, and so $\tilde{g}(\gamma )$ is exponentially localized with a rate that is uniform in $\gamma$ for $\gamma$ small. In fact, the exponential localization of the coefficients of $u^+$ and $u^-$ in Equation 3.33 together with Proposition 3.1 and Hypothesis 2 (which implies that $0$ is not in the spectrum of $\mathcal{L}_-$) allows us to conclude that $\tilde{g}(\gamma )$ is Lipschitz in $\gamma$ in exponentially localized spaces with small exponents. Recall the notation for exponentially weighted spaces $L^p_{\mathrm{exp}, \eta _-, \eta _+} (\mathbb{R})$ introduced in Section 1.2.
With exponential localization of $\tilde{f}(\gamma )$ in hand, we solve Equation 3.32 by making the far-field/core ansatz
where $a \in \mathbb{C}$ and we will require $w$ (the core piece of the solution) to be exponentially localized. Inserting this ansatz into Equation 3.32 results in an equation
We will solve Equation 3.36 by taking advantage of Fredholm properties of $F$ on exponentially weighted function spaces. First we state relevant Fredholm properties of $\mathcal{L}$ which follow readily from Morse index calculations Reference 30Reference 57. Throughout, for $\eta > 0$ we let $(X_\eta , Y_\eta )$ denote either pair of spaces
The proof of Proposition 3.5 is similar to the proof of Reference 4, Proposition 3.1 — the only additional ingredient is the use of the Sobolev embedding $\| g \|_{L^\infty } \leq C \| g \|_{W^{1, 1}}$ to account for the interplay of $L^1$- and $L^\infty$-based spaces, so we only give the details of the proof of Proposition 3.6.
4. Linear decay estimates
We translate the resolvent estimates of Section 3 into decay estimates on the semigroup $e^{\mathcal{L}t}$. We only sketch proofs, which follow very closely the analogous results in Reference 4, and point out modifications. We note that the estimates in this section rely on all of Hypotheses 1 through 4, as they depend on the full resolvent estimates of Section 3.2.
Therefore, $\mathcal{L}$ generates an analytic semigroup on $L^p (\mathbb{R})$, given by the inverse Laplace transform
for a suitably chosen contour $\Gamma$. Note that $\mathcal{L}$ is not densely defined on $L^\infty (\mathbb{R})$ and we rely on the construction of analytic semigroups in Reference 64 for not necessarily densely defined sectorial operators; in particular, strong continuity at time $t = 0$ holds only after regularizing with $(\mathcal{L}-\lambda )^{-1}$. We now begin stating decay estimates on this semigroup.
The previous results trade spatial localization for temporal decay. The next result obtains decay of derivatives without loss of localization.
Finally, we record useful small time regularity estimates for $e^{\mathcal{L}t}$.
Similarly, we obtain the following small time bounds on derivatives of solutions.
5. Stability argument
Recall from Section 2.4 that our goal is to use the sharp linear estimates obtained in Section 4 to control perturbations to the approximate solution. We start by letting $v$ solve $\mathrm{F}_{\mathrm{res}}[v] = 0$, where $\mathrm{F}_{\mathrm{res}}$ is given by Equation 2.2, which is the original equation Equation 1.6 in the co-moving frame with the logarithmic delay, and with the exponential weight $\omega$. We then define the perturbation $w = v -\psi$, which solves
where $R = F_{\mathrm{ref}}[\psi ]$ and $N(\omega ^{-1}w)$ is defined in Equation 2.51. Note that $\omega ^{-1} \psi$ is uniformly bounded, so by Taylor’s theorem,
$$\begin{equation} \omega | N (\omega ^{-1} w) | \leq C \omega ^{-1} w^2, \cssId{texmlid86}{\tag{5.2}} \end{equation}$$
Note that $\omega (\omega ^{-1})' (x) + \eta _* \equiv 0$ for $x > 1$, so that this term is essentially removed, which ultimately allows us to regain the $t^{-3/2}$ decay for $z$, equivalent to boundedness for $w$. This argument requires some care, however: we must track dependence on $T$, and compensate for the extra factor of $(t+T)^{3/2}$ now appearing with the nonlinearity. Tracking the $T$ dependence is necessary since we will treat linear terms such as $-\frac{3}{\eta _*} (t+T)^{-1} z_x$ perturbatively, and so we use largeness of $T$ to guarantee that these terms remain small, and because we need to track the $T$ dependence of $z$ in order to conclude that $w = (t+T)^{3/2} z$ is bounded. To handle the nonlinearity, note that, by Taylor’s theorem and exponential decay of $\omega ^{-1}$, there exists a nondecreasing function $K : \mathbb{R}_+ \to \mathbb{R}_+$ such that
$$\begin{equation} \| (t+T)^{-3/2} \omega N (\omega ^{-1} (t+T)^{3/2} z) \|_{L^\infty _r} \leq K(B) (t+T)^{3/2} \| z \|_{L^\infty _{-1}}^2, \cssId{texmlid58}{\tag{5.5}} \end{equation}$$
provided $(t+T)^{3/2} \| \omega ^{-1} z \|_{L^\infty } \leq B$. In summary, the nonlinearity gains spatial localization but carries a factor of $(t+T)^{3/2}$.
We rewrite Equation 5.4 in mild form via the variation of constants formula
By standard parabolic regularity Reference 44Reference 64, the equation for $z$ is locally well-posed in $L^\infty _r (\mathbb{R})$ for any $r \in \mathbb{R}$, in the sense that given any small initial data $z_0 \in L^\infty _r (\mathbb{R})$, the variation of constants formula Equation 5.6 defines a unique solution $z(t)$ for $t \in (0, t_*)$ to Equation 5.4 with
for any $\lambda$ in the resolvent set of $\mathcal{L}$. Furthermore, the maximal existence time $t_*$ depends only on $\| z_0 \|_{L^\infty _r}$, and there is a constant $C > 0$ such that for $t$ sufficiently small,
Note that $z_0 = T^{-3/2} w_0$, where $w_0$ is the initial data for Equation 2.52, such that Equation 5.15 only enforces smallness of $\| w_0 \|_{L^\infty _r}$, independent of $T$.
The remainder of this section is dedicated to proving Theorem 5.1 by estimating terms in the variation of constants formula. A first attempt would aim to control $(t+T)^{3/2} \| z (t) \|_{L^\infty _{-1}}$, but $z_x$ also enters via the term $\mathcal{I}_2 (t)$. Handling this term requires the most care: in order to obtain $t^{-3/2}$ decay there, one needs $z_x (s)$ to decay in $L^1_1(\mathbb{R})$, a norm that enforces localization. To close the argument, we then use the weaker decay of $z_x (s)$ in $L^\infty _r (\mathbb{R})$ according to the linear estimates in Proposition 4.4. We capture this bootstrapping procedure, as well as the small-time regularity of the solution in the norm template
where $\beta = \frac{3}{2} - \frac{r}{2} = \frac{1}{2} - \frac{\mu }{2}$. With the spatio-temporal decay, uniformly in $T$, encoded in $\Theta (t)$, we eventually obtain global-in-time control of $\Theta (t)$ from the following local-in-time estimates.
We break the proof of this proposition into several parts according to the estimates on $z(t)$ versus $z_x(t)$, and to the different terms in the variation of constants formula Equation 5.6. For the remainder of this section, we fix $0 < \mu < \frac{1}{8}$,$z_0 \in L^\infty _r (\mathbb{R})$, and $T$ large, and let $t_*$ be the maximal existence time of $z(t)$ in $L^\infty _{-1} (\mathbb{R})$. Unless otherwise noted, constants in this section are independent of $z_0 \in L^\infty _r(\mathbb{R})$ and $T \geq T_*$.
5.1. Heuristics for $\Theta (t)$
For $x$ large, the equation for $w$ at least formally resembles the equation $\mathrm{F}_{\mathrm{res}}[v] = 0$ considered in Section 2. Revisiting the scaling variables analysis therein (compare Equation 2.27), we expect $w$ to develop a diffusive tail as well, so that to leading order, at large $x$,
for $x$ large. We expect dynamics to be driven by this diffusive tail and infer the decay rates of $z(x,t)$ from this approximation. Indeed, we see decay in $L^\infty _{-1} (\mathbb{R}_+)$ with rate $(t+T)^{-3/2}$ on the right hand side of Equation 5.20, and decay of the derivative in $L^1_1(\mathbb{R})$ with rate $(t+T)^{-1/2}$, as captured in Equation 5.17.
However, we have $\| z_0 \|_{L^\infty _r} \lesssim T^{-3/2}$, for the initial data, which we wish to encode in the constant $c_0$. For small times, we expect to retain this $T^{-3/2}$ estimate at the price of some blowup in $t$ according to small-time regularity estimates in Section 4, which we capture in the terms in Equation 5.17 involving $0 < s <1$. We then track how this smallness of the initial data propagates to large times by incorporating $1_{s \geq 1} (s+T)^{1/2} T^{\tilde{\beta }} \| z_x (s)\|_{L^1_1}$ into the definition of $\Theta (t)$, leaving $\tilde{\beta }$ free at first. Throughout the course of the proof (see in particular Proposition 5.10), we find that $\tilde{\beta } = \frac{1}{2}$ is the optimal choice, and this control just suffices to close our argument.
At this point, we can also identify why we needed sharp $L^1$-$L^\infty$ estimates to replace the $L^2$-based linear estimates of Reference 4. In order to control $\mathcal{I}_2(t)$ and obtain $t^{-3/2}$ decay of $z(t)$, we need the integrand in Equation 5.8 to be bounded by $(t-s)^{-3/2} (s+T)^{-3/2}$. In order to extract $(t-s)^{-3/2}$ decay from $e^{\mathcal{L}(t-s)}$, we have to control $z_x (s)$ in a space $X$ for which an estimate $\| e^{\mathcal{L}t} \|_{X \to L^\infty _{-1}} \leq C t^{-3/2}$ holds. From Proposition 3.1, we see that the weakest such norm on the scale of algebraically weighted $L^p$ spaces is $L^1_1(\mathbb{R})$. We need this weakest norm, since we also have to extract decay from $z_x (s)$ to obtain the $(s+T)^{-3/2}$ estimate. Indeed, for $r > 2$ we have by Proposition 4.4
a decay is strictly slower than $t^{-1/2}$, which would not enable us to obtain the necessary $(s+T)^{-3/2}$ decay in the integrand and close the argument. This obstruction remains if we replace $L^\infty _r (\mathbb{R}), r > 2$, with the corresponding $L^2$-based localization, $L^2_{\tilde{r}} (\mathbb{R}), \tilde{r} > \frac{3}{2}$. Measuring the derivative instead in $L^1_1(\mathbb{R})$ gives the sharp $t^{-1/2}$ estimate which suffices to close the estimates on $z(t)$.
The proof now proceeds by estimating norms of $z$ and $z_x$ through the variation-of-constant formula invoking $\Theta$ to control the right-hand side.
5.2. Control of $z(t)$
We start with estimates on $z(t)$.
In the following, we estimate each term in the variation of constants formula. We prepare the proof with several lemmas. Throughout, we repeatedly use the following elementary inequality.
We next estimate the nonlinearity.
We finally estimate $\mathcal{I}_2(t)$, using the decay of $z_x(t)$ in $L^1_1(\mathbb{R})$ which we have encoded into the definition of $\Theta (t)$.
We emphasize that the proof of Lemma 5.8 shows that control of $z(t)$ necessitates good estimates on $\| z_x (t) \|_{L^1_1}$. We now gather the individual estimates on terms in $z(t)$ for the proof of Proposition 5.3, and thereby establish control of $\| z(t) \|_{L^\infty _{-1}}$ provided appropriate control of $z_x(t)$.
5.3. Control of derivatives
To complete the proof of Proposition 5.2, we now estimate $z_x (t)$. We therefore differentiate the variation of constants formula, obtaining
We need estimates on $\| z_x (t) \|_{L^1_1}$ to close the argument for $\| z(t) \|_{L^\infty _{-1}}$, and in turn we need estimates on $\| z_x (t)\|_{L^\infty _r}$ to close the argument for $\| z_x (t) \|_{L^1_1}$. We thereby rely on the sharp linear estimates on derivatives from Section 4.
The full control of $\Theta (t)$, Proposition 5.2, follows directly from the control of $z(t)$ and $z_x (t)$ in Propositions 5.3, 5.9, 5.10, and 5.11.
6. Consequences for front propagation — proof of Theorem 1
Let $u$ solve the original equation
$$\begin{equation} u_t = \mathcal{P}(\partial _y) u + f(u) \tag{6.1} \end{equation}$$
with initial data $u_0$. Here we use $y$ for the original stationary variable, since we will consider both the stationary and moving frames in this section. Indeed, we let
$$\begin{equation} x = y - c_* t + \frac{3}{2\eta _*} \log (t+T) - \frac{3}{2 \eta _*} \log (T), \tag{6.2} \end{equation}$$
and define $U(x,t) = u(y,t)$, so that $U$ solves the equation in the moving frame with initial data $U_0 (x) \coloneq U(x,0) = u_0 (x)$. We then let $v = \omega U$, so that $v$ solves $\mathrm{F}_{\mathrm{res}}[v] = 0$, and let $w(x,t) = v(x,t) - \psi (x, t; T)$ be a perturbation to $\psi$, so that $w$ solves Equation 2.52. The decay of $z(x,t)$ in Theorem 5.1 translates into the following stability result for $w$.
In other words, provided the initial data is close to $\psi (\cdot , 0; T)$, the solution $U$ in the co-moving frame with the logarithmic shift is well-approximated by $\psi (\cdot , t; T)$ for all times.
where $\mathcal{P} (\partial _x; \delta )$ is an elliptic operator of order $2m$ whose coefficients are smooth in $\delta$, and where $f$ is smooth in both its arguments, with $f(0; \delta ) = f(1; \delta ) = 0$ for $\delta$ small and $f'(0; \delta ) > 0, f'(1; \delta ) < 0$ for $\delta$ small. We assume that at $\delta = 0$, Hypotheses 1 through 4 are satisfied, and we conclude here that these assumptions hold for $\delta$ small. Our argument is essentially that of Reference 3 adapted to a general setting, without the additional technical difficulty of regularizing the singular perturbation.
We write the asymptotic dispersion relation on the right as
We note that Lemma 7.1 together with standard spectral perturbation theory implies that the essential spectrum of $\mathcal{L}(\delta )$ is marginally stable, touching the imaginary axis only at the origin, i.e. Hypothesis 1 is satisfied.
We now turn to the existence of the critical front. Let $q_0$ denote the critical front at $\delta = 0$ and $\mathcal{L}(0)$ the linearization about that front, in the weighted space with weight $\omega _0$, where
Recall from Section 3 that our assumptions imply that $\mathcal{L}(0)$ is Fredholm with index -1 when considered as an operator from $H^{2m}_{\mathrm{exp}, \eta } (\mathbb{R}) \to L^2_{\mathrm{exp}, \eta }( \mathbb{R})$ for $\eta > 0$ small. By Hypothesis 4, the kernel of $\mathcal{L}(0)$ is trivial on this space, so it has a one-dimensional cokernel.
We now state and prove the existence of the critical front, which solves
With the existence of the critical front in hand, we let $\mathcal{L}(\delta )$ denote the linearization about the critical front in the weighted space with weight $\omega _\delta$. We now show that there is no resonance for $\mathcal{L}(\delta )$ at $\lambda = 0$ for $\delta$ small. As in Section 3, we substitute the ansatz
to the equation $(\mathcal{L}(\delta ) - \gamma ^2) u = g$ for $g \in L^2_{\mathrm{exp}, \eta } (\mathbb{R})$. Decomposing the resulting equation as in Lemma 7.2, we obtain the system
so that the first equation in Equation 7.19 reads $\mathcal{F} (w, \beta ; \gamma , \delta ) = P g$. For $g = 0$, we have a trivial solution $F(0, 0; 0, 0) = 0$. Since $D_w F (0, 0; 0, 0) = P\mathcal{L}(0)$ is invertible, the implicit function theorem gives a solution $w(\beta ; \gamma , \delta ) \in H^{2m}_{\mathrm{exp}, \eta } (\mathbb{R})$ for $\beta , \gamma$, and $\delta$ small which is unique in a neighborhood of the origin. Using this uniqueness and the fact that $\mathcal{F}$ is linear in $w$ and $\beta$, one sees that $w (\beta ; \gamma , \delta ) = \beta \tilde{w}(\gamma , \delta )$ for some $\tilde{w}(\gamma , \nu ) \in H^{2m}_{\mathrm{exp}, \eta } (\mathbb{R})$. Inserting this solution into the second equation in Equation 7.19 for $g = 0$, we find an equation
The function $E$ is analytic in $\gamma$ and smooth in $\delta$, and by Lemma 7.2, we have $E(0, 0) \neq 0$, so that $E(\gamma , \delta )$ is nonzero for $\gamma , \delta$ in a neighborhood of the origin as well. From the construction of $E$, we see that we have a solution $(w, \beta )$ to Equation 7.19 if and only if $E(\gamma , \delta ) = 0$. One can further show that $(\mathcal{L}(\delta ) - \gamma ^2)$ is invertible if $\gamma$ is to the right of the essential spectrum of $\mathcal{L}(\delta )$ and $E(0, 0) \neq 0$, so that the zeros of $E$ precisely detect eigenvalues (and more generally resonances) of $\mathcal{L}(\delta )$. This results in Proposition 7.4 — again, see Reference 3 for more details in the case of the extended Fisher-KPP equation.
8. Examples and discussion
We discuss examples and limitations of our results.
Second-order equations
The simplest application of our results is to the Fisher-KPP equation
Spectral stability in the sense of Hypothesis 4 holds when, for instance, $0 < f(u) \leq f'(0) u$ for $u \in (0,1)$Reference 2Reference 78. As mentioned in Section 1, front selection results for this equation allow for large classes of initial data such as compactly supported perturbations of the step function — the “diffusive tail” does not have to be baked into the initial data. However, these results mostly restrict to positive solutions. In this regard, we believe that restricting to the class of initial data with well-formed “diffusive tail” is not merely a technical limitation. Considering for example Equation 8.1 with the bistable nonlinearity $f(u) = u - u^3$ with an additional negative invasion front connecting $-1$ to $0$, one can ask for a description of selection basins for both positive and negative fronts. Considering “step-like” initial data $u_0$ such that $u_0 (x) \equiv 1$ for $x \leq 0$ and $u_0$ is strongly localized on the right, we expect that the long time behavior is determined by the form of $u_0$ as $x \to \infty$. If the initial data has a negative diffusive tail, the solution will develop a roughly stationary kink between the two stable states $\pm 1$, while $u = -1$ is spreading into the unstable state $u = 0$. The tail dependence is more dramatic if one considers strongly asymmetric cubics $f(u)=(u-a)(1-u^2)$, with $a = 1 - \varepsilon$,$\varepsilon$ small, such that the invasion $-1$ to $a$ is pushed, while the $1$-to-$a$-invasion remains pulled Reference 39 (see below for further discussion of pushed fronts). In this case, not only the selected state in the wake but even the propagation speed depends in subtle ways on tail behavior since both pushed and pulled fronts are “selected”. Again, well-developed Gaussian tails appear to select front speed and state in the wake in the sense that open classes of initial conditions with such tails converge to the corresponding front. In summary, a description of the boundaries of the mutual basins of attraction is in many ways a question of global dynamics, which to our knowledge has not been explored.
The extended Fisher-KPP equation
The ideas laid out in Section 7 were previously developed and used in Reference 3 to show that Hypotheses 1 through 4 hold for the extended Fisher-KPP equation
for $\delta$ small. The perturbation in $\delta$ is singular such that, compared to the analysis in Section 7, an additional regularization step is required in Reference 3 in order to perturb from the second order Fisher-KPP equation to Equation 8.2. Once this regularization step has been carried out, establishing that Equation 8.2 satisfies Hypotheses 1 through 4 for some $\delta > 0$, Theorem 2 applies to show that all nearby fourth order equations satisfy these assumptions. Equation Equation 8.2 is of interest due to its role in describing the behavior of solutions to reaction-diffusion systems near certain higher co-dimension bifurcations Reference 76, and its ability to interpolate between “simple” invasion fronts and more complex phenomena Reference 19. This example also highlights that our approach is not restricted to second order equations and does not rely on the presence of a comparison principle. The set of fourth order equations to which our results apply is therefore both nonempty and open in the sense of Theorem 2. We expect that a perturbation analysis analogous to Reference 3 would carry over to singular perturbations of higher order, $-\delta ^2 (i\partial _x)^{2m}u$.
Systems of equations
We have focused here on a general framework for scalar equations for simplicity and clarity of presentation. However, we expect that our methods can be used to prove analogous results in systems of equations satisfying appropriate versions of our assumptions. In particular, the linear spreading speed analysis extends readily and yields diffusive dynamics in the leading edge: one finds an associated exponential weight and an associated eigenvector, which can be used to construct Gaussian tails. In the case where the linearization in the leading edge is diagonal, this Gaussian tail would be confined to one component in the system, a case which occurs in particular in Lotka-Volterra systems as considered by Faye and Holzer Reference 27, who obtained sharp local stability of pulled fronts in this system.
Pattern-forming systems
A common exception to our results is oscillatory pulled fronts, with pinched double roots $\lambda _*=i\omega _*$,$\nu _*\in \mathbb{C}\setminus \mathbb{R}$, and selected states in the wake that are not exponentially stable. Both difficulties combine in pattern-forming systems such as the Swift-Hohenberg, the Cahn-Hilliard, or phase-field equations Reference 15Reference 37Reference 80Reference 84. We expect that the key mechanism of front selection, exploited here, through matching of a diffusive tail with the main front profile can be adapted Reference 83Reference 84. However, even results on asymptotic stability of fronts appear to be known only for speeds above the linear spreading speed Reference 23. Related but somewhat simpler examples arise in pattern-forming mechanisms with a stationary mode $\lambda _*=0,\nu _*\in \mathbb{R}$, such as the Ginzburg-Landau modulation approximation to Swift-Hohenberg, where stability is known up to the critical speed Reference 24, or the FitzHugh-Nagumo equation Reference 14. Our results here do not directly apply to these fronts, despite stationary invasion, due to the diffusive stability of the patterns in the wake, so that Hypothesis 2 is not satisfied. However, we expect this to be mostly a technical issue, rather than a fundamental obstacle to extending the analysis here to these cases.
Pushed fronts and other instabilities of the front
Hypothesis 4 requires absence of unstable eigenvalues. Indeed, as mentioned above, nonlinearities can create instabilities of fronts propagating at the linear speed and lead to the selection of faster pushed fronts. Simple explicit examples occur in the cubic family $u_t=u_{xx}+u(u+\delta )(1-\delta -u)$, where fronts connecting $1-\delta$ to $0$ are stable for $1/3<\delta \leq 1/2$ but unstable with a single unstable eigenvalue for $0<\delta < 1/3$. In this latter regime, invasion is faster than the linear spreading, mediated by a steeper, pushed front. Compared to pulled fronts, it is much simpler to establish selection of pushed fronts, i.e. that pushed fronts attract open classes of steep initial data. Perturbations that cut off the tail of a pushed front are small in a space with an exponential weight that pushes the essential spectrum strictly into the left half plane, allowing selection to be established with a classical stability argument Reference 78. At the transition from pulled to pushed propagation, in this simple example at $\delta =1/3$, the linearization at the front propagating with the linear spreading speed possesses a resonance at $\lambda =0$, violating Hypothesis 4. In this boundary case, it is conjectured Reference 84 that the front still propagates with speed $c_*$, but with the $-\frac{3}{2 \eta _*} \log t$ delay replaced by $-\frac{1}{2 \eta _*} \log t$ due to the stronger effect of the nonlinearity.
Beyond pushed fronts, one also observes destabilization against complex eigenvalues or against essential spectrum induced by instabilities in the wake. All of those lead to more complex dynamics, for instance oscillatory invasion despite a zero frequency pinched double root. Specific examples are studied in Reference 48 in simple problems with comparison principles. Generalizing these examples to complex coefficient amplitude equations, one readily finds scenarios where invasion dynamics are periodic or even chaotic, despite predicted simple stationary dynamics from the pinched double root analysis; compare also the complexity of invasion dyanmics in the complex Ginzburg-Landau equation as described for instance in Reference 1Reference 81Reference 84. We are not aware of selection results in the presence of oscillatory or chaotic dynamics.
While we expect that our assumptions and results hold in open classes of systems of several equations, we caution that there are examples of selected fronts that do not satisfy our assumptions. Spreading in these examples relies on different pointwise instability mechanisms, which may preclude front stability in any fixed exponential weight or even necessitate linear speed selection criteria different from the pinched double root criterion Reference 28. Moreover, stability analysis and numerical evidence strongly suggest that such selection mechanisms occur in open classes of equations.
Specifically, such anomalous modes of invasion were first studied in systems of two coupled Fisher-KPP equations Reference 28Reference 46Reference 47, where a pointwise decaying second component accelerates the propagation in the first component through a weakly decaying exponential tail. This mechanism was interpreted in Reference 28 more broadly as a spreading behavior mediated by resonant couplings, present in open classes of equations. Such resonances can preclude stability of fronts in exponentially weighted spaces when the pinched double root criterion gives the correct spreading speed: the associated front is shown to be asymptotically stable in a model problem in Reference 29 and strong numerical evidence indicates that it is selected in our sense. Resonant couplings can also give rise to spreading speeds and selected fronts that are not predicted by pinched double roots but rather by other resonances in the complex dispersion relation. From this perspective, simple pinched double roots are branched 1:1 resonances. Unbranched 1:1 resonances are nongeneric but occur in Reference 46Reference 47. The phenomena in Reference 28Reference 29 are induced by 1:2 resonances.
Spreading speeds in these examples can be reliably determined from a formal linear marginal stability criterion for resonant modes Reference 28. However, even formulating a marginal stability conjecture for selection of nonlinear fronts in these examples is challenging since the invasion process crucially relies on modes that exhibit pointwise temporal decay in the frame with the observed spreading speed. It is then not clear how to formulate linear (or nonlinear) marginal stability near such a time-dependent profile.
$$\begin{equation} \partial _t u = \partial _{xx}u + u - u^2, \quad x \in \mathbb{R}, \quad t > 0, \cssId{texmlid1}{\tag{1.1}} \end{equation}$$
Equation (1.4)
$$\begin{equation} \partial _t u = - \delta ^2 \partial _{xxxx} u + \partial _{xx} u + f(u), \quad x \in \mathbb{R}, \quad t > 0, \cssId{texmlid2}{\tag{1.4}} \end{equation}$$
Equation (1.5)
$$\begin{equation} \partial _t u = -\partial _{xxxx}u - 2\partial _{xx}u + (\mu -1) u - u^3, \quad x \in \mathbb{R}, \quad t > 0, \quad \mu > 0; \cssId{texmlid3}{\tag{1.5}} \end{equation}$$
Equation (1.6)
$$\begin{equation} u_t = \mathcal{P}(\partial _x) u + f(u), \quad u = u (x,t) \in \mathbb{R}, \quad x \in \mathbb{R}, \quad t > 0, \cssId{texmlid8}{\tag{1.6}} \end{equation}$$
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This work was supported by the National Science Foundation through the Graduate Research Fellowship Program under Grant No. 00074041, as well as through NSF-DMS-1907391. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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