# On the Speed of Spread for Fractional Reaction-Diffusion Equations

Authors: Hans Engler
Journal: International Journal of Differential Equations
 (2010)

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## Abstract

The fractional reaction diffusion equation∂tu+Au=g(u)
is discussed, where A is a fractional differential operator on ℝ of order
α∈(0,2), the C1 function g vanishes at ζ=0 and ζ=1, and either
g≥0 on (0,1) or g<0 near ζ=0. In the case of nonnegative g,
it is shown that solutions with initial support on the positive half axis
spread into the left half axis with unbounded speed if g(ζ) satisfies some
weak growth condition near ζ=0 in the case α>1, or if g is merely
positive on a sufficiently large interval near ζ=1 in the case α<1. On the other hand, it shown that solutions spread with finite speed if
g′(0)<0. The proofs use comparison arguments and a suitable family
of travelling wave solutions.


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## Body

## 1. Introduction

The scalar reaction-diffusion equation(1.1)∂tu(x,t)-∂x2u(x,t)=g(u(x,t))
has been the subject of much study, beginning with the celebrated paper [1]. The authors of [1] proposed this equation, with g being positive and concave on (0,1) such that g(0)=g(1)=0, as a model for a population that undergoes logistic growth and Brownian diffusion. If u(x,0)=H(x), the Heaviside function, and g(u)=u-u2, the equation in fact has an exact probabilistic interpretation, given in [2]. Consider a population of particles that undergo independent Brownian motion and branching processes, with each child particle again following the same behavior. Then u(x,t) is the probability that there is a particle to the left of position x at time t, assuming that there was exactly one particle at position x=0 at time t=0. Equation (1.1) also can be derived heuristically for the mean behavior of an interacting particle process in which two types of particles (call them A- and B-particles) simultaneously undergo Brownian diffusion and conversion reactions(1.2)A+B→2A,A+B→2B
with suitable reaction rates. Then the volume density fraction u(x,t) of A-particles in the hydrodynamic limit of large particle numbers per unit volume formally satisfies (1.1) with g(u)=cu(1-u) where c depends on the reaction rates in (1.2); see [3] for a discussion of the underlying limit procedure and an exact connection to a stochastic version of (1.1). If instead the conversion reactions are(1.3)2A+B→3A,2A+B→A+2B,
then the equation for u becomes (1.1) with g(u)=cu2(1-u). Other polynomial reaction terms g occur in similar ways.For (1.1) with g(u)=u(1-u), it is known that solutions approach a wave profile ψ in the sense that(1.4)u(x+m(t),t)→ψ(x)(t→∞),
where m(t) is the median, u(m(t),t)=1/2. It turns out that m(t)=c*t+O(logt) for a suitable asymptotic finite wave speed c*. Larger asymptotic speeds are only possible if the initial data are supported on ℝ. A more general result, given in [4], implies that there is a critical speed c* such that for fairly general initial data u(·,0) that are nonnegative and supported on (0,∞),(1.5)lim supt→∞supx<-ctu(x,t)=0
whenever c>c* and(1.6)lim inft→∞infx<-ctu(x,t)=1
whenever c<c*. If u is interpreted as the density of a quantity whose spread is governed by (1.1), a runner may escape from it by running to -∞ at a speed c>c*, but this quantity will catch up with and engulf it if its speed is c<c*. In this sense, solutions of the (1.1) exhibit finite speed of spread.Equation (1.1) was also derived in [5] to describe antiphase domain coarsening in alloys. In this situation, g(0)=g(1)=g(u*)=0 for some u*∈(0,1), and g<0 on (0,u*), and g>0 on (u*,1). In this case there exists exactly one wave speed c* with associated wave profile. In particular, (1.5) and (1.6) still hold for this c*. The first of the two cases (the KPP case) corresponds to “pulled” fronts (the state u=0 is unstable) while the second case (the Allen-Cahn case) results in a “pushed” front (the state u=0 is stable). More on these two fundamentally different situations may be found in [6] and the references given there. A vast range of applications leading to related models is discussed in [7].The purpose of this note is a study of the fractional reaction-diffusion equation:(1.7)∂tu(x,t)+Au(x,t)=g(u(x,t)).
Here A=Aαρ is a pseudodifferential operator with symbol p=pαρ that is homogeneous of degree α∈(0,2], such that p(-λ)=p(λ)¯ and |p(1)|=1. Following the presentation in [8], we write p in the Riesz-Feller form:(1.8)p(λ)=e-i(π/2)sign(λ)ρ|λ|α,
where the skewness parameterρ must also satisfy |ρ|≤min{α,2-α}. The operator A is the infinitesimal generator of a stable Lévy process, that is, a continuous time stochastic process that has cádlág paths and independent stationary increments with stable distributions; see [9]. Paths of such a process must have jumps, and the variance of the displacement must be infinite.Forρ=0 we obtain fractional powers of the usual negative one-dimensional Laplacian, abbreviated often by (-Δ)α/2. There are various real variable representations of such operators, for example, as singular integral operators or as limits of suitable difference operators; see [8]. In the special case where α=1 and -1≤ρ≤1, there is the representation(1.9)A1,ρu(x)=ddx(cosρπ2ℋu(x)+sinρπ2u(x)),
where ℋu is the Hilbert transform of u. In particular, for ρ=±1, this is an ordinary first-order derivative, not a fractional derivative.The functiong is always assumed to satisfy g(0)=g(1)=0. We are interested in both the KPP-case, that is, g(ζ)≥0 for 0<ζ<1, and the Allen-Cahn case, that is, g(ζ)<0 for ζ near 0 and g(ζ)>0 for ζ near 1.The equation occurs in the heuristic hydrodynamic limit of interacting particle populations in which conversion reactions such as (1.2) or (1.3) occur together with motion by a stable Lévy process. It has been proposed in, for example, [10–15]. It should be noted that the term “anomalous diffusion” is also used for situations in which the first-order time derivative is replaced by a fractional order derivative; see [8] where the present case of a first-order time derivative is called space-fractional diffusion. Another generalization of (1.1) consists in allowing time delays; see [16]. These further generalizations will not be discussed here.There is strong evidence that (1.7) does not admit traveling wave solutions if 0<α<2 and g is positive and concave on (0,1). Rather, numerical results in [11, 17] suggest that for initial data that are supported on the positive half axis and increase there from 0 to 1, the median satisfies m(t)~-ect for some c>0. In [18], the estimates(1.10)lim supt→∞supx<-ectu(x,t)=0,lim inft→∞infx>-edtu(x,t)=1
are shown to hold for such initial data whenever c>c*>d>0, where c*=g′(0)/α. Thus the support of a solution grows asymptotically like an exponential. For the case where g<0 on some interval (0,u*), the results in [14, 19] suggest on the other hand that there exist wave profile solutions that move with constant speed, although no rigorous proofs are given there.The main results of this note are concerned with the existence and nonexistence of a finite speed of spread and take the form (1.5) and (1.6). It is shown that for a large class of right-hand sides g that are nonnegative on (0,1), the speed of spread is infinite; that is, the estimate (1.6) holds for all positive speeds c. It is not necessary to assume that g′(0)>0, and if α<1, one does not even have to assume that g is strictly positive on (0,1). On the other hand, if g′(0)<0 and therefore g is negative near 0, then it will be shown that there exists a finite speed of spread; that is, (1.5) holds for some finite positive c. These results are stated and proven in Section 3. Some basic existence and comparison results for (1.7) are sketched in Section 4.To prove these results, comparison arguments are employed which follow from integral representations of solutions of (1.7) and which are therefore extensions of similar arguments for the study of (1.1). The challenge then is to come up with suitable comparison solutions. Since (1.7) is nonlocal in nature, techniques from ordinary differential equations cannot be employed to construct such solutions. Instead, in this paper a set of travelling wave solutions is used that comes directly from the fundamental solution of the linear problem (see (2.1) below). These solutions are discussed in Section 2 and may be of independent interest.


## 2. A Class of Travelling Wave Solutions

In this section, it will be shown that fundamental solutions of fractional diffusion equations (without reaction terms) lead to traveling wave solutionsu(x,t)=U(x+ct) of (1.7), for suitable functions g. A two-parameter family will be constructed for each possible choice of α and ρ, one parameter being the speed c. The main contribution of this section is the characterization of the nonlinear function g that is required to make the equation hold. Throughout this section, let α∈(0,2) and |ρ|<min{α,2-α}.Consider the “free” equation(2.1)∂tu(x,t)+Aαρu(x,t)=0,
where Aαρ is the pseudodifferential operator with symbol pαρ defined in (1.8). It is known that (2.1) has the fundamental solution(2.2)(x,t)↦Wαρ(x,t)=t-1/αfαρ(xt-1/α)
with initial data Wαρ(x,0)=δ0(x), the delta distribution; see [8]. In particular, Wαρ(x,1)=fαρ(x). Here fαρ is the probability density function of a stable distribution; see [20, 21]. There are many ways to parametrize such distributions. For α∉{0,1,2} the version used here corresponds to form (B) in [21] with the same choice of α, scale parameter γ=1, location parameter δ=0, and skewness parameter:(2.3)β=±ρmin{α,2-α},
where the sign is positive if 0<α<1 and negative if 1<α<2. For α=1, the form used here corresponds to form (C) in [21] with β=ρ,γ=1,δ=0.There is also the special solution(2.4)(x,t)↦Vαρ(x,t)=Fαρ(xt-1/α),
where Fαρ(x)=∫-∞xfαρ(s)ds is a cumulative distribution function. Then Vαρ solves (2.1) with initial data Vαρ(x,0)=H(x), the Heaviside function. The equations hold in the sense of distributions, and the initial data are attained in this sense. It is known that fαρ is positive, infinitely differentiable, and unimodal. Also, as x→∞, there are expansions(2.5)1-Fαρ(x)~∑j≥1cjαρx-jα,fαρ(x)~∑j≥1c̃jαρx-1-jα,
and as x→-∞(2.6)Fαρ(x)~∑j≥1djαρ(-x)-jα,fαρ(x)~∑j≥1d̃jαρ(-x)-1-jα.
These are convergent expansions if 0<α<1 and asymptotic expansions if 1<α<2; see [8].For the remainder of this section, we suppress the subscriptsα and ρ in most formulae that involve Aαρ,fαρ, and Fαρ. For fixed c∈ℝ and τ>0 we consider the function(2.7)Uτ(ξ)=F(ξτ-1/α).
Set ucτ(x,t)=Uτ(x+ct), then(2.8)Aucτ(x,t)=-∂τF((x+ct)τ-1/α)=1α((x+ct)τ-1/α-1)f((x+ct)τ-1/α)∂tucτ(x,t)=cτ-1/αf((x+ct)τ-1/α),
and therefore(2.9)∂tucτ(x,t)+Aucτ(x,t)=(1α((x+ct)τ-1/α-1)+cτ-1/α)f((x+ct)τ-1/α).
Now (x+ct)τ-1/α=F-1(ucτ(x,t)) and consequently(2.10)∂tucτ(x,t)+Aucτ(x,t)=cτ-1/αg0(ucτ(x,t))+1ατg1(ucτ(x,t))
with(2.11)g0(ζ)=f(F-1(ζ)),g1(ζ)=F-1(ζ)f(F-1(ζ)).
Equation (2.10) is of the form (1.7), with g(ζ)=cτ-1/αg0(ζ)+1/(ατ)g1(ζ).In the caseα=1 and -1<ρ<1, everything is explicit. Let κ=cos(πρ)/2 and σ=sin(πρ)/2. Then by results in [21],(2.12)F(x)=12+1πarctanx-σκ,F-1(ζ)=σ-κcot(πζ),f(x)=1πκ(1+(x-σ)2/κ2),g0(ζ)=1κπsin2(πζ),g1(ζ)=σπκsin2(πζ)-1πcos(πζ)sin(πζ).It remains to characterize the functionsg0,g1 in the general case.Proposition 2.1.
Let0<α<2,|ρ|<min{α,2-α}. The functions g0,g1 have the following properties.(a)
g0 and g1 are infinitely differentiable on (0,1).(b)
The functiong0 is positive on (0,1). The function g1 is negative on (0,Fαρ(0)) and positive on (Fαρ(0),1). The function
(2.13)ζ↦cτ-1/αg0(ζ)+1ατg1(ζ)
is negative on (0,u*) and positive on (u*,1), where u*=Fαρ(-cατ-1/α+1).(c)
Asζ↓0, g0(ζ)=O(ζ1+1/α) and g0(1-ζ)=O(ζ1+1/α).(d)
Asζ↓0, g1(ζ)=-αζ+O(ζ1+1/α). As ζ↑1, g1(ζ)=α(1-ζ)+O((1-ζ)1+1/α).(e)
The functionsg0 and g1 can be represented as(2.14)g0(ζ)=ddζ∫-∞F-1(ζ)f2(s)ds,g1(ζ)=ddζ∫-∞F-1(ζ)sf2(s)ds.Proof.
Property (a) follows sincef and F together with its inverse are infinitely differentiable. Property (b) is obvious. Properties (c) and (d) follow from the asymptotic expansions (2.5) and (2.6). Finally (e) can be checked by differentiation.Property (e) will not be used in what follows. It should be noted thatg0 and g1 are of class C1 on [0,1] but are not infinitely differentiable at the interval endpoints, except if α=1. Clearly g0 and g1 do not depend on c or τ. We are therefore free to form fairly arbitrary linear combinations of g0 and g1 by choosing c and τ.The construction provides travelling wave solutions for (1.7) for a special class of functions for which g∈C1([0,1]), g(0)=g(u*)=g(1) for some u*∈(0,1), and g′(0)<0,g′(u*)>0,g′(1)<0. This suggests that (1.7) possesses travelling wave solutions for more general functions g with these properties.If the same construction is attempted for the caseα=2, it turns out that g0 and g1 are merely continuous on [0,1], with derivatives that have logarithmic singularities near ζ=0 and ζ=1. Therefore the arguments in the next section cannot be extended to the case α=2, and indeed the results of the next section do not hold in that case.


## 3. Results on the Speed of Spread

This section contains the main results of this paper. As before, the operatorA has symbol (1.8) with 0<α<2 and |ρ|<min{α,2-α}. We always assume that u is a solution of (1.7) and that g∈C1([0,1],ℝ) with g(0)=g(1)=0. Initial data u0 will be assumed to satisfy(3.1)u0∈C(ℝ,ℝ),0≤u0(x)≤1,limx→∞u0(x)=1,supp(u0)⊂[0,∞).
The results in Section 4 then imply that (1.7) has a unique mild solution u that exists for all x∈ℝ,t>0, and this solution satisfies 0≤u(x,t)≤1 for all (x,t). The notation of that section will also be used here.We first discuss the case whereg≥0 on (0,1). The main result in this case is the following.Theorem 3.1.
Letu be the solution of (1.7) with u0 satisfying (3.1).(a)
Letα>1. Assume that g>0 on (0,1) and that there are 0<γ<α/(α-1),c0>0 such that for all ζ∈[0,1/2](3.2)g(ζ)≥c0ζγ.
Then for all c>0
(3.3)lim inft→∞infx≥-ctu(x,t)=1.(b)
Letα=1. Assume that g>0 on (0,1). Then for all c>0
(3.4)lim inft→∞infx≥-ctu(x,t)=1.(c)
Letα<1. Assume that g≥0 on (0,1) and g(ζ)>0 for ζ∈[(α-ρ)/2α,1). Then for all c>0(3.5)lim inft→∞infx≥-ctu(x,t)=1.The result shows that the speed of spread is unbounded (i.e., (1.6) holds for all c>0), and it exhibits different mechanisms for this phenomenon. Recall that in the interpretation of [1], the function g is responsible for the growth of a substance whose density is given by u, while A describes the spread of this substance. If α∈(0,2), the substance spreads with a jump process, not with Brownian diffusion, and jumps of magnitude exceeding M occur with a probability that is O(M-α) for large M. If α>1, the mean jump distance is still finite. In this case, the growth rate g(u) at small densities (small u) is responsible for the unbounded speed of spread. If α is close to 1, this growth can be very weak (g(ζ)~ζγ with large γ) yet the speed of spread is still unbounded. If on the other hand α<1, then large jumps tend to be more frequent, and jump sizes have unbounded mean. In this case, the growth rate for small densities is no longer the reason why the speed of spread is unbounded; in fact there may be no growth at all for small densities (g(u)=0 for small u) for this to occur. Rather, the unbounded speed of spread results from growth that occurs solely for large densities (g(ζ)>0 only for ζ≥(α-ρ)/2α). The substance is transported towards -∞ due large (α<1) negative jumps, resulting in an unbounded speed of spread. It is known that in this case, (α-ρ)/2α is the probability that a jump is negative. If this fraction is large, then growth that occurs only for large densities, that is, g(u)>0 on [(α-ρ)/2α,1), already leads to an unbounded speed of spread. The case α=1 is intermediate: any growth for small densities (g(ζ)>0 for ζ>0) results in an unbounded speed of spread.In the caseα>1, it would be interesting to know if the speed of spread is still unbounded if γ≥α/(α-1) or if a finite speed of spread occurs ((1.5) holds for large c) if γ becomes sufficiently large, that is, if growth is extremely weak for small densities u. In the case α<1, it would be interesting to know if a finite speed of spread is possible at all if g≥0 and g is not identically equal to 0.The main result in the case whereg is negative near ζ=0 is the following.Theorem 3.2.
Letu be the solution of (1.7) with initial data u0 satisfying (3.1). Assume that g′(0)<0. Then there exists c>0 such that
(3.6)lim supt→∞supx≤-ctu(x,t)=0.The result shows that negative proportional growth at small densities (g′(0)<0) always limits the speed of spread of a substance whose growth and spread are governed by (1.7), even for processes whose jump sizes tend to be very large (α<1). I am not aware of an interpretation of this result in the context of material science, similar to the use of (1.1) in [5].The proofs will be given below. The main tools in the proofs are the comparison arguments given in the next section, together with the following crucial auxiliary result.Lemma 3.3.
Letg∈C1([0,1],ℝ) and let g0,g1 be defined as in (2.11), depending on α and ρ.(a)
Letα∈(1,2). Suppose that g(ζ)>0 for all ζ∈(0,1] and that there exist c0>0 and γ<α/(α-1) such that g(ζ)≥c0ζγ for all ζ∈[0,1/2]. Then given any c>0 there exists τ>0 such that for all ζ∈[0,1](3.7)g(ζ)≥cτ-1/αg0(ζ)+(ατ)-1g1(ζ).(b)
Letα=1. Suppose that g>0 on (0,1]. Then given any c>0 there exists τ>0 such that for all ζ∈[0,1](3.8)g(ζ)≥cτ-1g0(ζ)+τ-1g1(ζ).(c)
Letα∈(0,1). Suppose that g≥0 on [0,1] and g(ζ)>0 for all ζ∈[(α-ρ)/2α,1). Then given any c>0 there exists τ>0 such that for all ζ∈[0,1](3.9)g(ζ)≥cτ-1/αg0(ζ)+(ατ)-1g1(ζ).(d)
Suppose thatg′(0)<0 and g(ζ)=0 for ζ∈[1-ϵ,1] for some ϵ. Then there exist c∈ℝ and τ>0 such that for all ζ∈[0,1](3.10)g(ζ)≤cτ-1/αg0(ζ)+(ατ)-1g1(ζ).Proof.
Consider first statement (a). Let us writeF=Fαρ and f=fαρ. Let α>1 and let M>0 be large enough such that for some c1,c2>0 and all x≤-M(3.11)F(x)≥c1|x|-α,f(x)≤c2|x|-1-α.
This is possible by (2.6). Let c>0 be given, then we may increase M further such that also
(3.12)c0c1γMr≥c2α-1αcα/(α-1),
where r=(α/(α-1))-γ>0. Now set δ=F(-M). Then for 0<ζ=F(x)≤δ, that is, x<-M, and for all τ≥(M/c)α/(α-1)(3.13)cτ-1/αg0(F(x))+(ατ)-1g1(F(x))=(cτ-1/α+(ατ)-1x)f(x)≤α-1αcα/(α-1)|x|1/(1-α)f(x)≤α-1αcα/(α-1)|x|1/(1-α)c2|x|-1-α=α-1αcα/(α-1)c2|x|-α2/(α-1),
where a standard calculus argument has been used to see that the expression (cτ-1/α+(ατ)-1x) is maximal for τ=(|x|/c)α/(α-1). We estimate further, using the choice of M,
(3.14)cτ-1/αg0(F(x))+(ατ)-1g1(F(x))≤c0c1γMr|x|-α2/(α-1)≤c0c1γ|x|r|x|-α2/(α-1)=c0(c1|x|α)γ≤c0(F(x))γ≤g(F(x)).
Therefore, for all τ≥(M/c)α/(α-1) and all ζ<δ=F(-M),
(3.15)cτ-1/αg0(ζ)+(ατ)-1g1(ζ)≤g(ζ).
Since g>0 on [δ,1] by assumption, this inequality can be achieved also on [δ,1] by increasing τ even further. This proves part (a).
The proof of part (b) is straight forward: given c>0, note that cg0(ζ)+g1(ζ)≤0 on the interval [0,F(-c)]. Then (cg0(ζ)+g1(ζ))/τ≤g(ζ) is true if τ is sufficiently large.
To prove part (c), let againc>0 be given. Let u*=inf{u∈[0,1]∣g(u)>0}. Then u*<(α-ρ)/2α=F(0). Pick τ large enough such that F(-cατ1-1/α)>u*. This is possible since α<1. Then on [0,u*],
(3.16)cτ-1/αg0(ζ)+(ατ)-1g1(ζ)≤g(ζ)
since the left-hand side is nonpositive there by Proposition 2.1. By increasing τ further, we can obtain this estimate also for ζ∈[u*,1], using again that g is assumed to be positive on [(1-β)/2,1].
To prove part (d), note first that(3.17)cτ-1/αg0(ζ)+(ατ)-1g1(ζ)≥g(ζ)
on an interval [0,δ] as soon as τ-1+g′(0)>0, that is, for sufficiently small τ. Increasing c sufficiently and noting that g=0 near ζ=1 extends this inequality to the entire interval [0,1].Proof of Theorem3.1.
The proof uses the same argument for all three parts; so we give details only in part (a). Letϵ>0. We replace u with ũ=(1+ϵ)u and g with g̃, where g̃(ζ)=(1+ϵ)g((1+ϵ)-1ζ). Then
(3.18)∂tũ+Aũ=g̃(ũ).
Then g̃(ζ)≥c̃0ζγ for ζ∈[0,1/2], possibly with a changed c0, and additionally g̃>0 on (0,1]. Let c>0 be given, then there exists τ>0 such that
(3.19)g̃(ζ)≥(c+1)τ-1/αg0(ζ)+(ατ)-1g1(ζ)
for all ζ∈[0,1] by Lemma 3.3. By extending g0 and g1 to be zero on [1,1+ϵ], this inequality is true on [0,1+ϵ]. Now find a constant d such that v0(x)=H(x-d)≤ũ(x,0) for all x. This is possible since limx→∞ũ(x,0)=1+ϵ. By Proposition 4.3, we see that
(3.20)ũ(x,t)≥F((x-d)t-1/α)
for all x∈ℝ,t>0. This is in particular true for t=τ. Now use Proposition 4.1 and (3.19) to infer that
(3.21)ũ(x,t)≥F((x-d+(c+1)t)τ-1/α)
for all x∈ℝ,t≥τ. Therefore for t≥τ and x≥-ct,
(3.22)ũ(x,t)≥F((x-d+(c+1)t)τ-1/α)≥F((-ct-d+(c+1)t)τ-1/α)=F((t-d)τ-1/α).
Ast→∞, the right-hand side goes to 1. Rewriting this in terms of u, we see that
(3.23)lim inft→∞infx≥-ctũ(x,t)≥(1+ϵ)-1.
Since ϵ>0 was arbitrary, the desired result follows.
In case of part (b), the same argument can be used without changes, appealing to part (b) of Lemma3.3.
In case of part (c), we have to restrictϵ such that g̃>0 on [(α-ρ)/2α,1], that is, g>0 on [(α-ρ)/(2α(1+ϵ)),1]. The rest of the proof is again unchanged, using part (c) of Lemma 3.3.Proof of Theorem3.2.
We replaceu with ũ=(1/2)u and g with g̃, where g̃(ζ)=(1/2)g(2ζ) for 0≤ζ≤1/2 and g̃(ζ)=0 for ζ∈(1/2,1]. Then g′̃(0)=g′(0)<0 and
(3.24)∂tũ+Aũ=g̃(ũ).
By Lemma 3.3, part (d), there exist c>0 and τ>0 such that
(3.25)g̃(ζ)≤(c-1)τ-1/αg0(ζ)+(ατ)-1g1(ζ).
Since limx→∞ũ(x,0)=1/2 and u(x,0)=0 for x<0, we can find d>0 such that F(x+d)τ-1/α)≥ũ(x,0) for all x∈ℝ. Using Proposition 4.1, one sees that
(3.26)F((x+d+(c-1)t)τ-1/α)≥ũ(x,t)
for all x∈ℝ,t>0. Therefore for t>0 and x≤-ct,
(3.27)ũ(x,t)≤F((x+d+(c-1)t)τ-1/α)≤F((-ct+d+(c-1)t)τ-1/α)=F((-t+d)τ-1/α).
The right-hand side tends to 0 as t→∞. In terms of u, this implies
(3.28)lim supt→∞supx≤-ctu(x,t)=0.
This concludes the proof.


## 4. Facts about Fractional Reaction-Diffusion Equations

In this section we summarize some basic theory about (1.7) that is needed in this note. A broader and deeper discussion may be found in [10].We work in the Banach space(4.1)𝒞lim={w∈C(ℝ)∣limx→∞w(x)andlimx→-∞w(x)exist}
equipped with the supremum norm ∥·∥. Let A=Aαρ be the pseudodifferential operator with symbol (1.8) and parameters α,ρ, with 0<α<2 and |ρ|<min{α,2-α}. Subscripts α,ρ will again frequently be suppressed. Solutions of the free equation (2.1) with initial data u(·,0)=φ∈𝒞lim then can be written in terms of the fundamental solution given in (2.2), namely,(4.2)u(x,t)=∫ℝWαρ(x-y,t)φ(y)dy.For fixedα and ρ and φ∈𝒞lim, we therefore define(4.3)S(t)φ(x)=u(x,t),
where u is given by (4.2). Then S(t)t≥0 is a positive C0 semigroup on 𝒞lim and a Feller semigroup on the subspace of functions in 𝒞lim that vanish at ±∞. If ψ is a continuous function from [0,T] to 𝒞lim, then solutions of the inhomogeneous equation(4.4)∂tu(x,t)+Au(x,t)=ψ(x,t),u(·,0)=φ
can be written with the variation-of-constants formula:(4.5)u(·,t)=S(t)φ+∫0tS(t-s)ψ(·,s)ds.
A continuous curve u:[0,T]→𝒞lim that satisfies (4.5) is commonly called a mild solution of (4.4). Next let g:[0,∞)×ℝ→ℝ be locally Lipschitz continuous in both variables and let φ∈𝒞lim. Then the equation ∂tu(x,t)+Au(x,t)=g(t,u(x,t)) (for which (1.7) is a special case) has a unique mild solution u∈C([0,T),𝒞lim), where 0<T≤∞ is maximal. Either T=∞, or ∥u(·,t)∥→∞ as t↑T. The solution can be obtained as the locally in time uniform limit of the iteration scheme:(4.6)un+1(·,t)=S(t)φ+∫0tS(t-s)g(s,un(·,s))ds(n=0,1,…)
with u0 being arbitrary, for example, u0(·,t)=S(t)φ. It is possible to set up a more general solution theory, but this is not needed for the purposes of this paper.Solutions of (1.7) satisfy comparison theorems. Results of this type are true for all Feller semigroup. A systematic study of such semigroups and their generators was carried out in [22], following the seminal work on this topic in [23]. For the sake of completeness, a comparison result is stated here, and its proof is sketched.Proposition 4.1.
Letu,v∈C([0,T],𝒞lim) be mild solutions of the equations
(4.7)∂tu+Au=g(u),∂tv+Av=h(v),
where g,h:ℝ→ℝ are locally Lipschitz continuous. If
(4.8)g(ζ)≤h(ζ),∀ζ∈ℝ,u(·,0)≤v(·,0),
then
(4.9)u(x,t)≤v(x,t)∀(x,t)∈ℝ×[0,T].Proof.
LetM=max[0,T](∥u(·,t)∥+∥v(·,t)∥+1). Let λ>|g′(ζ)|+|h′(ζ)| for all |ζ|≤M. Without loss of generality we may assume that g and h are constant outside [-M,M]. Set
(4.10)U(x,t)=eλtu(x,t),V(x,t)=eλtv(x,t),
and observe that U and V satisfy
(4.11)∂tU+AU=g̃(t,U),∂tV+AV=h̃(t,V)
with g̃(t,ζ)=λζ+eλtg(e-λtζ) and h̃(t,ζ) defined similarly. Clearly, g̃(t,ζ)≤h̃(t,ζ) for all ζ. The function g̃ is nondecreasing in its second argument, since for almost all ζ(4.12)∂ζg̃(t,ζ)=λ+g′(e-λζ)≥0.
Consider the iteration scheme:
(4.13)Un+1(·,t)=S(t)u(·,0)+∫0tS(t-s)g̃(s,Un(·,s))ds
and similarly for Vn and h̃. The scheme for the Un converges to the limit U, and the scheme for the Vn converges to the limit V.
We now employ a standard induction argument to show thatUn≤Vn on ℝ×[0,T] for all n. Let U0(·,t)=S(t)u(·,0) and V0(·,t)=S(t)v(·,0), then U0≤V0 on ℝ×[0,T] since S is a positive semigroup and u(·,0)≤v(·,0). Suppose Un≤Vn on ℝ×[0,T], then
(4.14)Un+1(·,t)=S(t)u(·,0)+∫0tS(t-s)g̃(s,Un(·,s))ds≤S(t)v(·,0)+∫0tS(t-s)g̃(s,Vn(·,s))ds≤S(t)v(·,0)+∫0tS(t-s)h̃(s,Vn(·,s))ds=Vn+1(·,t),
which completes the induction step. Taking the limit, this implies that U≤V and therefore also u≤v on ℝ×[0,T]. This proves the proposition.Corollary 4.2.
Consider a mild solutionu∈C([0,T),𝒞lim) of (1.7) and assume that g is locally Lipschitz continuous. If g(γ)≥0 for some γ and u(·,0)≥γ, then u(·,t)≥γ for all t. If g(γ)≥0≥g(δ) for some γ<δ and γ≤u(·,0)≤δ, then γ≤u(·,t)≤δ for all t, and the solution can be continued to ℝ×[0,∞).The proof consists in observing that the constant functionsv(x,t)=γ and w(x,t)=δ solve (1.7) with right-hand sides 0 and therefore must be pointwise bounds for the solution, by Proposition 4.1. If the solution remains bounded between two constants, then its supremum norm remains bounded and it can be continued to ℝ×[0,∞).Also required is a comparison result for solutions whose initial data are step functions. Since such initial data are not in𝒞lim, a separate argument is required.Proposition 4.3.
Letu∈C([0,T],𝒞lim) be a mild solution of (1.7), with locally Lipschitz continuous g. Assume that
(4.15)u(x,0)≥v0(x)=a0+∑j=1NajH(x-cj)∀x∈ℝ,
where aj∈ℝ,c1<c2⋯<cN, and H is the Heaviside function. Let γ=minℝv0(x) and δ=maxℝv0(x). Assume also that g≥0 on [γ,δ]. Then
(4.16)u(x,t)≥a0+∑j=1NajVαρ(x-cj,t)∀x∈ℝ,0<t≤T,
where Vαρ is defined in (2.4).Proof.
We know thatu(x,t)≥γ and thus may assume that g(ζ)≥0 also for ζ<γ. For arbitrary ϵ,σ>0, we set
(4.17)vϵσ(x)=1σ∫0σv0(x-z)dz-ϵ.
Then vϵσ is piecewise linear and constant outside the interval [c1,cN+σ]; in particular, vϵσ∈𝒞lim. Solving (2.1) with initial data vϵ,σ gives the solution
(4.18)Gϵσ(x,t)=1σ∫0σ∑j=1NajVαρ(x-z-cj,t)+a0-ϵ.
Given ϵ>0, it is possible to find σ>0 such that vϵσ<u0(x) on ℝ, since u0 is uniformly continuous. Clearly, γ-ϵ≤Gϵσ≤δ. We may therefore view Gϵσ as a solution of (1.7) with a right-hand side h that satisfies h(ζ)=0≤g(ζ) for ζ≤δ and h(ζ)≤g(ζ) also for ζ>δ. Then by Proposition 4.1(4.19)u(x,t)≥Gϵ,σ(x,t)∀x∈ℝ,0<t≤T.
Send δ to 0, then since Vαρ is uniformly continuous, (4.16) is obtained with a0 replaced by a0-ϵ on the right-hand side. Now send ϵ to 0 and (4.16) follows.


## 5. Conclusion

In this note, conditions for the speed of spread of solutions of fractional scalar reaction-diffusion equations to be finite or infinite have been derived. If the reaction term is positive for all positive arguments, then this speed is shown to be infinite as soon as the reaction term describes some very weak growth for low densities. This is in contrast to the corresponding problem for standard diffusion, where the speed of spread is always finite for such reaction terms. On the other hand, if the reaction term is negative for small positive arguments, then the speed of spread is finite, just as it is for the case of standard diffusion.



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