On the Speed of Spread for Fractional Reaction-Diffusion Equations

The fractional reaction diffusion equation u_t + Au = g(u) is discussed, where A is a fractional differential operator on the real line with order \alpha between 0 and 2, the C^1 function g vanishes at 0 and 1, and either g is non-negative on (0,1) or g<0 near 0. In the case of non-negative 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 satisfies some weak growth condition near 0 in the case \alpha>1, or if g is merely positive on a sufficiently large interval near 1 in the case \alpha<1. On the other hand, it shown that solutions spread with finite speed if g'(0)<0. The proofs use comparison arguments and a new family of traveling wave solutions for this class of problems.


Introduction
The scalar reaction-diffusion equation (1) ∂ t u(x, t) − ∂ 2 x u(x, t) = g(u(x, t)) has been the subject of long study, beginning with the celebrated paper [11]. The authors of [11] proposed this equation, with g 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 − u 2 , the equation in fact has an exact probabilistic interpretation, see [14]. For this problem, is known that solutions approach a wave profile ψ in the sense that where m(t) is the median, u(m(t), t) = 1 2 . It turns out that m(t) = c * t + O(log t) 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 [2], implies that there is a critical speed c * such that for fairly general initial data u(·, 0) that are non-negative and supported on (0, ∞), whenever c < c * . If u denotes a quantity that is to be avoided and whose spread is governed by (1), a runner may escape from it by running to −∞ at a speed c > c * , but this quantity will catch up with and engulf her if her speed is c < c * .
Equation (1) was also derived in [1] 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 * ), g > 0 on (u * , 1). In this case there exists exactly one wave speed c * with associated wave profile. In particular, (2) still holds 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 [8] and the references given there. A vast range of applications leading to related models is discussed in [9].
The purpose of this note is a study of the fractional reaction-diffusion equation Here A is a pseudo-differential operator with symbol p that is homogeneous of degree α ∈ (0, 2], such that p(−λ) = p(λ) and |p(1)| = 1. Then we can write p in the form where ρ ∈ R. There will be additional restrictions on the parameter ρ in section 2. For ρ = 0 we obtain fractional powers of the usual negative onedimensional Laplacian, abbreviated often by (−∆) α/2 . There are various real variable representations of such operators, e.g. as a singular integral operator or as a limit of suitable difference operators; see [3] where this is explained in more detail. The function g is always assumed to satisfy g(0) = g(1) = 0. We are interested in both the KPP-case, i.e. g(ζ) ≥ 0 for 0 < ζ < 1, and the Allen-Cahn case, i.e. g(ζ) < 0 for ζ near 0 and g(ζ) > 0 for ζ near 1.
This class of equations has recenty been proposed as a model for reaction and anomalous diffusion; see [3,7,12,13,19,20]. 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. Another generalization of (1) consists in allowing time delays; see [17]. These further generalizations will not be discussed here.
There is strong evidence that the equation (4) does not admit traveling wave solutions if 0 < α < 2 and g is positive and concave on (0, 1). Rather, numerical results in [7] and [4] 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) ∼ −e ct for some c > 0. In [6], the estimates are shown to hold for such initial data whenever c > c * > d, where c * = g ′ (0)/α. Thus the asymptotic speed of spread grows exponentially. For the case where g < 0 on some interval (0, u * ), the results in [15] and [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 take the form (2) and (3). It is shown that for a large class of right hand sides g that are non-negative on (0, 1), the estimate (3) holds for all speeds c; that is, the speed of spread is unbounded. It is not necessary to assume that g ′ (0) > 0, and if α < 1, g may even be zero near 0. On the other hand, if g ′ (0) < 0, then it is shown that (2) holds for some finite c, that is, there is always a bound on the speed of spread. These results are stated and proven in section 3. This is done by employing comparison arguments together with a set of travelling wave solutions that is constructed in section 2 and that may be of independent interest. Some basic existence and comparison results for (4) are sketched in section 4.

A Class of Travelling Wave Solutions
In this section, it will be shown that cumulative distribution functions of stable probability distributions with parameters α, β lead to traveling wave solutions u(x, t) = U (x + ct) of (4), 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.
The corresponding cumulative distribution function is denoted by F αβ . It is known that f αβ is positive, infinitely differentiable, and unimodal. Also, as x → ∞, there are asymptotic representations αc jαβ x −1−jα (9) and as x → −∞ Consider now the "free" equation (12) ∂ t u(x, t) + Au(x, t) = 0 where the symbol of A is given by (5), with ρ given by Throughout the rest of the paper, we shall only consider operators A with symbol p in (5) for which ρ is of the form (13) for some β ∈ (−1, 1). The relation (13) between β and ρ will always be assumed. Taking the Fourier transform then shows that equation (12) has the fundamental solution with initial data W (x, 0) = δ 0 (x), the delta distribution. In particular, W (x, 1) = f αβ (x). There is also the special solution , the Heaviside function. The equations hold in the sense of distributions, and the initial data are attained in this sense.
Proposition 2.1. Let 0 < α < 2, −1 < β < 1. The functions g 0 , g 1 have the following properties. a) g 0 and g 1 are infinitely differentiable on (0, 1). b) The function g 0 is positive on (0, 1). The function g 1 is negative on (0, F αβ (0)) and positive on (F αβ (0), 1). The function e) The functions g 0 and g 1 can be represented as Proof. Property a) follows since f αβ and F αβ together with its inverse are infinitely differentiable. Property b) is obvious. Properties c) and d) follow from the asymptotic expansions (9) and (11). Finally e) can be checked by differentiation.
Property e) will not be used in what follows. It should be noted that g 0 and g 1 are of class C 1 on [0, 1], but are not infinitely differentiable at the interval endpoints, except if α = 1. Clearly g 0 and g 1 do not depend on c or τ . We are therefore free to form fairly arbitrary linear combinations of g 0 and g 1 by choosing c and τ .
If the same construction is attempted for the case α = 2, it turns out that g 0 and g 1 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.

Results on the Speed of Spread
This section contains the main results of this paper. As before, the operator A has symbol (5) with 0 < α < 2 and ρ satisfying (13) with −1 < β < 1. We always assume that u is a solution of (4) and that g ∈ C 1 ([0, 1], R) with g(0) = g(1) = 0. Initial data u 0 will be assumed to satisfy (17) u The results in section 4 then imply that (4) 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 where g ≥ 0 on (0, 1). The main result in this case is the following. b) Let α < 1. Assume that g ≥ 0 on (0, 1) and g(ζ) > 0 for ζ ∈ [ 1−β 2 , 1). Then for all c > 0 lim inf The result shows that the speed of spread is unbounded (that is, (3) holds for all c > 0), and it exhibits different mechanisms for this phenomenon. Recall that in the interpretation of [11], 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 h occur with a probability that is O(h −α ) for large h. 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) ∼ ζ γ with large γ), yet the speed of spread is still unbounded. If on the other hand α < 1, i.e. jump sizes have unbounded mean, the growth rate for small densities does not matter any more for the speed of spread to be unbounded; in fact there may be no growth at all for small densities (g(u) = 0 for small u), and yet the speed of spread is unbounded. In this case, the unbounded speed of spread results from growth that occurs solely for large densities (g(ζ) > 0 only for ζ ≥ 1−β 2 ). 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, 1−β 2 is the fraction of negative jumps. If this fraction is large, then it is sufficient that growth occurs only for densities close to the maximal value, i.e. g(u) > 0 on [ 1−β 2 , 1) already implies that the speed of spread is unbounded. 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 case α > 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 ((2) holds for large c) if γ becomes sufficiently large, i.e. 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 where g is negative near ζ = 0 is the following. 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 (4), even for processes whose jump sizes tend to be very large (α < 1). I am not aware of an interpretation of (4) in the context of material science, similar to the use of (1) in [1].
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 auxiliary result.
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).
In case of part b), the same argument can be used without changes, appealing to part b) of Lemma 3.1.
In case of part c), we have to restrict ǫ such thatg > 0 on [ 1−β 2 , 1], that is, The rest of the proof is again unchanged, using part c) of Lemma 3.1.

Facts About Fractional Reaction-Diffusion Equations
In this section we summarize some basic theory about (4) that is needed in this note. A broader and deeper discussion may be found in [3].
We work in the Banach space equipped with the supremum norm · . Let A be the pseudodifferential operator with symbol (5) and parameters α, ρ. As always, let ρ, β be related by (13) and −1 < β < 1. Solutions of the free equation (12) with initial data u(·, 0) = ϕ ∈ C lim then can be written in terms of the fundamental solution (x, t) → t −1/α f αβ For fixed α and ρ and ϕ ∈ C lim , define where u is given by (19). This is a positive C 0 semigroup on C lim and a Feller semigroup on the subspace of functions in C lim that vanish at ±∞. If ψ is a continuous function from [0, T ] to C lim , then solutions of the inhomogeneous equation (20) ∂ t u(x, t) + Au(x, t) = ψ(x, t), u(·, 0) = ϕ can be written with the variation-of-constants formula (21) u(·, t) = S(t)ϕ + t 0 S(t − s)ψ(·, s) ds .
A continuous curve u : [0, T ] → C lim that satisfies (21) is commonly called a mild solution of (20). Next let g : [0, ∞) × R → R be locally Lipschitz continuous in both variables and let ϕ ∈ C lim . Then the equation ∂ t u(x, t) + Au(x, t) = g(t, u(x, t)) (for which (4) is a special case) has a unique mild The solution can be obtained as the locally in time uniform limit of the iteration scheme u n+1 (·, t) = S(t)ϕ + t 0 S(t − s)g(s, u n (·, s)) ds (n = 0, 1, . . . ) with u 0 arbitrary, e.g. u 0 (·, 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 (4) 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 [5], following the seminal work on this topic in [18]. For the sake of completeness, a comparison result is stated here, and its proof is sketched.
Consider the iteration scheme U n+1 (·, t) = S(t)u(·, 0) + t 0 S(t − s)g(s, U n (·, s)) ds and similarly for V n andh. The scheme for the U n converges to the limit U , and the scheme for the V n converges to the limit V .
We now employ a standard induction argument to show that U n ≤ V n on R × [0, T ] for all n. This implies that U ≤ V and therefore also u ≤ v on R × [0, T ]. Let U 0 (·, t) = S(t)u(·, 0) and V 0 (·, t) = S(t)v(·, 0), then U 0 ≤ V 0 on R × [0, T ] since S is a positive semigroup and u(·, 0) ≤ v(·, 0). Suppose which completes the induction step. This proves the proposition.
The proof consists in observing that the constant functions v(x, t) = γ and w(x, t) = δ solve (4) 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 in C lim , a separate argument is required. Proposition 4.3. Let u ∈ C ([0, T ], C lim ) be a mild solution of (4), with locally Lipschitz continuous g. Assume that where a j ∈ R, c 1 < c 2 · · · < c N , and H is the Heaviside function. Let γ = min R v 0 (x) and δ = max R v 0 (x). Assume also that g ≥ 0 on [γ, δ]. Then where F αβ is the cumulative distribution function of the associated stable distribution.
Send δ to 0, then since F αβ is uniformly continuous, (22) is obtained with a 0 replaced by a 0 − ǫ on the right hand side. Now send ǫ to 0 and (22) follows.