1 Introduction

In this paper we consider the Cauchy problem for the reaction-fractional-diffusion equation

$$\begin{aligned} u_t+(-\Delta )^{s} u=f(u), \end{aligned}$$
(1.1)

with \((t,x)\in [0,\infty )\times \mathbb {R}^d\) and f a Lipschitz reaction function. One frequently assumes that \(f(0)=f(1)=0\), and considers solutions \(0\le u\le 1\) that model transitions between two equilibrium states (i.e., \(u\equiv 0\) and \(u\equiv 1\)), driven by the interplay of the two physical processes involved: reaction and diffusion. Our goal is to obtain optimal estimates on the speed of invasion of one equilibrium (\(u\equiv 0\)) by the other (\(u\equiv 1\)), so we will study the speeds of propagation of solutions with front-like (see (1.10) below) and localized initial data. Note that the comparison principle shows that in the case of front-like initial data, it suffices to consider (1.1) in one spatial dimension \(d=1\), that is,

$$\begin{aligned} u_t+(-\partial _{xx})^{s} u=f(u) \end{aligned}$$
(1.2)

on \([0,\infty )\times \mathbb {R}\). We will do that here when we discuss such initial data, while for localized data we will consider (1.1) with \(d\ge 1\). The distinction between these two cases is marginal when \(s=1\), but this is not anymore the case when \(s\in (0,1)\) and diffusion has long range kernels.

The classical diffusion case \(s=1\) goes back to pioneering works by Kolmogorov, Petrovskii, and Piskunov [28], and Fisher [19], and it is now well-known that solutions with both types of initial data propagate ballistically for all reaction functions of interest — monostable, ignition, as well as (unbalanced) bistable [4].

We will therefore concentrate here on the fractional diffusion case \(s\in (0,1)\), with the fractional Laplacian given by

$$\begin{aligned} (-\Delta )^s u(x) = c_{s,d} \,\,\mathrm{p.v.} \int _{\mathbb {R}^d} \frac{u(x)-u(y)}{|x-y|^{d+2s}}dy, \end{aligned}$$
(1.3)

where \(c_{s,d}:=c_s \left( \int _{\mathbb {R}^{d-1}} (1+h^2)^{-\frac{d}{2} -s} dh \right) ^{-1}\) and \(c_s=c_{s,1}>0\) is an appropriate constant. Then (1.1) models reactive processes subject to non-local diffusion, mediated by Lévy stochastic processes with jumps (see, e.g., [33] and references therein), and the question of propagation of solutions turns out to be much more complicated. Its study was initiated by Cabré and Roquejoffre in [10], who considered (1.1) with Fisher-KPP reactions (specifically, concave ones with \(f'(0)>f(0)=0=f(1)>f'(1)\)), which are a special case of 1-monostable reactions from Definition 1.1 below. They proved that solutions to (1.2) with front-like initial data propagate exponentially, in the sense that \(u(t,\cdot )\rightarrow 1\) uniformly on \(\{x\le e^{\sigma t}\}\) for each \(\sigma <\frac{f'(0)}{2s}\), while \(u(t,\cdot )\rightarrow 0\) uniformly on \(\{x\ge e^{\sigma t}\}\) for each \(\sigma >\frac{f'(0)}{2s}\). They also considered localized (non-zero non-negative fast-decaying) initial data for (1.1) and showed that in that case one has \(u(t,\cdot )\rightarrow 1\) uniformly on \(\{|x|\le e^{\sigma t}\}\) for each \(\sigma <\frac{f'(0)}{d+2\,s}\), while \(u(t,\cdot )\rightarrow 0\) uniformly on \(\{|x|\ge e^{\sigma t}\}\) for each \(\sigma >\frac{f'(0)}{d+2s}\). We note that prior to [10], exponential propagation for Fisher-KPP reactions and continuous diffusion kernels with algebraically decreasing tails (from compactly supported initial data in one dimension) was established by Garnier [20]. While there are many other papers studying such questions for various diffusion operators (see, e.g., [2, 8] and references therein), we will restrict our presentation here to (1.1).

The exponential propagation rates for Fisher-KPP reactions and \(s<1\) are due to interaction between the long range kernels of the fractional diffusion and a strong hair-trigger effect of the reaction. These contrast with the case \(s=1\), when level sets of solutions are located in an o(t) neighborhood of the point \(x=ct\) (for front-like data) resp. the sphere \(\partial B_{ct}(0)\) (for localized data), with the spreading speed c depending only on f (for all the above types of reactions [4]). It turns out that they are in fact a special feature of 1-monostable reactions, and the situation is very different for all the other reaction types. Let us now define these.

Definition 1.1

Let \(f:[0,1] \rightarrow \mathbb {R}\) be a Lipschitz continuous function with \(f(0)=f(1)=0\).

  1. (i)

    If there is \(\theta _0\in (0,1)\) such that \(f(u)=0\) for all \(u\in (0,\theta _0]\) and \(f(u)>0\) for all \(u\in (\theta _0,1)\), then f is an ignition reaction and \(\theta _0\) is the ignition temperature.

  2. (ii)

    If \(f(u)>0\) for all \(u\in (0,1)\), then f is a monostable reaction. If we also have that

    $$\begin{aligned} \gamma u^\alpha \le f(u) \le \gamma ' u^\alpha \qquad \text { for all }u\in (0,\theta _0] \end{aligned}$$
    (1.4)

    and some \(\alpha \ge 1\), \(\theta _0\in (0,1)\), and \(\gamma ,\gamma '>0\), we say that f is an \(\alpha \)-monostable reaction.

  3. (iii)

    If there is \(\theta _0\in (0,1)\) such that \(f(u)<0\) for all \(u\in (0,\theta _0)\) and \(f(u)>0\) for all \(u\in (\theta _0,1)\), as well as \(\int _0^1 f(u) du>0\), then f is an (unbalanced) bistable reaction.

Ignition reactions are used to model combustive processes, while monostable reactions model phenomena such as chemical kinetics and population dynamics [6, 18, 22, 34]. Bistable reactions are used in models of phase transitions and nerve pulse propagation [3, 5, 31], and the unbalanced condition \(\int _0^1 f(u) du>0\) guarantees at least ballistic propagation for all non-negative solutions that are initially larger than \(\theta \) (for any \(\theta >\theta _0\)) on some large-enough ball (of \(\theta \)-dependent radius). This is also the case for all ignition and monostable reactions (with \(\theta >0\) in the latter case).

In particular, the following holds for all the reactions from Definition 1.1: if \(0\le u\le 1\) is a solution to (1.2) with either \(\liminf _{x\rightarrow -\infty } u(0,x)>\theta _0\) for ignition and bistable f, or \(\liminf _{x\rightarrow -\infty } u(0,x)>0\) for monostable f, then for any \(\lambda \in (0,1)\), we have

$$\begin{aligned} \liminf _{t\rightarrow \infty } \frac{{{\underline{x}}}_\lambda (t;u)}{t}>0, \end{aligned}$$
(1.5)

where, for \(t\ge 0\), we let

$$\begin{aligned} {{\underline{x}}}_\lambda (t;u):=\inf \left\{ x\in \mathbb {R}\,\big |\, u(t,x)\le \lambda \right\} \end{aligned}$$
(1.6)

be the left end of the \(\lambda \)-level set of \(u(t,\cdot )\). If we instead let

$$\begin{aligned} {{\underline{x}}}_\lambda (t;u):=\inf \left\{ |x| \,\big |\, u(t,x)\le \lambda \right\} , \end{aligned}$$
(1.7)

this claim also extends to solutions to (1.1) in any dimension and with \(\inf _{|x|\le R_\theta } u(0, x)\ge \theta \), where \(\theta \) must satisfy either \(\theta >\theta _0\) for ignition and bistable f, or \(\theta >0\) for monostable f (and \(R_\theta \) also depends on s, f, d). Both of these claims easily follow from the proof of the last claim in Lemma 2.6 below. (We note that when f is sufficiently small near \(u=0\), solutions with small enough initial data may be quenched in the sense that \(\lim _{t\rightarrow \infty } \Vert u(t,\cdot )\Vert _\infty =0\).)

Ballistic propagation for u is therefore equivalent to

$$\begin{aligned} \limsup _{t\rightarrow \infty } \frac{{{\overline{x}}}_\lambda (t;u)}{t} <\infty \end{aligned}$$

for all \(\lambda \in (0,1)\), where for front-like data and (1.2) we let

$$\begin{aligned} {{\overline{x}}}_\lambda (t;u):=\sup \left\{ x\in \mathbb {R}\,\big |\, u(t,x)\ge \lambda \right\} \qquad (\ge {{\underline{x}}}_\lambda (t;u)) \end{aligned}$$
(1.8)

be the right end of the \(\lambda \)-level set of \(u(t,\cdot )\), and for localized data and (1.1) we let

$$\begin{aligned} {{\overline{x}}}_\lambda (t;u):=\sup \left\{ |x| \,\big |\, u(t,x)\ge \lambda \right\} \qquad (\ge {{\underline{x}}}_\lambda (t;u)). \end{aligned}$$
(1.9)

The comparison principle shows that this holds whenever we have either \(u(0,\cdot )\le \theta \chi _{(-\infty ,R)}\) or \(u(0,\cdot )\le \theta \chi _{B_R(0)}\) for some \(\theta <1\) and \(R\in \mathbb {R}\), provided (1.2) has a traveling front. The latter is a solution of the form \({\tilde{u}}(t,x)=U(x-ct)\), with \(\lim _{x\rightarrow -\infty } U(x)=1\) and \(\lim _{x\rightarrow \infty } U(x)=0\) (i.e., U must satisfy \(-cU_x+(-\partial _{xx})^{s} U=f(U)\)). In fact, it suffices to have \(u(0,\cdot )\le \chi _{(-\infty ,R)}\) or \(u(0,\cdot )\le \chi _{B_R(0)}\) as long as a traveling front exists for (1.2) with some \({\tilde{f}}\ge f\) in place of f, such that \({\tilde{f}}(1+\delta )=0\) and \({\tilde{f}}>0\) on \([1,1+\delta )\) for some \(\delta >0\) (then of course \(\lim _{x\rightarrow -\infty } U(x)=1+\delta \), and one only needs \(u(0,\cdot )\) to be dominated by some shift of U). Hence, in the rest of this discussion, we will assume that

$$\begin{aligned} \theta \chi _{(-\infty ,0)}\le u(0,\cdot ) \le \chi _{(-\infty ,R)} \end{aligned}$$
(1.10)

for front-like data and

$$\begin{aligned} \theta \chi _{B_{R'}(0)} \le u(0,\cdot ) \le \chi _{B_R(0)} \end{aligned}$$
(1.11)

for localized data, with some \(\theta ,R,R'>0\) (and \(\theta >\theta _0\) when f is ignition or bistable).

Proving the existence of traveling fronts for (1.2), and hence ballistic propagation of solutions, requires one to solve only a (non-local) ODE, and this was indeed achieved in a number of cases. These include all the above reactions with \(s=1\) [4, 5], where diffusion is local, as well as all \(C^2\) bistable reactions with any \(s\in (0,1)\) [1, 14, 24], where the negative values of f near \(u=0\) suppress the effects of long range dispersal. The cases of ignition and monostable reactions with \(s\in (0,1)\) are more delicate, and depend intimately on the interplay of the long range diffusion and the strength of f near \(u=0\). Nevertheless, Mellet, Roquejoffre, and Sire proved that traveling fronts still exist for ignition reactions with \(f'(1)<0\) when \(s>\frac{1}{2}\) [30], while Gui and Huan later showed that they do not exist when \(s\le \frac{1}{2}\), as well as that they exist for \(\alpha \)-monostable reactions (and \(s\in (0,1)\)) precisely when \(s\ge \frac{\alpha }{2(\alpha -1)}\) [23] (f was assumed to satisfy additional hypotheses in [23] when \(s>\frac{1}{2}\)). We note that since the comparison principle can be used to estimate propagation of solutions for monostable reactions that lie between multiples of two distinct powers of u near \(u=0\), it makes sense to concentrate only on \(\alpha \)-monostable reactions among the monostable ones; we will do so here.

This leaves one with an expectation of super-ballistic (i.e., accelerating) propagation in the cases of ignition reactions with \(s\le \frac{1}{2}\) and \(\alpha \)-monostable reactions with \(s< \min \{\frac{\alpha }{2(\alpha -1)},1\}\). For front-like initial data and (1.2), this has indeed been confirmed in all these cases except for ignition reactions with \(s=\frac{1}{2}\). The result for concave Fisher-KPP reactions in [10] immediately yields exponential propagation for all 1-monostable reactions and \(s\in (0,1)\), albeit with the lower and upper exponential rates being \(\frac{\gamma }{2s}\) and \(\frac{\gamma '}{2s}\), respectively. More recently, Coville, Gui, znd Zhao [15] proved for \(\alpha \)-monostable reactions with \(\alpha >1\) and \(s< \min \{\frac{\alpha }{2(\alpha -1)},1\}\) that

$$\begin{aligned} 0< \liminf _{t\rightarrow \infty } t^{-\frac{\max \{\alpha -1,1\}}{2s(\alpha -1)}} \, {{\overline{x}}}_\lambda (t;u) \qquad \text {and}\qquad \limsup _{t\rightarrow \infty } t^{-\frac{\alpha }{2s(\alpha -1)}} \, {{\overline{x}}}_\lambda (t;u) <\infty \end{aligned}$$

for all \(\lambda \in (0,1)\) (assuming in addition that f is \(C^1\) and \(f'(1)<0\)), which then also yields, for ignition reactions with \(s\in (0,\frac{1}{2}]\), that

$$\begin{aligned} \limsup _{t\rightarrow \infty } t^{-\frac{1}{2s}-\varepsilon } \, {{\overline{x}}}_\lambda (t;u) <\infty \end{aligned}$$
(1.12)

for all \(\varepsilon >0\) and \(\lambda \in (0,1)\). We note that while [15, Theorem 1.3] may appear to also imply \(\liminf _{t\rightarrow \infty } t^{-\frac{1}{2\,s}} \, {{\overline{x}}}_\lambda (t;u) >0\) for ignition reactions and all \(\lambda \in (0,1)\), Proposition 3.1 in its proof in fact assumes f to be monostable. Nevertheless, we still have (1.5) in this case.

While these results cover all the cases of interest in which traveling fronts do not exist, in all of them there is a gap between the powers of time resp. exponential rates in the best available lower and upper bounds on the dynamic: an infinitesimal one for concave Fisher-KPP reactions and for ignition reactions with \(s=\frac{1}{2}\), and a positive one in all the other cases.

In the first main result of the present paper, we fully close this gap in all the latter cases, proving that \({{\underline{x}}}_\lambda (t;u)\) and \({{\overline{x}}}_\lambda (t;u)\) both have the exact power behavior \(O(t^{\frac{\alpha }{2s(\alpha -1)}})\) in time for all \(\alpha \)-monostable reactions with \(\alpha >1\), and \(O(t^{\frac{1}{2s}})\) for all ignition reactions (so we improve both the lower and upper bounds in the ignition case). We do so for all the values of s for which traveling fronts do not exist, except for ignition reactions with \(s=\frac{1}{2}\), where almost-ballistic propagation (which follows from (1.5) and (1.12)), remains the best result.

Theorem 1.2

Let \(0\le u\le 1\) be a solution to (1.2) such that (1.10) holds for some \(\theta ,R>0\), and let \({{\underline{x}}}_\lambda (t;u)\) and \({{\overline{x}}}_\lambda (t;u)\) be from (1.6) and (1.8). Then,

  1. (i)

    If f is an ignition reaction with ignition temperature \(\theta _0\in (0,\theta )\) and \(s\in (0,\frac{1}{2})\), for each \(\lambda \in (0,1)\) we have

    $$\begin{aligned} 0< \liminf _{t\rightarrow \infty } t^{-\frac{1}{2s}} \, {{\underline{x}}}_\lambda (t;u) \le \limsup _{t\rightarrow \infty } t^{-\frac{1}{2s}} \, {{\overline{x}}}_\lambda (t;u) <\infty ; \end{aligned}$$
  2. (ii)

    If f is an \(\alpha \)-monostable reaction for some \(\alpha >1\) and \(s\in (0,\min \{\frac{\alpha }{2(\alpha -1)},1\})\), then for each \(\lambda \in (0,1)\) we have

    $$\begin{aligned} 0< \liminf _{t\rightarrow \infty } t^{-\frac{\alpha }{2s(\alpha -1)}} \, {{\underline{x}}}_\lambda (t;u) \le \limsup _{t\rightarrow \infty } t^{-\frac{\alpha }{2s(\alpha -1)}} \, {{\overline{x}}}_\lambda (t;u) <\infty . \end{aligned}$$

Remark

  1. 1.

    The leading orders of the propagation rates in both (i) and (ii) only depend on s and the qualitative behavior of f near 0, and in (ii) they are independent of \(\gamma , \gamma '\) from (1.4). In contrast, for Fisher-KPP reactions they also depend on \(f'(0)\) [10], and so their dependence on f for general 1-monostable reactions will be much more sensitive.

  2. 2.

    As Theorem 3.4 below shows, (i) extends to the case when \(u(0,\cdot ) \le \chi _{(-\infty ,R)}\) is replaced by \(\limsup _{x\rightarrow \infty } x^{-2s}u(0,x)<\infty \). The supersolutions constructed in [15] show that in (ii) we can instead allow \(\limsup _{x\rightarrow \infty } x^{-\frac{2\,s}{\alpha }}u(0,x)<\infty \).

  3. 3.

    One-sided bounds in (1.4) obviously yield one-sided bounds in (ii).

  4. 4.

    For any \(s\in (0,\frac{1}{2})\), (i) can be regarded as the \(\alpha \rightarrow \infty \) limit of (ii).

To the best of our knowledge, these are the first qualitatively optimal propagation results for front-like solutions in situations where no traveling fronts exist.

When it comes to localized initial data and (1.1), the corresponding Fisher-KPP result from [10] was improved by Coulon and Yangari in [16]. They proved for each \(\lambda \in (0,1)\) that

$$\begin{aligned} 0< \liminf _{t\rightarrow \infty } e^{-\frac{f'(0)}{d+2s} t} \, {{\underline{x}}}_\lambda (t;u) \le \limsup _{t\rightarrow \infty } e^{-\frac{f'(0)}{d+2s} t} \, {{\overline{x}}}_\lambda (t;u) <\infty \end{aligned}$$

for solutions with fast-decaying initial data when \(s\in (0,1)\) and f is any \(C^1\) 1-monostable reaction with \(f(u)-f'(0)u=O(u^{1+\delta })\) (for some \(\delta >0\)). This also implies exponential propagation for general 1-monostable reactions, albeit with the lower and upper exponential rates being \(\frac{\gamma }{d+2\,s}\) and \(\frac{\gamma '}{d+2\,s}\), respectively.

An interesting feature of the results in [10, 16] is that, for Fisher-KPP reactions and \(s<1\), the exponential propagation rates for localized initial data differ from those for front-like data, and they also depend on the dimension. These phenomena happen neither when \(s=1\) (for any reaction), because the diffusion kernel is short range, nor for bistable reactions and \(s\in (0,1)\), when ballistic propagation from localized data at the same speed as from front-like data follows from existence of traveling fronts.

It is therefore not obvious which propagation rates one should expect for ignition and \(\alpha \)-monostable reactions with \(\alpha >1\) when \(s\in (0,1)\) and initial data are localized. One obviously has the ballistic lower bound (1.5), and the same upper bounds as for front-like data (which follow immediately by comparison). However, we are not aware of other relevant prior results for (1.1) in this setting. Our second main theorem therefore appears to provide the first non-trivial such result, and is again also qualitatively optimal. It shows that in all the cases from Theorem 1.2, propagation rates for localized data do coincide with those for front-like data. In particular, unlike for 1-monostable reactions, they do not depend on the dimension.

Theorem 1.3

Let \(0\le u\le 1\) be a solution to (1.1) such that (1.11) holds for some \(\theta ,R>0\) and large enough \(R'\) (depending on \(f,s,\theta \)), and let \({{\underline{x}}}_\lambda (t;u)\) and \({{\overline{x}}}_\lambda (t;u)\) be from (1.7) and (1.9). Then both parts of Theorem 1.2 hold.

Remark

We can obviously again replace \(u(0,\cdot ) \le \chi _{B_R(0)}\) by \(\limsup _{|x|\rightarrow \infty } |x|^{-2\,s} \)\( u(0,x)<\infty \) in (i), and by \(\limsup _{|x|\rightarrow \infty } |x|^{-\frac{2\,s}{\alpha }}u(0,x)<\infty \) in (ii).

The proofs of Theorems 1.2 and 1.3 rest on finding appropriate sub- and supersolutions \(\Phi \) satisfying \({{\underline{x}}}_\lambda (t;\Phi )=O(t^\beta )={{\overline{x}}}_\lambda (t;\Phi )\), with \(\beta =\frac{1}{2s}\) for ignition reactions and \(\beta =\frac{\alpha }{2s(\alpha -1)}\) for \(\alpha \)-monostable reactions. Since these will accelerate in time, one cannot use the traveling front ansatz \(\Phi (t,x)\sim \varphi (x-ct)\) in their construction, because then their transition regions (where they decrease from values close to 1 to those close to 0) would only travel with constant speed c. One might instead hope to have \(\Phi (t,x)\sim \varphi (x-ct^\beta )\), which does travel with the right speed \(O(t^{\beta -1})\). However, it turns out that the acceleration of propagation also forces sub- and supersolutions to have transition regions that stretch in time (see also [21]).

We will therefore construct localized subsolutions of the form \(\Phi (t,x)\sim \varphi (ct^{-\frac{1}{2s}} x)\) in the proof of Theorems 1.2(i) and 1.3(i). Such functions propagate with speeds \(O(t^{\frac{1}{2s}-1})\) but also “flatten” in space, having transition regions of widths \(O(t^\frac{1}{2s})\) (so the latter stretch with roughly the same speeds \(O(t^{\frac{1}{2s}-1})\)). This will be sufficient for subsolutions, but we will have to employ a much more complicated construction for front-like supersolutions. Their propagation speeds will again be \(O(t^{\frac{1}{2s}-1})\), but we will need their stretching speeds to also depend on the value of \(\Phi \), and they will in fact grow from \(O(t^{\frac{1}{2s}-2})\) where \(\Phi \ge \theta _0\) to \(O(t^{\frac{1}{2s}-1})\) where \(\Phi \sim 0\). Of course, this stretching will then accumulate over time to transition regions between different values of \(\Phi \) having lengths from \(O(t^{\frac{1}{2s}-1})\) to \(O(t^{\frac{1}{2s}})\), meaning that these supersolutions will be flattened in a spatially non-uniform manner. This approach, which seems to be necessary in the hunt for qualitatively optimal supersolutions in the ignition case (and hence optimal upper bounds in Theorems 1.2(i) and 1.3(i)), makes this effort significantly more challenging and explains the complexity in our construction in Sect. 3 below.

This type of construction appears to be new, as all previous ones that we are aware of involve spatially uniform stretching rates. In particular, the supersolutions for \(\alpha \)-monostable reactions with \(\alpha >1\) constructed in [15], which propagate with optimal speeds \(O(t^{\frac{\alpha }{2s(\alpha -1)}-1})\), have transition regions that stretch with speeds \(O(t^{\frac{\alpha }{2s(\alpha -1)}-2})\). The subsolutions we construct below in this case will have the same propagation and stretching speeds. However, unlike for ignition reactions, we are only able to achieve these optimal speeds with localized but not compactly supported subsolutions. This, and the fact that we need to find them in all dimensions \(d\ge 1\), further complicate this part of our work.

Finally, we note that smoothing properties of the fractional parabolic dynamic of (1.1) mean that the sense in which our functions solve the PDE is not consequential here. While we consider below mild solutions with uniformly continuous initial data, Theorem 2.5 shows that these immediately become classical. This is also true for bounded weak solutions, via an argument as in the proof of Theorem 2.5 (based on the regularity results in [17, 27]), so these three notions of solutions coincide here. This means that our propagation results hold as well for not necessarily uniformly continuous initial data, due to the comparison principle. Since we were not able to locate a suitable version of the latter in the literature, we prove it in Theorem 2.4 below (which is hence of independent interest). We in fact state it for distributional sub- and supersolutions (see Definition 2.3) because the supersolutions we construct here will only be Lipschitz continuous.

We also highlight here Lemma 2.6, below, which constructs compactly supported stationary subsolutions to (1.1). These then provide initially compactly supported time-increasing solutions, which can be very convenient in the analysis of long-time dynamics of solutions (and specifically, construction of subsolutions in Sect. 4). We are not aware of such a result for (1.1) with \(s\in (0,1)\) prior to our work, although its \(s=1\) version is well known, and in [9] it was also obtained for diffusion operators with integrable kernels.

Remark

Shortly before we finished writing this paper in May 2021, we informed E.  Bouin, J. Coville, and G. Legendre about it. They posted the preprints [7, 8] on arXiv immediately afterwards, just days before we posted ours. The main results claimed in [7, 8] correspond to the first inequalities in Theorem 1.2(i,ii), respectively, which are our optimal lower bounds for front-like initial data (both [7, 8] consider more general diffusion kernels in one dimension, with \(x^{-1-2s}\) decay at \(\pm \infty \)). Unlike our constructions in Sects. 4 and 5 below, the subsolution candidate functions presented in [7, 8] are front-like, so they would not yield localized initial data results such as Theorem 1.3 (even when \(d=1\) because the diffusion kernels are long range for all \(s\in (0,1)\)). However, the 23-page May 2021 version of [8] is incomplete, and was replaced in July 2022 by a 45-page version with a much longer proof containing many changes and additions. Moreover, the May 2021 version of [7] is clearly very preliminary and no other version seemed to be available at the time the present paper went into press in August 2023. Neither preprint appears to have been peer reviewed by that time either.

Organization of the Paper In Sect. 2 we collect various preliminary results, including a comparison principle for (1.1). We then prove parts (i) of Theorems 1.2 and 1.3 in Sects. 3 and 4, and parts (ii) in Sect. 5 (these three sections are completely independent and can be read in any order).

2 Well-posedness and a Comparison Principle

In this section we collect some basic well-posedness and regularity results for (1.1). We also prove two important (and to the best of our knowledge new) results here. The first is a comparison principle, Theorem 2.4, which removes certain restrictive hypotheses from previous results (see the paragraph before Definition 2.3). The second is Lemma 2.6, which constructs initially compactly supported time-increasing solutions to (1.1) for ignition (and therefore also for \(\alpha \)-monostable) reactions and all \(s\in (0,1)\).

We start with the notion of mild solutions, defined via Duhamel’s formula (see [10, 32]). We use \(C_{b,u}(X)\) to denote the space of bounded uniformly continuous functions on X (with the supremum norm), and \(S_t\) to denote the semigroup generated by \((-\Delta )^s\) on \(\mathbb {R}^d\).

Definition 2.1

We say that \(u\in C([0,T);C_{b,u}(\mathbb {R}^d))\) is a mild solution to (1.1) (and that it is global if \(T=\infty \)) if for each \(t\in [0,T)\) we have

$$\begin{aligned} u(t,\cdot )=S_t [u(0,\cdot )]+\int _0^t S_{t-\tau } [f(u(\tau ,\cdot ))] d\tau . \end{aligned}$$

Remark

Notice that if \(u\in C([T_0,T_1];C_{b,u}(\mathbb {R}^d))\) for some \(T_0<T_1\), then it is also uniformly continuous in time on \([T_0,T_1]\) and it follows that in fact \(u\in C_{b,u}([T_0,T_1] \times \mathbb {R}^d)\).

We now have the following global well-posedness result:

Theorem 2.2

If \(s\in (0,1)\) and f is Lipschitz continuous, then for any \(u_0\in C_{b,u}(\mathbb {R}^d)\), there is a unique global mild solution u to (1.1) with \(u(0,\cdot )=u_0\).

Remark

Theorem 2.2 can be proved via a standard fixed point argument using that

$$\begin{aligned} N[u] (t,\cdot ):=S_t [u(0,\cdot )]+\int _0^t S_{t-\tau } [f(u(\tau ,\cdot ))] d\tau \end{aligned}$$

defines a contraction mapping N on the subspace of \(u\in C([0,T];C_{b,u}(\mathbb {R}^d))\) with \(u(0,\cdot )=u_0\) when T is sufficiently small, see for instance [10, 29] (both these papers concern more general diffusion operators than \((-\Delta )^s\)). We note that although \(f'\in C_{b,u}(\mathbb {R})\) is assumed in [10, Sections 2.3 and 2.4], this can easily be relaxed to f being Lipschitz (we prefer to consider here this case instead of \(f\in C^1([0,1])\)). We also mention that a viscosity-solutions-based approach to well-posedness, via maximum principles for general non-local nonlinear PDEs and Perron’s method, was used in [11, 13, 25].

We next turn to a comparison principle for (1.1), and the related definition of sub- and supersolutions. We were not able to use in this work comparison principles that we found in the literature, as these do not quite apply to the Lipschitz continuous sub- and supersolutions we construct below (mild solutions have better regularity, see Theorem 2.5). For instance [10, Proposition 2.8] only applies to classical sub- and supersolutions that satisfy an extra hypothesis on their order as \(|x|\rightarrow \infty \) at all times \(t\ge 0\), while [10, Proposition 2.11] only applies to mild solutions to \(u_t+(-\Delta )^su=h(t,x)\) with uniformly continuous h. We therefore prove here a comparison principle without extra hypotheses and in the more general distributional sense, which then also applies to mild solutions due to Remark 2 below.

Definition 2.3

We say that \(u\in C((T_0,T_1); C_{b,u}(\mathbb {R}^d))\) is a subsolution (supersolution) to (1.1) if for each \(0\le \varphi \in C_c^\infty ((T_0,T_1)\times \mathbb {R}^d)\) we have

$$\begin{aligned}{} & {} \int _{T_0}^{T_1} \int _{\mathbb {R}^d} \left[ - u(t,x)\varphi _t(t,x)+u(t,x)(-\Delta )^s\varphi (t,x)-f(u(t,x))\varphi (t,x) \right] \, dxdt\\{} & {} \quad \le 0 \quad (\ge 0). \end{aligned}$$

Remark

  1. 1.

    Note that when \(s\in (0,\frac{1}{2})\), any bounded Lipschitz continuous function u has bounded \((-\Delta )^{s} u\); and if it satisfies \( u_t+(-\Delta )^{s} u-f(u) \le 0\) (\(\ge 0\)) for a.e. (t, x), then it is clearly a subsolution (supersolution) to (1.1).

  2. 2.

    It is easy to show that a mild solution u to (1.1) on time interval [0, T) is both a sub- and a supersolution on time interval (0, T). Indeed, let \(u_\varepsilon :=\phi _\varepsilon *u\), with \(\phi _\varepsilon \) a smooth space-time mollifier as in the following proof. Then \(u_\varepsilon \) is a classical solution to \(u_t+(-\Delta )^su=f_\varepsilon (t,x)\) on time interval \((T_0+\varepsilon ,T_1-\varepsilon )\), where \(f_\varepsilon :=\phi _\varepsilon *(f\circ u)\). Since \(u_\varepsilon \rightarrow u\), and \(f_\varepsilon \rightarrow f\circ u\) uniformly on \([t_0,t_1]\times \mathbb {R}^d\) for any \([t_0,t_1]\subseteq (T_0,T_1)\) (see the remark after Definition 2.1), this yields the claim.

Theorem 2.4

Let \(s\in (0,1)\) and f be Lipschitz. If \(u,v\in C([0,T); C_{b,u} (\mathbb {R}^d))\) are, respectively, a subsolution and a supersolution to (1.1) on time interval (0, T) and satisfy \(u(0,\cdot )\le v(0,\cdot )\) on \(\mathbb {R}^d\), then \(u\le v\) on \([0,T)\times \mathbb {R}^d\).

Proof

It suffices to prove that \(u\le v\) on \([0,T']\times \mathbb {R}^d\) for any \(T'<T\), so we can assume that \(T<\infty \) and \(u,v\in C([0,T]; C_{b,u}(\mathbb {R}^d))\). Let \(K:=\max \{\Vert f'\Vert _\infty , \)\( \Vert u\Vert _\infty ,\Vert v\Vert _\infty \}\), and then let

$$\begin{aligned} \tilde{u}(t,x):=e^{Kt}u(t,x),\quad \tilde{v}(t,x):=e^{Kt}v(t,x),\quad g(t,u):=K{u}+e^{Kt}f(e^{-Kt}{u}). \end{aligned}$$

Then \(\tilde{u}\) and \(\tilde{v}\) are, respectively, a subsolution and a supersolution to (1.1) with f(u) replaced by g(t, u), on time interval (0, T). Moreover, \(\max \{\Vert {\tilde{u}}\Vert _\infty ,\Vert {\tilde{v}}\Vert _\infty \}\le Ke^{KT}\), and if we let \(S:=[0,T]\times [-Ke^{KT},Ke^{KT}]\), then for all \((t,u)\in S\) we have

$$\begin{aligned} 0\le g_u(t,u)\le 2K \qquad \text { and }\qquad |g_t(t,u)|\le Ke^{KT} \Vert f|_{[-K,K]}\Vert _{{\infty }}+K^3=:K'.\nonumber \\ \end{aligned}$$
(2.1)

Finally, we have \(\tilde{u}(0,\cdot )\le \tilde{v}(0,\cdot )\), and proving \(u\le v\) is equivalent to proving \(\tilde{u}\le \tilde{v}\). We will therefore slightly abuse notation, and write below u, v instead of \(\tilde{u}, \tilde{v}\).

For any small \(\varepsilon \in (0,1)\), fix a smooth mollifier \(\phi _\varepsilon \ge 0\) with \(\textrm{supp}\, \phi _\varepsilon \subseteq B_\varepsilon (0) \subseteq \mathbb {R}^{d+1}\) and \(\int _{\mathbb {R}^{d+1}}\phi _\varepsilon (t,x)d(t,x)=1\). Then for \((t,x)\in [0,T-2\varepsilon ]\times \mathbb {R}^d\) let

$$\begin{aligned} u_\varepsilon (t,x):=(\phi _\varepsilon *u)(t+\varepsilon ,x)\qquad \text {and}\qquad v_\varepsilon (t,x):=(\phi _\varepsilon *v)(t+\varepsilon ,x) \end{aligned}$$

(the \(\varepsilon \)-shifts in time allow us to define \(u_\varepsilon ,v_\varepsilon \) at \(t=0\)). The remark after Definition 2.1 shows that there is \(\omega _\varepsilon \ge 0\) such that \(\lim _{\varepsilon \rightarrow 0^+}\omega _\varepsilon =0\) and

$$\begin{aligned} \sup _{\begin{array}{c} (t,x),(\tau ,y)\in [0,T]\times \mathbb {R}^d,\\ \max \{ |t-\tau |, |x-y|\}\le 2\varepsilon \end{array}} \max \{ |u(t,x)-u(\tau ,y)|, |v(t,x)-v(\tau ,y)|\} \le \omega _\varepsilon . \end{aligned}$$
(2.2)

Then on \([0,T-2\varepsilon ]\times \mathbb {R}^d\) we have

$$\begin{aligned} \max \{ |u_\varepsilon -u|,|v_\varepsilon -v| \} \le \omega _\varepsilon . \end{aligned}$$
(2.3)

Now consider any \((t,x)\in [0,T-2\varepsilon ]\times \mathbb {R}^d\). If there is \((t',x')\in B_\varepsilon (t+\varepsilon ,x)\) such that \(u(t',x')\le v(t',x')\), then it follows from (2.1) and (2.2) that

$$\begin{aligned}&(\phi _\varepsilon * [g(\cdot , u(\cdot ,\cdot )) - g(\cdot , v(\cdot ,\cdot )) ] ) \, (t+\varepsilon ,x) \\&\quad \le g(t',u(t',x'))-g(t',v(t',x')) + 4\varepsilon \Vert g_t |_{S} \Vert _{\infty } + 2\omega _{\varepsilon } \Vert g_u |_{S}\Vert _{\infty } \\&\quad \le 4\varepsilon K' + 4K\omega _{\varepsilon }. \end{aligned}$$

If, instead, \(u\ge v\) on \(B_\varepsilon (t+\varepsilon ,x)\), then \(g_u\le 2K\) yields

$$\begin{aligned}{} & {} (\phi _\varepsilon * [g(\cdot , u(\cdot ,\cdot )) - g(\cdot , v(\cdot ,\cdot )) ] ) \, (t+\varepsilon ,x)\le 2K(\phi _\varepsilon * (u-v))(t+\varepsilon ,x)\\{} & {} \quad \le 2K(u_\varepsilon (t,x)-v_\varepsilon (t,x)). \end{aligned}$$

From these estimates, and from u and v being, respectively, a sub- and a supersolution, we get for \(w_\varepsilon :=u_\varepsilon -v_\varepsilon \) and \(\omega '_\varepsilon :=4\varepsilon K' + 4K\omega _{\varepsilon }\) (\(\rightarrow 0\) as \(\varepsilon \rightarrow 0\)) that

$$\begin{aligned}{} & {} h_\varepsilon :=(w_\varepsilon )_t+(-\Delta )^s w_\varepsilon \le (\phi _\varepsilon *[g(\cdot , u(\cdot ,\cdot )) - g(\cdot , v(\cdot ,\cdot )) ]) \,(\cdot +\varepsilon ,\cdot )\\{} & {} \quad \le \max \{2K w_\varepsilon ,\omega '_{\varepsilon }\}. \end{aligned}$$

Duhamel’s principle for smooth solutions to the linear PDE

$$\begin{aligned} w_t+(-\Delta )^s w=h(t,x) \end{aligned}$$

(see [10, 29, 32]) now yields

$$\begin{aligned} w_\varepsilon (t,x)\le S_t [w_\varepsilon (0,\cdot )](x)+\int _0^t S_{t-\tau } [\max \{2K w_\varepsilon (\tau ,\cdot ),\omega '_{\varepsilon } \} ](x)\,d\tau \end{aligned}$$

for all \((t,x)\in [0,T-2\varepsilon ]\times \mathbb {R}^d\). Since \(S_t\) preserves order and \(S_t[1]\equiv 1\), we have \(S_t[w_\varepsilon (0,\cdot )]\le 2\omega _\varepsilon \) on \(\mathbb {R}^d\) (by \(u(0,\cdot )\le v(0,\cdot )\) and (2.3)), and if we let \(\xi (t):=\max \{\sup _{x\in \mathbb {R}^d}w_\varepsilon (t,x),0\}\), then

$$\begin{aligned} \xi (t)\le 2\omega _\varepsilon +T\omega '_{\varepsilon }+ \int _0^t 2K \xi (\tau )\, d\tau \end{aligned}$$

for each \(t\in [0,T-2\varepsilon ]\). Grönwall’s inequality now yields

$$\begin{aligned} \xi (t) \le (2\omega _\varepsilon +T\omega '_{\varepsilon })\,e^{2KT} \end{aligned}$$

for \(t\in [0,T-2\varepsilon ]\), hence

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0} \sup _{(t,x)\in [0,T-2\varepsilon ]\times \mathbb {R}^d} w_\varepsilon (t,x) = 0. \end{aligned}$$

This shows that \(u\le v\) on \([0,T]\times \mathbb {R}^d\), finishing the proof. \(\quad \square \)

Similarly to parabolic PDE with classical diffusion, the dynamics of (1.1) provides certain smoothing, which is the basis of relevant regularity results. The following theorem, in which we suppress dependence of all constants on d in the notation, is consequence of results from [17, 27] (we also refer the reader to [12, 13, 26] for the viscosity solutions setting):

Theorem 2.5

Let \(s\in (0,1)\) and f be Lipschitz, and let \(u\in C([0,T); C_{b,u}(\mathbb {R}^d))\) be a bounded mild solution to (1.1). There is \(\sigma =\sigma (s)>0\) such that for any \(\tau \in (0,T)\) there is \(C=C(s,f,\Vert u\Vert _\infty ,\tau )>0\) such that

$$\begin{aligned} \Vert u\Vert _{ C^{1+\sigma /2s,2s+\sigma }([\tau ,T)\times \mathbb {R}^d)}\le C. \end{aligned}$$

In particular, u is a classical solution to (1.1).

Proof

For any \(\varepsilon \in (0, \frac{\tau }{4})\), let \(\phi _\varepsilon \) be the space-time mollifier from the proof of Theorem 2.4. If \(f_\varepsilon :=\phi _\varepsilon *(f\circ u)\), then \(u_\varepsilon :=\phi _\varepsilon *u\) satisfies in the classical sense the linear PDE

$$\begin{aligned} (u_\varepsilon )_t+(-\Delta )^su_\varepsilon =f_\varepsilon (t,x) \end{aligned}$$
(2.4)

on \((\frac{\tau }{4},T-\varepsilon )\times \mathbb {R}^d\). Since \(\Vert u\Vert _\infty <\infty \), we have

$$\begin{aligned} \max \{\Vert u_\varepsilon \Vert _\infty , \Vert f_\varepsilon \Vert _\infty \} \le C_1:= |f(0)|+(1+\Vert f'\Vert _\infty )(1+\Vert u\Vert _\infty ). \end{aligned}$$

The interior Hölder estimate [27, Theorem 1.1] now yields \(\sigma =\sigma (s)\in (0,\min \{1,2s\})\) and \(C_2=C_2(s,C_1,\tau )>0\) such that

$$\begin{aligned} \Vert u_\varepsilon \Vert _{C^{{\sigma /2s},\sigma } ([{\tau }/{2},T-\varepsilon )\times \mathbb {R}^d )}\le C_2. \end{aligned}$$

Since \(u_\varepsilon \rightarrow u\) uniformly on \([\frac{\tau }{2},T-\delta ]\times \mathbb {R}^d\) for any \(\delta >0\) by the remark after Definition 2.1, taking \(\varepsilon \rightarrow 0\) shows that

$$\begin{aligned} \Vert u\Vert _{C^{{\sigma /2s},\sigma }([{\tau }/{2},T)\times \mathbb {R}^d)} \le C_2. \end{aligned}$$

This and Lipschitz continuity of f yield \(C_3=C_3(C_1,C_2)>0\) such that

$$\begin{aligned} \Vert f_\varepsilon \Vert _{C^{{\sigma }/{2s},\sigma }([{\tau }/{2},T-\varepsilon )\times \mathbb {R}^d) } \le C_3 \end{aligned}$$

for all \(\varepsilon \in (0,\frac{\tau }{4})\). It now follows from (2.4) and [17, Theorem 1.1] that

$$\begin{aligned} \Vert u_\varepsilon \Vert _{C^{1+{\sigma }/{2s},2s+\sigma }([\tau ,T-\varepsilon )\times \mathbb {R}^d)}\le C_4 \end{aligned}$$

for these \(\varepsilon \), with \(C_4=C_4 (s,C_1,C_2,\tau )>0\). Taking \(\varepsilon \rightarrow 0\) finishes the proof. \(\quad \square \)

Finally, to obtain the lower bounds in Theorems 1.2 and 1.3, we will need to use certain non-trivial initial data \( u_0:\mathbb {R}^d\rightarrow [0,1]\) satisfying

$$\begin{aligned} -(-\Delta )^s u_0 +f(u_0)\ge 0 \end{aligned}$$
(2.5)

(note that the comparison principle then shows that solutions to (1.1) with such initial data are time-increasing). The following lemma, whose proof we postpone to the appendix, provides such functions for ignition reactions (see the remark below for monostable reactions):

Lemma 2.6

Let f be an ignition reaction with ignition temperature \(\theta _0\in (0,1)\), and let \(s\in (0,1)\). For \(\theta \in (\theta _0,1)\), there are \(R_\theta \ge 1\) and a non-increasing smooth function \(u_\theta \) on \(\mathbb {R}\) (both depending also on s, f, d) such that

$$\begin{aligned}{} & {} \theta \chi _{(-\infty ,0]}\le u_\theta \le \theta \chi _{(-\infty ,R_\theta ]}, \end{aligned}$$
(2.6)
$$\begin{aligned}{} & {} \inf _{x\in \mathbb {R}} \left[ -(-\partial _{xx})^su_\theta (x)+f(u_\theta (x)) \right] \ge 0, \end{aligned}$$
(2.7)
$$\begin{aligned}{} & {} \inf _{x\le R_\theta } \left[ -(-\partial _{xx})^su_\theta (x)+f(u_\theta (x)) \right] > 0, \end{aligned}$$
(2.8)

and the function \({\bar{u}}_\theta (x):=u_\theta (|x|)\) on \(\mathbb {R}^d\) satisfies

$$\begin{aligned}{} & {} \inf _{x\in \mathbb {R}^d} \left[ -(-\Delta )^s {\bar{u}}_\theta (x)+f({\bar{u}}_\theta (x)) \right] \ge 0, \end{aligned}$$
(2.9)
$$\begin{aligned}{} & {} \inf _{|x|\le R_\theta } \left[ -(-\Delta )^s {\bar{u}}_\theta (x)+f({\bar{u}}_\theta (x)) \right] > 0. \end{aligned}$$
(2.10)

Moreover, if \(0\le u\le 1\) is a global mild solution to (1.1) and \(u(0,\cdot )\ge {\bar{u}}_\theta (\cdot -x_0)\) for some \(x_0\in \mathbb {R}^d\), then \(u(t,\cdot )\rightarrow 1\) locally uniformly on \(\mathbb {R}^d\) as \(t\rightarrow \infty \).

Remark

Since for each monostable reaction f and each \(\theta \in (0,1)\), there is an ignition reaction \(g\le f\) with ignition temperature smaller than \(\theta \), the lemma extends to such \(f,\theta \).

3 Supersolutions for Ignition Reactions

In this section we prove the upper bound in Theorem 1.2(i), which then automatically provides the same bound in Theorem 1.3(i) via the comparison principle. The main step is construction of a family of supersolutions \(\Phi \) to (1.2) that satisfy \({{\overline{x}}}_\lambda (t;\Phi )=O(t^{\frac{1}{2s}})\).

We start with a simple fractional Laplacian estimate. For \(A_1<A_2\) and \(\theta \in (0,1)\), let

$$\begin{aligned} \psi _{A_1,A_2,\theta }(x):=\left\{ \begin{aligned}&1{} & {} \text { if }x\le A_1,\\&1- (1-\theta ) \frac{x-A_1}{A_2-A_1}{} & {} \text { if }x\in ( A_1,A_2],\\&\theta{} & {} \text { if }x> A_2. \end{aligned}\right. \end{aligned}$$

Note that \(s<\frac{1}{2}\) and Lipschitz continuity of \(\psi _{A_1,A_2,\theta }\) show that \((-\partial _{xx})^{s}\psi _{A_1,A_2,\theta }\) is bounded.

Lemma 3.1

Let \(s\in (0,\frac{1}{2})\), \(C_{\alpha }:= \max \{\frac{c_s}{2\,s(1-2\,s)},1\}\), \(A_1<A_2\), and \(\theta \in (0,1)\). If \(\varphi \le \psi _{A_1,A_2,\theta }\) and \(\varphi (x)=\psi _{A_1,A_2,\theta }(x)\) for some \(x\in \mathbb {R}\), then

$$\begin{aligned} (-\partial _{xx})^s\varphi (x)\ge -C_s(1-\theta ) (A_2-A_1)^{-2s}. \end{aligned}$$

Proof

By (1.3), it suffices to assume that \(\varphi \equiv \psi _{A_1,A_2,\theta }\). Then

$$\begin{aligned} \varphi (x)-\varphi (x+h)\ge \varphi (A_2)-\varphi (A_2+h) \end{aligned}$$

for each \(h\in \mathbb {R}\), so it suffices to assume that \(x=A_2\). Now a direct computation yields

$$\begin{aligned} (-\partial _{xx})^s\varphi (A_2)&=-c_s(1-\theta )\int _{-\infty }^{A_1-A_2} |h|^{-1-2s} dh- \frac{c_s(1-\theta )}{A_2-A_1}\int _{A_1-A_2}^{0} |h|^{-2s}dh\\&\quad = -c_s(1-\theta ) \left( (2s)^{-1}+(1-2s)^{-1}\right) (A_2-A_1)^{-2s}, \end{aligned}$$

finishing the proof. \(\quad \square \)

We will now construct an infinite family of supersolutions to (1.2) indexed by \(k\in \mathbb {N}\) (see (3.6) below), each defined on a finite time interval of length \(e^{e^{O(k)}}\) and obtained by gluing together \(k+3\) separate pieces (all but one of them linear in space); see the introduction for a discussion of the reasons for such a complicated construction.

Let us take any \(k\in \mathbb {N}\), and for any \(n\in \mathbb {N}_k:=\{0,1,...,k\}\) let

$$\begin{aligned} \alpha ^k_n:=\Sigma _{j=1}^{n} (k-n+j){(2s)^{j-1}}\qquad \text { and }\qquad \beta ^k_n:=2^{-\alpha _n^k}, \end{aligned}$$

where \(\sum _{j=1}^0 a_j:=0\) (so \(\alpha ^k_0=0\) and \(\beta ^k_0=1\)). We then have \(\alpha ^k_k\le \sum _{j=1}^\infty j(2\,s)^{j-1} = \frac{1}{(1-2\,s)^{2}}\), so

$$\begin{aligned} \beta ^k_k\ge 2^{-\frac{1}{(1-2s)^2}}. \end{aligned}$$
(3.1)

(In fact, \(\alpha ^k_{k-n}\) is increasing in k for each fixed \(n\ge 0\) and converges to \(\frac{1}{(1-2\,s)^{2}} + \frac{n}{1-2\,s}\).) Also, from \(\alpha _n^k\ge k-n+1\) we have

$$\begin{aligned} \Sigma _{n=1}^k\beta _n^k\le 1, \end{aligned}$$
(3.2)

and a simple computation yields

$$\begin{aligned} \beta _n^k = 2^{-k+n-1} (\beta _{n-1}^k)^{2s}. \end{aligned}$$
(3.3)

We also have the simple lemma

Lemma 3.2

If \(t\ge 2^{\frac{k}{1-2s}}\), then for any \(n\in \mathbb {N}_{k}\setminus \{0\}\),

$$\begin{aligned} \beta ^k_{n} t^{\frac{1}{2s}-(2s)^{n}} \ge 2\beta ^k_{n-1} t^{\frac{1}{2s}-(2s)^{n-1}}. \end{aligned}$$
(3.4)

Proof

A direct computation shows that (3.4) is equivalent to

$$\begin{aligned} t^{(2s)^{n-1}-(2s)^{n}}\ge 2^{k(2s)^{n-1}-\Sigma _{j=1}^{n-1}(2s)^{j-1}}. \end{aligned}$$

This is equivalent to

$$\begin{aligned} t^{1-2s}\ge 2^{k-((2s)^{1-n}-1)/(1-2s)}, \end{aligned}$$

which clearly holds, by the hypothesis. \(\quad \square \)

In the rest of this section, and in the next section, we will assume the following:

  1. (I)

    Let f be an ignition reaction with ignition temperature \(\theta _0\in (0,1)\), and let \(s\in (0,\frac{1}{2})\).

Let us now define \(\theta _*:=\frac{\theta _0}{2}\). For \(n\in \mathbb {N}_k\) we let

$$\begin{aligned} \theta _n^k:=(1-2^{-k+n-1})\theta _* \qquad \text {and}\qquad \theta _{-1}^k:=1, \end{aligned}$$

and then for \(t\ge 1\)

$$\begin{aligned} l^k_{n}(t):=\Sigma _{j=0}^{n}\beta _j^k\,t^{\frac{1}{2s}-(2s)^j} \qquad \text {and}\qquad l_{-1}^k(t):=0 \end{aligned}$$

as well as

$$\begin{aligned} L^k_n(t,x):=\theta _{n-1}^k-(\theta _{n-1}^k-\theta _n^k)\frac{x-l^k_{n-1}(t)}{l^k_{n}(t)-l^k_{n-1}(t)}. \end{aligned}$$

Therefore, the function

$$\begin{aligned} {\hat{L}}^k(t,x):= L^k_n(t,x) \qquad \text { when }x\in ( l^k_{n-1}(t),l^k_{n}(t)]\text { for some }n\in \mathbb {N}_k \end{aligned}$$

is continuous on its domain \(\left\{ (t,x)\in \mathbb {R}^2 \,| \, t\ge 1 \text { and } x\in (0,l^k_{k}(t)]\right\} \), and piecewise linear in x for each fixed \(t\ge 1\).

Finally, with \(C_s\) from Lemma 3.1, let

$$\begin{aligned} c_*:=\max \left\{ C_s+ 2 {\Vert f\Vert _\infty }, (C_s\gamma _{0})^{\frac{1}{s}}, (C_s \gamma _{0} \gamma _{1}^{2s} )^2 \right\} , \end{aligned}$$
(3.5)

where

$$\begin{aligned} \gamma _{0}:=(4\theta _*^{-1})^{\frac{1}{s}} \qquad \text {and} \qquad \gamma _{1}:= 2^{1+\frac{1}{(1-2s)^2}}, \end{aligned}$$

and define the Lipschitz continuous function

$$\begin{aligned} \Phi ^k(t,x):=\left\{ \begin{aligned}&1{} & {} x\le c_* t^\frac{1}{2s},\\&{\hat{L}}^k(t,x-c_* t^\frac{1}{2s}){} & {} x\in ( c_* t^\frac{1}{2s},c_* t^\frac{1}{2s}+l^k_{k}(t)],\\&\left[ (\theta _k^k)^{-\frac{1}{2s}}+ c_* ^{-\frac{1}{2}}t^{-\frac{1}{2s}} \left( x-c_* t^\frac{1}{2s}-l^k_{k}(t) \right) \right] ^{-2s}{} & {} x> c_* t^\frac{1}{2s}+l^k_{k}(t). \end{aligned}\right. \nonumber \\ \end{aligned}$$
(3.6)

We now show that \(\Phi ^k\) is a supersolution to (1.2) on some (long for large k) time interval.

Theorem 3.3

Let f and s satisfy (I), and let \(\Phi ^k\) be from (3.6) for each \(k\in \mathbb {N}\). Then \(\Phi ^k\) is a supersolution to (1.2) on the time interval \((2^\frac{k}{1-2s},2^{(2s)^{-k}})\).

Proof

We write \(\Phi =\Phi ^k\) for simplicity. Let \(T_k:=2^{(2s)^{-k}}\) and fix any \(t\in [2^\frac{k}{1-2s},T_k]\). At any \(x< c_* t^\frac{1}{2\,s}\) we clearly have \(\Phi _t=0\le (-\partial _{xx})^s\Phi \), and so \(\Phi _t+(-\partial _{xx})^s\Phi -f(\Phi )\ge 0\) because \(f(1)=0\). By Remark 1 after Definition 2.3, it suffices to extend this claim to a.e. \(x> c_* t^\frac{1}{2s}\).

Next we claim that \(\Phi (t,\cdot )\) is convex on \([c_* t^\frac{1}{2\,s},\infty )\). The slope of \(L_n^k(t,\cdot )\) for \(n\in \mathbb {N}_{k}{\setminus }\{0\}\) is

$$\begin{aligned} -\frac{\theta _{n-1}^k-\theta _n^k}{l^k_{n}(t)-l^k_{n-1}(t)}=-\frac{2^{-k+n-2}\theta _*}{\beta _n^k\,t^{\frac{1}{2s}-(2s)^n}}, \end{aligned}$$

while for \(n=0\) it is \(-(1-\theta _0^k)t^{1-\frac{1}{2s}}\). Since \(t\ge 2^\frac{k}{1-2\,s}\) and \(t^{1-2\,s}\ge 1\ge \frac{\theta _*}{2(1-\theta _*)}\), Lemma 3.2 shows that \(\Phi (t,\cdot )\) is convex on \([c_* t^\frac{1}{2\,s},c_* t^\frac{1}{2\,s}+l^k_{k}(t)]\). It is clearly also convex on \([c_* t^\frac{1}{2\,s}+l^k_{k}(t),\infty )\), so we only need to check one-sided derivatives at \(y:=c_* t^\frac{1}{2s}+l^k_{k}(t)\). We have

$$\begin{aligned} \lim _{x\rightarrow y+}\Phi _x(t,x)=-2s(\theta ^k_k)^{1+\frac{1}{2s}}c_* ^{-\frac{1}{2}}t^{-\frac{1}{2s}}=-2s(2^{-1}\theta _*)^{1+\frac{1}{2s}}c_* ^{-\frac{1}{2}}t^{-\frac{1}{2s}}, \end{aligned}$$
(3.7)

as well as (using also (3.2))

$$\begin{aligned} \lim _{x\rightarrow y-}\Phi _x(t,x)=-\frac{\theta _{k-1}^k-\theta _k^k}{l^k_{k}(t)-l^k_{k-1}(t)}=-\frac{\theta _*}{4} (\beta _k^k)^{-1}\,t^{-\frac{1}{2s}+(2s)^k} \le -\frac{\theta _*}{4} t^{-\frac{1}{2s}}, \end{aligned}$$

which is no more than (3.7) because \(s<\frac{1}{2}\) and so \(c_* \ge 1\ge 16\,s^2(2^{-1}\theta _*)^\frac{1}{s}\). The claim follows.

Next consider any \(x\in ( c_* t^\frac{1}{2\,s},c_* t^\frac{1}{2\,s}+l^k_0(t))\). It follows from the definition of \(\Phi \) that

$$\begin{aligned} \Phi _t(t,x)=(1-\theta _0^k)(c_* + (1-2s)(2s)^{-1} xt^{-\frac{1}{2s}})\ge c_* (1-\theta _0^k). \end{aligned}$$

Lemma 3.1 with \(A_1:=c_* t^\frac{1}{2\,s}\), \(A_2:=c_* t^\frac{1}{2\,s}+l^k_0(t)\), \(\theta :=\theta ^k_0\), together with \(c_* \ge C_s+2{\Vert f\Vert _\infty }\), show that at such x we have

$$\begin{aligned} \Phi _t+(-\partial _{xx})^s\Phi -f(\Phi )\ge c_* (1-\theta _0^k)-C_s(1-\theta _0^k) t^{-1+2s}-{\Vert f\Vert _\infty }\ge 0 . \end{aligned}$$

Convexity of \(\Phi (t,\cdot )\) on \([c_* t^{\frac{1}{2}},\infty )\) shows that for any \(n\in \mathbb {N}_k\setminus \{0\}\) and any \(x\in ( c_* t^\frac{1}{2\,s}+l^k_{n-1}(t),c_* t^\frac{1}{2\,s}+l^k_{n}(t))\) we have

$$\begin{aligned} \Phi (t,\cdot )\le \psi _{c_* t^{1/2s},x,\Phi (t,x)} \end{aligned}$$

on \(\mathbb {R}\), and thus Lemma 3.1 and the definition of \(l^k_{n-1}(t)\) yield

$$\begin{aligned} (-\partial _{xx})^s \Phi (t,x)\ge -C_s(1-\theta _{n}^k)(x-c_* t^\frac{1}{2s})^{-2s} \ge -C_s(\beta _{n-1}^k)^{-2s}\,t^{-1+(2s)^{n}}.\nonumber \\ \end{aligned}$$
(3.8)

Since \(l^k_{n-1}(t)\) is increasing in \(t\ge 1\) and \(\beta _n^k\le 1\) by (3.2), we obtain

$$\begin{aligned} \Phi _t(t,x)&=\frac{d}{dt} L_n^k(t,x-c_* t^\frac{1}{2s})\\&\quad \ge 2^{-k+n-2}\theta _* (\beta _n^k)^{-1} \left[ - t^{-\frac{1}{2s}+(2s)^n}\partial _t(-c_* t^{\frac{1}{2s}})\right. \\&\quad \left. - (x-c_* t^\frac{1}{2s}-l^k_{n-1}(t)) \partial _t(t^{-\frac{1}{2s}+(2s)^n}) \right] \\&\quad \ge 2^{-k+n-2}\theta _*(\beta _n^k)^{-1} c_* ({2s})^{-1} t^{-1+(2s)^n}. \end{aligned}$$

From this, (3.8), (3.3), and \(c_* \ge \frac{4sC_s}{\theta _*}\) (due to \(c_* \ge (C_s\gamma _{0})^{\frac{1}{s}}\) and \(C_s\ge 1\)), we obtain \(\Phi _t+(-\partial _{xx})^s\Phi \ge 0\) at the x in question (notice that \(f(\Phi (t,x))=0\) for all \(x\ge c_* t^\frac{1}{2\,s}+l^k_0(t)\)).

Finally we need to consider any \(x> c_* t^\frac{1}{2s}+l^k_{k}(t)\), and this is the region where we will use that \(t\le T_k\). Since \(l^k_{k}(t)\) is increasing in \(t\ge 1\), with \(y_{t,x}:=x-c_* t^\frac{1}{2\,s}-l^k_{k}(t)\) we have

$$\begin{aligned} \Phi _t (t,x)&\ge \left[ (\theta _k^k)^{-\frac{1}{2s}}+c_* ^{-\frac{1}{2}}t^{-\frac{1}{2s}} y_{t,x}\right] ^{-1-2s} \left( c_* ^{-\frac{1}{2}}t^{-\frac{1}{2s}-1}y_{t,x}+c_* ^{\frac{1}{2}}t^{-1} \right) . \end{aligned}$$

Thus if \(c_* ^{-\frac{1}{2}}t^{-\frac{1}{2s}} y_{t,x} \le 1\), then \((\theta _k^k)^{-\frac{1}{2\,s}}+1 \le 2 (\frac{\theta _*}{2})^{-\frac{1}{2\,s}}\) and so by \(s<\frac{1}{2}\) we have

$$\begin{aligned} \Phi _t (t,x) \ge 2^{-\frac{(1+2s)^2}{2s}}\theta _*^{\frac{1+2s}{2s}} c_* ^{\frac{1}{2}} t^{-1} \ge \gamma _0^{-1} c_* ^{\frac{1}{2}} t^{-1}. \end{aligned}$$
(3.9)

And if \(c_* ^{-\frac{1}{2}}t^{-\frac{1}{2s}} y_{t,x}\ge 1\), we obtain

$$\begin{aligned} \Phi _t (t,x){} & {} \ge \left[ (\theta _k^k)^{-\frac{1}{2s}}+1\right] ^{-1-2s} \left( c_* ^{-\frac{1}{2}}t^{-\frac{1}{2s}}y_{t,x} \right) ^{-1-2s}c_* ^{-\frac{1}{2}} t^{-\frac{1}{2s}-1}y_{t,x}\nonumber \\{} & {} \ge \gamma _0^{-1} c_* ^{s} y_{t,x}^{-2s}. \end{aligned}$$
(3.10)

Since \(\Phi (t,\cdot )\) is convex on \([c_* t^\frac{1}{2s},\infty )\), Lemma 3.1 shows that

$$\begin{aligned} (-\partial _{xx})^s\Phi (t,x)\ge -C_s(x-c_* t^\frac{1}{2s})^{-2s}. \end{aligned}$$
(3.11)

Therefore, at all \(x> c_* t^\frac{1}{2s}+l^k_{k}(t)\) such that \(c_* ^{-\frac{1}{2}}t^{-\frac{1}{2\,s}} y_{t,x} \ge 1\) we have

$$\begin{aligned} \Phi _t+(-\partial _{xx})^s\Phi \ge (\gamma _0^{-1} c_* ^{s}-C_s) y_{t,x}^{-2s} \ge 0 \end{aligned}$$

by (3.10) and (3.5). If instead \(c_* ^{-\frac{1}{2}}t^{-\frac{1}{2\,s}} y_{t,x} \le 1\), we note that \(t\le T_k\) implies \(t^{(2s)^k}\le 2\), so

$$\begin{aligned} l^k_{k}(t)\ge \beta _k^k\,t^{\frac{1}{2s}-(2s)^k}\ge 2^{-\frac{1}{(1-2s)^2}-1}t^\frac{1}{2s}=\gamma _1^{-1}t^\frac{1}{2s} \end{aligned}$$

by (3.1). Hence (3.11) yields

$$\begin{aligned} (-\partial _{xx})^s\Phi (t,x)\ge -C_s l^k_{k}(t)^{-2s}\ge -C_s \gamma _1^{2s}t^{-1}, \end{aligned}$$

and then (3.9) and (3.5) again show that \(\Phi _t+(-\partial _{xx})^s\Phi \ge 0\) at such x. Therefore, \(\Phi \) is indeed a supersolution to (1.2) on the time interval \((2^\frac{k}{1-2s},T_k)\). \(\quad \square \)

We can now use the supersolutions from Theorem 3.3 to obtain an upper bound for general solutions to (1.2).

Theorem 3.4

Let f and s satisfy (I), and let \(0\le u\le 1\) solve (1.2). If

$$\begin{aligned} u(0,x)\le Ax^{-2s} \end{aligned}$$

for some \(A\ge 1\) and all \(x>0\), then for each \(\lambda \in (0,1)\) there is \(C_{\lambda ,A}>0\) (depending also on s, f) such that for all \(t\ge 0\) we have

$$\begin{aligned} {{\overline{x}}}_\lambda (t;u)\le C_{\lambda ,A}(1+t)^\frac{1}{2s}. \end{aligned}$$

Remark

It is easy to see from this that one also has \({{\overline{x}}}_\lambda (t;u)\le C_{\lambda }t^\frac{1}{2s}\) for \(t\ge \tau _{\lambda , A}\), with some \(\tau _{\lambda , A}=\tau _{\lambda , A}(s,f)\) but \(C_{\lambda }=C_{\lambda }(s,f)\) independent of A.

Proof

Let \(k_0\) be the smallest positive integer such that \({(2s)^{-k}}\ge {\frac{k+1}{1-2s}+1}\) for all \(k\ge k_0\). Then we have

$$\begin{aligned} \bigcup _{k\ge k_0}\left[ 2^{\frac{k}{1-2s}},2^{(2s)^{-k}}-2^{\frac{k}{1-2s}}\right] =[2^{\frac{k_0}{1-2s}},\infty ). \end{aligned}$$
(3.12)

Next let \(k_1\ge k_0\) be such that

$$\begin{aligned} 2^{\frac{k_1}{1-2s}} \left( 2^{\frac{1}{2s}} \theta _*^{-\frac{1}{2s}} c_* ^{-1} +c_* ^{-\frac{1}{2}} \right) ^{-2s}\ge A. \end{aligned}$$

It follows from (3.12) that for any \(T\ge 2^{\frac{k_1}{1-2\,s}}\), there is \(k\ge k_1\) such that

$$\begin{aligned} 2^{\frac{k}{1-2s}}\le T\le 2^{(2s)^{-k}}-2^{\frac{k}{1-2s}}. \end{aligned}$$

Fix this T and k, and let \(\Phi ^k\) be from (3.6). Then \(\Phi ^k(2^{\frac{k}{1-2\,s}},\cdot )\equiv 1\) on \((-\infty ,c_* 2^{\frac{k}{(1-2\,s)2\,s}}]\), while \(\Phi ^k\) being non-increasing and \(\theta _k^k=\frac{\theta _*}{2}\) show that for \(x> c_* 2^{\frac{k}{(1-2\,s)2\,s}}\) we have

$$\begin{aligned} \Phi ^k(2^{\frac{k}{1-2s}},x)\ge & {} \Phi ^k(2^{\frac{k}{1-2s}},x+l_k^k(2^{\frac{k}{1-2s}}))\ge \left( 2^{\frac{1}{2s}} \theta _*^{-\frac{1}{2s}}+ c_* ^{-\frac{1}{2}} 2^{-\frac{k}{(1-2s)2s}} x \right) ^{-2s}\\\ge & {} Ax^{-2s}. \end{aligned}$$

Thus we have \(\Phi ^k(2^{\frac{k}{1-2s}},\cdot )\ge u(0,\cdot )\) on \(\mathbb {R}\). So if we let \(\varphi (t,\cdot )\!:=\!\Phi ^k(t\!+\!2^{\frac{k}{1-2s}},\cdot )\), then \(\varphi \) is a supersolution to (1.2) on the time interval [0, T] with \(\varphi (0,\cdot )\ge u(0,\cdot )\). It now follows from the comparison principle (Theorem 2.4; see also Remark 2 after Definition 2.3) that \(u\le \phi \) on \([0,T]\times \mathbb {R}\).

Since \(l_k^k(t)\le 2t^\frac{1}{2s}\) by (3.2), from (3.6) we obtain for any \(\lambda \in (0,\frac{\theta _*}{2})\),

$$\begin{aligned} {{\overline{x}}}_\lambda (t;\Phi ^k)\le c_* t^\frac{1}{2s}+l_k^k(t)+ \left( \lambda ^{-\frac{1}{2s}}- ( \theta _k^k)^{-\frac{1}{2s}} \right) c_* ^\frac{1}{2}t^\frac{1}{2s} \le \left( c_* +2 + \sqrt{c_* } \lambda ^{-\frac{1}{2s}} \right) t^\frac{1}{2s} \end{aligned}$$

for all \(t\ge 1\). This and \(T\ge 2^{\frac{k}{1-2s}}\) show that for all \(t\in [0,T]\) we have

$$\begin{aligned} {{\overline{x}}}_\lambda (T;u)\le {{\overline{x}}}_\lambda (T+2^{\frac{k}{1-2s}};\Phi ^k) \le \left( c_* +2 + \sqrt{c_* } \lambda ^{-\frac{1}{2s}} \right) 2^\frac{1}{2s} T^\frac{1}{2s}. \end{aligned}$$

Since \(T\ge 2^{\frac{k_1}{1-2s}}\) was arbitrary, the result now clearly follows for any \(\lambda \in (0,1)\), with \(C_{\lambda ,A}\) depending also on \(s,K,\theta _0\) (since \(c_* \) and \(k_0\) depend on them). \(\quad \square \)

4 Subsolutions for Ignition Reactions

In this section we prove the lower bound in Theorem 1.3(i), which then automatically provides the same bound in Theorem 1.2(i) via the comparison principle. We do so by constructing appropriate subsolutions to (1.1) in the following counterpart to Theorem 3.4:

Theorem 4.1

Let f and s satisfy (I), and let \(0\le u\le 1\) solve (1.1). If

$$\begin{aligned} u(0,\cdot ) \ge \theta \chi _{B_{R_\theta }(0)} \end{aligned}$$

for some \(\theta >\theta _0\) and \(R_\theta \) from Lemma 2.6, then for each \(\lambda \in (0,1)\) there are \(C_{\lambda }, \tau _{\lambda ,\theta }>0\) (depending also on s, f, d) such that for all \(t\ge \tau _{\lambda ,\theta }\) we have

$$\begin{aligned} {{\underline{x}}}_\lambda (t;u)\ge C_{\lambda }\,t^\frac{1}{2s}. \end{aligned}$$

Proof

The comparison principle and Lemma 2.6 show that it suffices to prove the result with \(C_\lambda \) also depending on \(\theta \), which we will do.

Let \(u_\theta , {\bar{u}}_\theta \) be from Lemma 2.6, and let \(L:=\Vert u_\theta '\Vert _\infty <\infty \) and \(\textrm{supp}\,u_\theta =(-\infty ,a]\) (so \(a\in (0,R_\theta ]\)). By (2.10), there is \(\varepsilon >0\) such that for all \(x\le a\) we have

$$\begin{aligned} (-\Delta )^s {\bar{u}}_\theta (x)-f({\bar{u}}_\theta (x))\le -\varepsilon . \end{aligned}$$
(4.1)

Next let

$$\begin{aligned} b:= ((2s\varepsilon )^{-1} L a)^{\frac{1}{2s}} \qquad \text {and}\qquad \Psi (t,x):= {\bar{u}}_\theta (b t^{-\frac{1}{2s}}x). \end{aligned}$$

Since \(\Psi (t,\cdot )\ge 0\) vanishes on \(B_{b^{-1}a t^{1/2s}} (0)\) for each \(t>0\), we have

$$\begin{aligned} \Psi _t (t,\cdot )+ (-\Delta )^s\Psi (t,\cdot )-f(\Psi (t,\cdot ))\le 0 \end{aligned}$$

there. From (4.1) and \(f\ge 0\) we obtain for any \(t\ge b^{2\,s}\) and \(|x|\le b^{-1}a t^\frac{1}{2\,s}\),

$$\begin{aligned}&\Psi _t (t,x)+ (-\Delta )^s\Psi (t,x)-f(\Psi (t,x))\\&\quad \le (2s)^{-1} b t^{-\frac{1}{2s}-1} |x| \, \Vert u_\theta '\Vert _\infty + b^{2s}t^{-1} \left[ (-\Delta )^s {\bar{u}}_\theta (bt^{-\frac{1}{2s}}x) - f \left( {\bar{u}}_\theta (bt^{-\frac{1}{2s}}x) \right) \right] \\&\quad \le \left( (2s)^{-1} L a -b^{2s}\varepsilon \right) t^{-1}, \end{aligned}$$

which is \(\le 0\) by the definition of b. Hence \(\Psi \) is a subsolution to (1.1) on time interval \((b^{2s},\infty )\).

Since clearly \( u(0,\cdot )\ge {\bar{u}}_\theta =\Psi (b^{2\,s},\cdot ), \) the comparison principle (Theorem 2.4) yields

$$\begin{aligned} u(t,\cdot )\ge \Psi (t+b^{2s},\cdot ) \end{aligned}$$

for all \(t\ge 0\). Hence for each \(\lambda \in (0,\theta )\) and \(t\ge 0\) we have

$$\begin{aligned} {{\underline{x}}}_\lambda (t;u)\ge {{\underline{x}}}_\lambda (t;\Psi (\cdot +b^{2s},\cdot ))= C_{\lambda ,\theta } b^{-1}(t+b^{2s})^\frac{1}{2s}, \end{aligned}$$
(4.2)

where \(C_{\lambda ,\theta }\in (0,R_\theta )\) is such that \(u_\theta (C_{\lambda ,\theta })=\lambda \). The claim now follows for each \(\lambda \in (0,\theta )\).

Moreover, it follows from (4.2) that there are \(\tau ',C'>0\) such that for all \(t\ge \tau '\) we have

$$\begin{aligned} \inf _{|x|\le C't^{1/2s}}u(t,x)\ge \frac{\theta +\theta _0}{2} \end{aligned}$$

Then the last claim in Lemma 2.6 shows that for any \(\lambda \in [ \theta ,1)\) there is \(\tau >0\) such that

$$\begin{aligned} \inf _{|x|\le C't^{1/2s}} u(t+\tau ,x)\ge \lambda \end{aligned}$$

for all \(t\ge \tau '\). It follows that for all \(t\ge \tau +\tau '\) we have

$$\begin{aligned} {{\underline{x}}}_\lambda (t;u)\ge C'(t-\tau )^\frac{1}{2s} \end{aligned}$$

(with \(C',\tau ',\tau \) depending on \(s,f,d,\lambda ,\theta \)), which proves the claim for \(\lambda \in [ \theta ,1)\). \(\quad \square \)

5 Subsolutions for Monostable Reactions

In this section we prove Theorems 1.2(ii) and 1.3(ii), so we will assume the following:

  1. (M)

    Let f be an \(\alpha \)-monostable reaction for some \(\alpha >1\), and let \(s\in (0,\frac{\alpha }{2(\alpha -1)})\).

The relevant upper bound on \({{\overline{x}}}_\lambda (t;u)\) for front-like data was already obtained in [15] (for \(\alpha \)-monostable reactions \({\tilde{f}}\) that are also \(C^1\) and have \({\tilde{f}}'(1)<0\), but there always exists such \({\tilde{f}}\ge f\), so the same bound for f follows by the comparison principle). This immediately provides the bounds in Theorems 1.2(ii) and 1.3(ii) via the comparison principle.

Hence it remains to show that for any \(\lambda \in (0,1)\) and all large t we have

$$\begin{aligned} {{\underline{x}}}_\lambda (t;u)\ge C t^{\frac{\alpha }{2s(\alpha -1)}}. \end{aligned}$$
(5.1)

It suffices to do this in the setting of Theorem 1.3(ii), because then the same bound in Theorem 1.2(ii) follows via the comparison principle. We will do this by constructing appropriate subsolutions to (1.1).

Let \(\alpha ,\theta _0,\gamma \) be from Definition  1.1(ii) for f, and let

$$\begin{aligned} \beta :=(d+2s)(\alpha -1) \qquad \text {and}\qquad \kappa := \frac{\beta \alpha }{2s(\alpha -1)} \quad (> \beta ) \end{aligned}$$

(with the inequality due to (M)). Fix any \(\theta \in (0,1)\), let

$$\begin{aligned} \theta _1:=\min \left\{ \theta _0, \frac{\theta }{2} \right\} \qquad \text {and}\qquad \theta _2:=\frac{1+\theta }{2}. \end{aligned}$$

Then let

$$\begin{aligned} \nu :=\alpha -1 \qquad \text {and}\qquad \tau :=\min \left\{ \tau _0,\,\frac{c_* }{2^{(d+2s)/\beta }C_*\theta _1}\right\} , \end{aligned}$$
(5.2)

where \(\tau _0,c_*, C_*\) are from Lemma 5.1 below with \(\beta ,\nu ,\theta _1\) as above (so \(\tau _0,c_*, C_*\) are independent from a, b in the lemma). Next let

$$\begin{aligned} \delta :=\inf _{u\in [\tau \theta _1,\theta ]} f(u)>0, \end{aligned}$$

define

$$\begin{aligned} a_3:=\min \left\{ 1,\,\frac{(\alpha -1) \gamma }{2\kappa -1},\,\frac{(\alpha -1) \delta }{4\kappa \theta _2^\alpha }\right\} , \end{aligned}$$
(5.3)

and then

$$\begin{aligned} a_1:=\left( \frac{(\alpha -1)c_* }{2^{1+(d+2s)/\beta }(2\kappa -1)}\right) ^{\frac{\beta }{2s}}{a_3^\frac{d}{2s}} \qquad \text {and}\qquad a_2:=a_1a_3. \end{aligned}$$
(5.4)

Now let \(\phi _\theta :[0,1]\rightarrow [0,\theta ]\) be smooth and such that

$$\begin{aligned} \phi _\theta (y)=y \, \text { on } [0,\theta _1]\qquad \text {and}\qquad \phi _\theta (y)=\theta \, \text { on }[\theta _2,1], \end{aligned}$$
(5.5)

as well as for some \(C_\theta >0\) we have

$$\begin{aligned} 0\le \phi _\theta '\le 1 \qquad \text {and}\qquad -C_\theta \le \phi _\theta ''\le 0. \end{aligned}$$
(5.6)

Finally, define

$$\begin{aligned} \psi _\theta (t,r):= ( a_1^{-1} t^{1-\kappa }(r^{\beta }- a_2t^\kappa ))^{-\frac{1}{\alpha -1}} \end{aligned}$$
(5.7)

for \( t>0\) and \(r> ( a_2t^\kappa )^{\frac{1}{\beta }}\), and then let

$$\begin{aligned} \Psi _\theta (t,x):= {\left\{ \begin{array}{ll} \phi _\theta \left( \psi _\theta (t,|x|)\right) &{}|x|\ge ( \theta _2^{1-\alpha }a_1t^{\kappa -1}+ a_2t^\kappa )^{\frac{1}{\beta }},\\ \theta &{}\text {otherwise. } \end{array}\right. } \end{aligned}$$

For any \(u\in (0,\theta _2]\), we also let

$$\begin{aligned} X_t(u):=(u^{1-\alpha }a_1 t^{\kappa -1}+ a_2t^\kappa )^\frac{1}{\beta } \end{aligned}$$

(hence \(\psi _\theta (t,X_t(u))=u\)). This construction shows that \(\Psi _\theta \) is a smooth function. We will now show that it is also a subsolution to (1.1) at all large times. We note that since we may have \(s>\frac{1}{2}\) here, this would be difficult if graphs of \(\Psi _\theta (t,\cdot )\) (as functions of x) had “concave” corners, which is the reason for the introduction of the function \(\phi _\theta \) above.

We start with a technical lemma, whose proof is easy when \(d=1\), but somewhat more involved when \(d>1\). We postpone the proof to the appendix.

Lemma 5.1

Let \(\theta _1\in (0,1]\) and \(\beta>\nu >0\) be such that \(\frac{\beta }{\nu }\ge d-2\). Let \(0<a\le 1\le b\), let \(X(u):=( a^{-1}(u^{-{\nu }}+b))^{\frac{1}{\beta }}\) for \(u>0\), and let \(\varphi :\mathbb {R}^d\rightarrow [0,1]\) be smooth and such that

$$\begin{aligned} \varphi (x)= ( a |x|^{\beta }- b)^{-\frac{1}{\nu }}\qquad \text { when } |x|\ge X(\theta _1) \end{aligned}$$

as well as \(\varphi (x)\ge \theta _1\) when \(|x|\le X(\theta _1)\). Then for any \(s\in (0,1)\), there are \(c_*,C_*>0\) and \(\tau _0\in (0,\frac{1}{4}]\) (depending only on \(s,\beta ,\nu ,\theta _1,d\)) such that for all \(|x|\ge X(\tau _0\theta _1)\) we have

$$\begin{aligned} (-\Delta )^s\varphi (x)\le -c_* X(\theta _1)^d |x|^{-d-2s}+ C_*|x|^{-2s} \varphi (x). \end{aligned}$$

We are now ready to construct a localized subsolution for (1.1) with monostable f.

Theorem 5.2

Let f and s satisfy (M). Then for any \(\theta \in (0,1)\), there is \(T_\theta \ge 1\) such that \(\Psi _\theta \) above is a subsolution to (1.1) on the time interval \([T_\theta ,\infty )\).

Proof

We drop \(\theta \) from \(\phi _\theta ,\psi _\theta ,\Psi _\theta ,T_\theta ,C_\theta \) in the proof. The desired T we obtain here will depend on the various constants in the above setup, and we will always assume that \(t\ge T \ge 1\).

Let us start with estimating \(| \psi _r|\) from above. Since \(\psi _{rr}>0>\psi _r\), we see that for all \(r\ge X_t(\theta _2)\) we have

$$\begin{aligned} |\psi _r(t,r)|\le -\psi _{r}(t,X_t(\theta _2))=\frac{\beta }{\alpha -1} a_1^{-1}t^{1-\kappa } \theta _2^\alpha X_t(\theta _2)^{\beta -1} \end{aligned}$$

When \(\beta <1\), from \(\theta _2<1\) and \( a_1\ge a_2\) we obtain

$$\begin{aligned} -\psi _{r}(t,X_t(\theta _2)) \le \frac{\beta }{\alpha -1} a_1^{-1}t^{1-\kappa } ( a_2t^\kappa )^\frac{\beta -1}{\beta } \le \frac{\beta }{\alpha -1} a_2^{-\frac{1}{\beta }}t^{\frac{\beta -\kappa }{\beta }}. \end{aligned}$$

When \(\beta \ge 1\), we again use \(\theta _2<1\) to get

$$\begin{aligned} \theta _2^\alpha X_t(\theta _2)^{\beta -1}= & {} \theta _2^{\frac{\alpha -1+\beta }{\beta }} ( a_1t^{\kappa -1}+ \theta _2^{\alpha -1} a_2t^\kappa )^\frac{\beta -1}{\beta } \le ( a_1t^{\kappa -1}+ a_2t^\kappa )^\frac{\beta -1}{\beta } \\\le & {} ( a_1t^{\kappa -1})^\frac{\beta -1}{\beta } + ( a_2t^\kappa )^\frac{\beta -1}{\beta }, \end{aligned}$$

so then \( a_2\le a_1\) and \(t\ge 1\) yield

$$\begin{aligned} -\psi _{r}(t,X_t(\theta _2)) \le \frac{\beta }{\alpha -1}\left( a_1^{-\frac{1}{\beta }}t^{\frac{1-\kappa }{\beta }}+ a_1^{-1} a_2^\frac{\beta -1}{\beta }t^{\frac{\beta -\kappa }{\beta }}\right) \le \frac{2\beta }{\alpha -1} a_2^{-\frac{1}{\beta }}t^{\frac{\beta -\kappa }{\beta }}. \end{aligned}$$

It follows that if T is such that \(\frac{2\beta }{\alpha -1} a_2^{-\frac{1}{\beta }}T^{\frac{\beta -\kappa }{\beta }}\le 1\) (recall that \(\kappa >\beta \)) and \(X_T(\theta _2)\ge 1\), then (5.6) and \(\psi _{rr}>0\) yield for any \(e\in \mathbb {S}^{d-1}\) and \(\rho _t:=\frac{2\beta }{\alpha -1} a_2^{-\frac{1}{\beta }}t^{\frac{\beta -\kappa }{\beta }}\le 1\),

$$\begin{aligned} \begin{aligned} D_{ee}^2\Psi (t,x)&=\frac{(x\cdot e)^2}{|x|^2}\phi '' \big (\psi (t,|x|) \big )|\psi _r(t,|x|)|^2+\frac{(x\cdot e)^2}{|x|^2}\phi ' \big (\psi (t,|x|) \big )\psi _{rr}(t,|x|)\\&\quad +\phi ' \big (\psi (t,|x|) \big )\psi _r(t,|x|) \frac{|x|^2-(x\cdot e)^2}{|x|^3} \ge -C \rho _t^2-\rho _t\ge -(C+1)\rho _t, \end{aligned} \end{aligned}$$

where we also used that \(\phi ' (\psi (t,|x|) ) =0\) when \(|x|\le 1\) (due to \(X_t(\theta _2)\ge 1\)). From this, \(\theta <1\), and \(s\in (0,1)\) we obtain (with \(\omega _d\) be the surface area of \(\mathbb {S}^{d-1}\))

$$\begin{aligned} \sup _{x\in \mathbb {R}^d}\, (-\Delta )^s\Psi (t,x)\le & {} \frac{c_{s,d}}{2}\int _{|h|\le \rho _t^{-1/2}} \frac{2\Psi (t,x)-\Psi (t,x+h)-\Psi (t,x-h)}{|h|^{d+2s}}dh\nonumber \\{} & {} + c_{s,d} \int _{|h|>\rho _t^{-1/2}} \frac{\Psi (t,x)-\Psi (t,x+h)}{|h|^{d+2s}} dh \nonumber \\\le & {} \frac{c_{s,d}}{2}\int _{|h|\le \rho _t^{-1/2}} \frac{(C+1) \rho _t}{|h|^{d-2+2s}}dh + c_{s,d} \int _{|h|>\rho _t^{-1/2}} \frac{\theta }{|h|^{d+2s}} dh\nonumber \\\le & {} \frac{c_{s,d}\omega _d (C+1)\rho _t^{s}}{2s(1-s)}=:C'\rho _t^s. \end{aligned}$$
(5.8)

We will need another estimate for (t, x) such that \(\Psi (t,x)\) is small. When \(\Psi (t,x)\le \tau \theta _1\) (then \(\Psi (t,x)=\psi (t,x)\) because \(\tau \le \tau _0\le 1\)), we can apply Lemma 5.1 with \(\varphi (\cdot )= \Psi (t,\cdot )\) and \(\beta ,\nu ,\theta _1\) as above, provided T is large enough so that \(a:= a_1^{-1}t^{1-\kappa }<1\) and \(b:=a_2a_1^{-1}t\ge 1\). So when \(\Psi (t,x)\le \tau \theta _1\), then we have

$$\begin{aligned} (-\Delta )^s \Psi (t,x)\le -c_* |x|^{-d-2s}X_t(\theta _1)^d+ C_* |x|^{-2s} \psi (t,x). \end{aligned}$$
(5.9)

Finally, we also note that \(\Psi _t(t,x)=0\) when \(|x|\le X_t(\theta _2)\), while for \(|x|>X_t(\theta _2)\) we have

$$\begin{aligned} \Psi _t(t,x)= & {} \phi '(\psi (t,x))\frac{\psi (t,x)^\alpha a_1^{-1}}{\alpha -1} \left( (\kappa -1)t^{-\kappa }|x|^{\beta }+ a_2\right) \nonumber \\\le & {} \frac{\psi (t,x)^\alpha a_1^{-1}}{\alpha -1} \left( (\kappa -1) a_1t^{-1} \psi (t,x)^{1-\alpha }+\kappa a_2 \right) \nonumber \\= & {} \frac{\kappa -1}{\alpha -1} \, \frac{\psi (t,x)}{t}+\frac{ a_2 \kappa }{ a_1(\alpha -1)} \, \psi (t,x)^{\alpha }. \end{aligned}$$
(5.10)

We are now ready to show that \(\Psi _t+(-\Delta )^s\Psi -f(\Psi )\le 0\) at all (t, x) with \(t\ge T\) (if T is large enough). When \(|x|\le X_t(\theta _2)\), then (5.8) and \(f(\Psi (t,x))=f(\theta )\ge \delta \) show that this follows from \(C' \rho _t^s\le {\delta }\), which holds if T is large. Since (5.4) and (5.3) also yield

$$\begin{aligned} \max \left\{ \frac{\kappa -1}{\alpha -1} \frac{\theta _2}{t},\, \frac{ a_2\kappa }{ a_1(\alpha -1)} \theta _2^\alpha ,\, C' \rho _t^s \right\} \le \frac{\delta }{3} \end{aligned}$$

if T is large, and since for \(|x|\in (X_t(\theta _2), X_t(\tau \theta _1)]\) we still have \(f(\Psi (t,x))\ge \delta \), it follows from (5.8) and (5.10) that for these x we again have

$$\begin{aligned} \Psi _t(t,x)+ (-\Delta )^s\Psi (t,x) -f(\Psi (t,x))\le & {} \frac{\kappa -1}{\alpha -1}\frac{\theta _2}{t} + \frac{ a_2\kappa }{ a_1(\alpha -1)} \theta _2^\alpha +C' \rho _t^{s} -\delta \\\le & {} 0. \end{aligned}$$

It therefore remains to consider \(|x|>X_t(\tau \theta _1)\) (when \(\Psi (t,x)= \psi (t,x)\) because \(\tau \le 1\)). If also \(t \Psi (t,x)^{\alpha -1}\le \frac{ a_1}{ a_2}\), it follows from (5.9), (5.10), \(|x|=(\psi (t,x)^{1-\alpha } a_1t^{\kappa -1}+ a_2t^\kappa )^{-\frac{1}{\beta }}\), and \(X_t(\theta _1)\ge (a_2t^\kappa )^\frac{1}{\beta }\) that (we drop (t, x) from the notation for simplicity)

$$\begin{aligned} \Psi _t+ (-\Delta )^s\Psi -f(\Psi )&\le \frac{\kappa -1}{\alpha -1}\frac{\psi }{t}+\frac{ a_2\kappa }{ a_1(\alpha -1)} \psi ^{\alpha }-c_* |x|^{-d-2s}X_t(\theta _1)^d \\&\quad +C_* |x|^{-2s} \psi \\&\le \frac{2\kappa -1}{\alpha -1}\frac{\psi }{t}-c_* (2\psi ^{1-\alpha } a_1t^{\kappa -1})^{-\frac{d+2s}{\beta }}(a_2t^\kappa )^\frac{d}{\beta }\\&\quad +C_*(\psi ^{1-\alpha } a_1t^{\kappa -1})^{-\frac{2s}{\beta }} \psi \\&= \frac{2\kappa -1}{\alpha -1} \frac{\psi }{t}-c_* (2 a_1)^{-\frac{d+2s}{\beta }}a_2^\frac{d}{\beta }\frac{\psi }{t}\\&\quad +C_* a_1^{-\frac{2s}{\beta }} t^{-\frac{(\kappa -1)2s}{\beta }} \psi ^{1+\frac{(\alpha -1)2s}{\beta }}. \end{aligned}$$

Using again \(t\psi ^{\alpha -1}\le \frac{a_1}{a_2}\), we obtain

$$\begin{aligned} t^{-\frac{(\kappa -1)2s}{\beta }} \psi ^{1+\frac{(\alpha -1)2s}{\beta }} = t^{\frac{2s}{\beta }} \psi ^{\frac{(\alpha -1)2s}{\beta }} t^{-\frac{\alpha }{\alpha -1}} \psi \le (a_1a_2^{-1})^\frac{2s}{\beta } t^{-\frac{1}{\alpha -1}}\frac{\psi }{t}. \end{aligned}$$

Therefore \(\Psi _t+ (-\Delta )^s\Psi -f(\Psi )\le 0\) by (5.4) if T is large enough.

When instead \(|x|>X_t(\tau \theta _1)\) and \(t \Psi (t,x)^{\alpha -1}> \frac{ a_1}{ a_2}\), then

$$\begin{aligned} (a_2t^\kappa )^\frac{1}{\beta } \le |x|=(\psi (t,x)^{1-\alpha } a_1t^{\kappa -1}+ a_2t^\kappa )^{-\frac{1}{\beta }} \le (2a_2t^\kappa )^\frac{1}{\beta }, \end{aligned}$$

so (5.9), \(X_t(\theta _1)\ge (a_2t^\kappa )^\frac{1}{\beta }\), \(\psi (t,x) \le \tau \theta _1\) (due to \(|x|>X_t(\tau \theta _1)\)), and (5.2) show that

$$\begin{aligned} (-\Delta )^s\Psi&\le -c_* (2a_2t^\kappa )^{-\frac{d+2s}{\beta }}(a_2t^\kappa )^\frac{d}{\beta }+ C_* (a_2t^\kappa )^{-\frac{2s}{\beta }} \psi \\&\le \left( C_* \tau \theta _1 -c_* 2^{-\frac{d+2s}{\beta }} \right) a_2^{-\frac{2s}{\beta }}t^{-\frac{2s\kappa }{\beta }} \le 0. \end{aligned}$$

This, (5.10), \(\frac{\psi (t,x)}{t}<\frac{a_2}{a_1}\psi (t,x)^\alpha \), and (M) show that

$$\begin{aligned} \Psi _t+ (-\Delta )^s\Psi -f(\Psi )&\le \frac{ a_2(2\kappa -1)}{ a_1(\alpha -1)} \psi ^{\alpha }-\gamma \psi ^\alpha \end{aligned}$$

which is again \(\le 0\) due to (5.3). This finishes the proof. \(\quad \square \)

We can now use the constructed subsolutions to prove (5.1).

Theorem 5.3

Let f and s satisfy (M), and let \(0\le u\le 1\) solve (1.1). If

$$\begin{aligned} u(0,\cdot ) \ge \theta \chi _{B_{R_\theta }(0)} \end{aligned}$$

for some \(\theta > 0\) and \(R_\theta \) from Lemma 2.6, then for each \(\lambda \in (0,1)\) there are \(C_{\lambda },\tau _{\lambda ,\theta }>0\) (depending also on s, f, d) such that for all \(t\ge \tau _{\lambda ,\theta }\) we have

$$\begin{aligned} {{\underline{x}}}_\lambda (t;u)\ge C_{\lambda }t^{\frac{\alpha }{2s(\alpha -1)}}. \end{aligned}$$

Proof

The comparison principle and Lemma 2.6 show that it suffices to prove the result with \(C_\lambda \) also depending on \(\theta \), which we will do.

Let \({\bar{u}}_\theta \) be from Lemma 2.6 (see the remark after that lemma), let \({\bar{u}}\) be the solution to (1.1) with initial data \({\bar{u}}_\theta \), and let \(t_0\) be such that \({\bar{u}}(t_0,\cdot )\ge \theta \chi _{B_1(0)}\). Since \(u_0\ge {\bar{u}}_\theta \), we have \(u(t_0,\cdot )\ge {\bar{u}}(t_0,\cdot )\) by the comparison principle (Theorem 2.4), and then comparison principle shows that it suffices to consider \(u_0={\bar{u}}(t_0,\cdot )\) without loss. The proof of Lemma 2.6 now shows that u is time increasing.

Similarly to [15, Theorem 3.1], since u dominates the solution to \(v_t+(-\partial _{xx})^s v=0\) with initial data \(\theta \chi _{B_{1}(0)}\), there is \(C>0\) (depending only on s, d) such that if \(t\ge 1 \) and \(|x|\ge t^\frac{1}{2\,s}+1\), then

$$\begin{aligned}&u(t,x)\ge C\theta \int _{B_{1}(x)} t^{-\frac{d}{2s}}(1+|t^{-\frac{1}{2s}}y|^{d+2s})^{-1}dy \\&\quad \ge \frac{C\theta \omega _d}{2d}\, t(|x|+1)^{-d-2s} \ge c\, t|x|^{-d-2s}, \end{aligned}$$

where \(\frac{\omega _d}{d}\) is the volume of \(B_1(0)\) and \(c:= 2^{-d-2\,s-1}d^{-1}C\theta \omega _d\).

Now let \(\Psi _\theta ,T_\theta \) be from Theorem 5.2 and let \({T}:=1+ c^{-1}a_1^{\frac{1}{\alpha -1}}T_\theta ^\frac{\kappa -1}{\alpha -1}\). If |x| is large enough, we then have \({u}({T},x)\ge \Psi _\theta (T_\theta ,x)\), which then yields \({u}({t},x)\ge \Psi _\theta (T_\theta ,x)\) for all these x and all \(t\ge T\) because u increases in time. But we also have \({u}({t},x)\ge \Psi _\theta (T_\theta ,x)\) for all the other x and some t by the last claim in Lemma 2.6. Hence there is \({T}'\ge {T}\) such that \({u}({T}',\cdot )\ge \Psi _\theta (T_\theta ,\cdot )\). Comparison principle now yields

$$\begin{aligned} {u}(t+{T'},\cdot )\ge \Psi _\theta (t+T_\theta ,\cdot ) \end{aligned}$$

for all \(t\ge 0\), so for any \(\lambda \in (0,\theta )\) and \(t\ge {T'}\) we have

$$\begin{aligned} {{\underline{x}}}_\lambda (t;u) \ge {{\underline{x}}}_\lambda (t-{T'}+T_\theta ;\Psi _\theta )\ge C_{\lambda ,\theta } (t-{T'}+T_\theta )^{\frac{\alpha }{2s(\alpha -1)}} \end{aligned}$$

for some time-independent \(C_{\lambda ,\theta }>0\). This proves the claim for each \(\lambda \in (0,\theta )\), and for \(\lambda \in [\theta ,1)\) it now follows as at the end of the proof of Theorem 4.1. \(\quad \square \)