doi:10.1155/2010/315421
Research Article
On the Speed of Spread for Fractional Reaction-Diffusion Equations
Hans Engler
Department of Mathematics, Georgetown University, Box 571233, Washington, DC 20057, USA
Correspondence should be addressed to Hans Engler,[email protected] Received 12 August 2009; Revised 12 October 2009; Accepted 25 October 2009 Academic Editor: Om Agrawal
Copyrightq2010 Hans Engler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The fractional reaction diffusion equation∂tuAu guis discussed, whereAis a fractional differential operator onRof orderα∈0,2, theC1functiongvanishes atζ0 andζ 1, and eitherg ≥0 on0,1org <0 nearζ0. 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 gζsatisfies some weak growth condition nearζ0 in the caseα >1, or ifgis merely positive on a sufficiently large interval nearζ1 in the caseα <1. On the other hand, it shown that solutions spread with finite speed ifg0<0. The proofs use comparison arguments and a suitable family of travelling wave solutions.
1. Introduction
The scalar reaction-diffusion equation
∂tux, t−∂2xux, t gux, t 1.1
has been the subject of much study, beginning with the celebrated paper1 . The authors of1 proposed this equation, withg being positive and concave on0,1such thatg0 g1 0, as a model for a population that undergoes logistic growth and Brownian diffusion.
If ux,0 Hx, the Heaviside function, and gu 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. Thenux, tis the probability that there is a particle to the left of positionxat timet, assuming that there was exactly one particle at positionx0 at timet 0. Equation1.1also 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
AB−→2A, AB−→2B 1.2
with suitable reaction rates. Then the volume density fractionux, t of A-particles in the hydrodynamic limit of large particle numbers per unit volume formally satisfies1.1with gu cu1−uwherecdepends on the reaction rates in1.2; see3 for a discussion of the underlying limit procedure and an exact connection to a stochastic version of1.1. If instead the conversion reactions are
2AB−→3A, 2AB−→A2B, 1.3
then the equation forubecomes1.1withgu cu21−u. Other polynomial reaction terms goccur in similar ways.
For1.1withgu u1−u, it is known that solutions approach a wave profileψin the sense that
uxmt, t−→ψx t−→ ∞, 1.4
wheremtis the median,umt, t 1/2. It turns out thatmt c∗tOlogtfor a suitable asymptotic finite wave speedc∗. Larger asymptotic speeds are only possible if the initial data are supported onR. A more general result, given in4 , implies that there is a critical speed c∗such that for fairly general initial datau·,0that are nonnegative and supported on0,∞,
lim sup
t→ ∞ sup
x<−ctux, t 0 1.5
wheneverc > c∗and
lim inf
t→ ∞ inf
x<−ctux, t 1 1.6
wheneverc < c∗. Ifuis 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 speedc > c∗, but this quantity will catch up with and engulf it if its speed isc < c∗. In this sense, solutions of the1.1exhibit finite speed of spread.
Equation 1.1 was also derived in 5 to describe antiphase domain coarsening in alloys. In this situation,g0 g1 gu∗ 0 for someu∗ ∈ 0,1, andg < 0 on0, u∗, andg >0 onu∗,1. In this case there exists exactly one wave speedc∗with associated wave profile. In particular,1.5and1.6still hold for thisc∗. The first of the two casesthe KPP casecorresponds to “pulled” frontsthe stateu 0 is unstablewhile the second casethe Allen-Cahn caseresults in a “pushed” frontthe stateu 0 is stable. More on these two fundamentally different situations may be found in6 and the references given there. A vast range of applications leading to related models is discussed in7 .
The purpose of this note is a study of the fractional reaction-diffusion equation:
∂tux, t Aux, t gux, t. 1.7
HereA Aαρis a pseudodifferential operator with symbolp pαρ that is homogeneous of degreeα∈0,2 , such thatp−λ pλand|p1|1. Following the presentation in8 , we writepin the Riesz-Feller form:
pλ e−iπ/2signλρ|λ|α, 1.8
where the skewness parameterρ must also satisfy|ρ| ≤ min{α,2−α}. The operatorAis the infinitesimal generator of a stable L´evy process, that is, a continuous time stochastic process that has c´adl´ag 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.
For ρ 0 we obtain fractional powers of the usual negative one-dimensional Laplacian, abbreviated often by−Δα/2. There are various real variable representations of such operators, for example, as singular integral operators or as limits of suitable difference operators; see8 . In the special case whereα1 and−1≤ρ≤1, there is the representation
A1,ρux d dx
cosρπ
2 Hux sinρπ
2 ux
, 1.9
whereHuis the Hilbert transform ofu. In particular, forρ±1, this is an ordinary first-order derivative, not a fractional derivative.
The functiongis always assumed to satisfyg0 g1 0. We are interested in both the KPP-case, that is,gζ ≥0 for 0 < ζ <1, and the Allen-Cahn case, that is,gζ <0 forζ near 0 andgζ>0 forζnear 1.
The equation occurs in the heuristic hydrodynamic limit of interacting particle populations in which conversion reactions such as1.2or1.3occur together with motion by a stable L´evy 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; see8 where the present case of a first-order time derivative is called space-fractional diffusion. Another generalization of1.1 consists in allowing time delays; see16 . These further generalizations will not be discussed here.
There is strong evidence that1.7does not admit traveling wave solutions if 0< α <2 andgis positive and concave on0,1. Rather, numerical results in11,17 suggest that for initial data that are supported on the positive half axis and increase there from 0 to 1, the median satisfiesmt∼ −ectfor somec >0. In18 , the estimates
lim sup
t→ ∞ sup
x<−ectux, t 0, lim inf
t→ ∞ inf
x>−edtux, t 1 1.10
are shown to hold for such initial data wheneverc > c∗> d >0, wherec∗g0/α. Thus the support of a solution grows asymptotically like an exponential. For the case whereg <0 on
some interval0, u∗, the results in14,19 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 form1.5and1.6. It is shown that for a large class of right-hand sidesgthat are nonnegative on0,1, the speed of spread is infinite; that is, the estimate1.6holds for all positive speedsc. It is not necessary to assume thatg0>0, and ifα <1, one does not even have to assume thatgis strictly positive on0,1. On the other hand, ifg0<0 and thereforegis negative near 0, then it will be shown that there exists a finite speed of spread; that is,1.5holds for some finite positivec. These results are stated and proven inSection 3. Some basic existence and comparison results for1.7are sketched inSection 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 of1.1. The challenge then is to come up with suitable comparison solutions. Since1.7is 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 problemsee 2.1below. These solutions are discussed inSection 2and 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 solutionsux, t Uxct of 1.7, for suitable functions g. A two-parameter family will be constructed for each possible choice of αand 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, letα∈0,2and|ρ|<min{α,2−α}.
Consider the “free” equation
∂tux, t Aαρux, t 0, 2.1
whereAαρ is the pseudodifferential operator with symbolpαρ defined in1.8. It is known that2.1has the fundamental solution
x, t−→Wαρx, t t−1/αfαρ
xt−1/α
2.2
with initial dataWαρx,0 δ0x, the delta distribution; see8 . In particular, Wαρx,1 fαρx. Here fαρ is the probability density function of a stable distribution; see 20, 21 . There are many ways to parametrize such distributions. Forα /∈ {0,1,2}the version used here corresponds to form Bin21 with the same choice ofα, scale parameterγ 1, location parameterδ0, and skewness parameter:
β± ρ
min{α,2−α}, 2.3
where the sign is positive if 0< α <1 and negative if 1< α <2. Forα1, the form used here corresponds to formCin21 withβρ, γ 1, δ0.
There is also the special solution
x, t−→Vαρx, t Fαρ xt−1/α
, 2.4
whereFαρx x
−∞fαρsdsis a cumulative distribution function. ThenVαρsolves2.1with initial data Vαρx,0 Hx, the Heaviside function. The equations hold in the sense of distributions, and the initial data are attained in this sense. It is known thatfαρ is positive, infinitely differentiable, and unimodal. Also, asx → ∞, there are expansions
1−Fαρx∼
j≥1
cjαρx−jα, fαρx∼
j≥1
cjαρx−1−jα, 2.5
and asx → −∞
Fαρx∼
j≥1
djαρ−x−jα, fαρx∼
j≥1
djαρ−x−1−jα. 2.6
These are convergent expansions if 0< α <1 and asymptotic expansions if 1< α <2; see8 . For the remainder of this section, we suppress the subscriptsαandρin most formulae that involveAαρ, fαρ, andFαρ. For fixedc∈Randτ >0 we consider the function
Uτξ F ξτ−1/α
. 2.7
Setucτx, t Uτxct, then
Aucτx, t −∂τF
xctτ−1/α 1
α
xctτ−1/α−1 f
xctτ−1/α
∂tucτx, t cτ−1/αf
xctτ−1/α ,
2.8
and therefore
∂tucτx, t Aucτx, t 1
α
xctτ−1/α−1
cτ−1/α
f
xctτ−1/α
. 2.9
Nowxctτ−1/α F−1ucτx, tand consequently
∂tucτx, t Aucτx, t cτ−1/αg0ucτx, t 1
ατg1ucτx, t 2.10
with
g0ζ f F−1ζ
, g1ζ F−1ζf F−1ζ
. 2.11
Equation2.10is of the form1.7, withgζ cτ−1/αg0ζ 1/ατg1ζ.
In the case α 1 and−1 < ρ < 1, everything is explicit. Letκ cosπρ/2 and σsinπρ/2. Then by results in21 ,
Fx 1
2 1
π arctanx−σ κ , F−1ζ σ−κcotπζ,
fx 1
πκ
1 x−σ2/κ2, g0ζ 1
κπsin2πζ, g1ζ σ
πκsin2πζ− 1
π cosπζsinπζ.
2.12
It remains to characterize the functionsg0, g1in the general case.
Proposition 2.1. Let 0 < α < 2,|ρ| < min{α,2 −α}. The functions g0, g1 have the following properties.
ag0andg1are infinitely differentiable on0,1.
bThe functiong0is positive on0,1. The functiong1is negative on0, Fαρ0and positive onFαρ0,1. The function
ζ−→cτ−1/αg0ζ 1
ατg1ζ 2.13
is negative on0, u∗and positive onu∗,1, whereu∗Fαρ−cατ−1/α1. cAsζ↓0,g0ζ Oζ11/αandg01−ζ Oζ11/α.
dAsζ↓0,g1ζ −αζOζ11/α. Asζ↑1,g1ζ α1−ζ O1−ζ11/α. eThe functionsg0andg1can be represented as
g0ζ d dζ
F−1ζ
−∞ f2sds, g1ζ d
dζ
F−1ζ
−∞ sf2sds.
2.14
Proof. Propertyafollows sincefandFtogether with its inverse are infinitely differentiable.
Propertybis obvious. Propertiescanddfollow from the asymptotic expansions2.5 and2.6. Finallyecan be checked by differentiation.
Propertyewill not be used in what follows. It should be noted thatg0andg1are of classC1on0,1 but are not infinitely differentiable at the interval endpoints, except ifα1.
Clearlyg0andg1do not depend oncorτ. We are therefore free to form fairly arbitrary linear combinations ofg0andg1by choosingcandτ.
The construction provides travelling wave solutions for 1.7 for a special class of functions for which g ∈ C10,1 ,g0 gu∗ g1for some u∗ ∈ 0,1, and g0 <
0, gu∗>0, g1<0. This suggests that1.7possesses travelling wave solutions for more general functionsgwith these properties.
If the same construction is attempted for the caseα2, it turns out thatg0andg1are merely continuous on0,1 , with derivatives that have logarithmic singularities nearζ 0 andζ1. Therefore the arguments in the next section cannot be extended to the caseα2, 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 operatorAhas symbol 1.8with 0 < α < 2 and|ρ|< min{α,2−α}. We always assume thatuis a solution of1.7 and thatg∈C10,1 ,Rwithg0 g1 0. Initial datau0will be assumed to satisfy
u0∈CR,R, 0≤u0x≤1, lim
x→ ∞u0x 1, suppu0⊂0,∞. 3.1 The results inSection 4then imply that1.7has a unique mild solutionuthat exists for all x∈R, t >0, and this solution satisfies 0≤ux, t≤1 for allx, 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.
Theorem 3.1. Letube the solution of1.7withu0satisfying3.1.
aLetα >1. Assume thatg >0 on0,1and that there are 0< γ < α/α−1, c0 >0 such that for allζ∈0,1/2
gζ≥c0ζγ. 3.2
Then for allc >0
lim inf
t→ ∞ inf
x≥−ctux, t 1. 3.3
bLetα1. Assume thatg >0 on0,1. Then for allc >0 lim inf
t→ ∞ inf
x≥−ctux, t 1. 3.4
cLetα < 1. Assume thatg ≥0 on0,1andgζ>0 forζ∈α−ρ/2α,1. Then for all c >0
lim inf
t→ ∞ inf
x≥−ctux, t 1. 3.5
The result shows that the speed of spread is unboundedi.e.,1.6holds for allc >0, and it exhibits different mechanisms for this phenomenon. Recall that in the interpretation of1 , the functiongis responsible for the growth of a substance whose density is given by u, whileAdescribes the spread of this substance. Ifα∈0,2, the substance spreads with a jump process, not with Brownian diffusion, and jumps of magnitude exceedingMoccur with a probability that isOM−αfor largeM. Ifα > 1, the mean jump distance is still finite. In this case, the growth rateguat small densitiessmalluis responsible for the unbounded speed of spread. Ifαis close to 1, this growth can be very weakgζ∼ζγ with largeγyet the speed of spread is still unbounded. If on the other handα <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 densitiesgu 0 for smallufor this to occur. Rather, the unbounded speed of spread results from growth that occurs solely for large densities gζ>0 only forζ≥α−ρ/2α. The substance is transported towards−∞due largeα <1 negative jumps, resulting in an unbounded speed of spread. It is known that in this case, α−ρ/2αis the probability that a jump is negative. If this fraction is large, then growth that occurs only for large densities, that is,gu > 0 onα−ρ/2α,1, already leads to an unbounded speed of spread. The caseα 1 is intermediate: any growth for small densities gζ>0 forζ >0results 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 ifγ ≥ α/α−1or if a finite speed of spread occurs 1.5holds for largecif γbecomes sufficiently large, that is, if growth is extremely weak for small densitiesu. In the caseα <1, it would be interesting to know if a finite speed of spread is possible at all ifg≥0 andgis not identically equal to 0.
The main result in the case wheregis negative nearζ0 is the following.
Theorem 3.2. Letube the solution of1.7with initial datau0satisfying3.1. Assume thatg0<
0. Then there existsc >0 such that
lim sup
t→ ∞ sup
x≤−ctux, t 0. 3.6
The result shows that negative proportional growth at small densities g0 < 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 largeα <1. I am not aware of an interpretation of this result in the context of material science, similar to the use of1.1in5 . 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 ∈C10,1 ,Rand letg0, g1be defined as in2.11, depending onαandρ.
aLetα ∈ 1,2. Suppose thatgζ > 0 for allζ ∈ 0,1 and that there exist c0 > 0 and γ < α/α−1such thatgζ≥c0ζγfor allζ∈0,1/2 . Then given anyc >0 there exists τ >0 such that for allζ∈0,1
gζ≥cτ−1/αg0ζ ατ−1g1ζ. 3.7
bLetα1. Suppose thatg >0 on0,1 . Then given anyc >0 there existsτ >0 such that for allζ∈0,1
gζ≥cτ−1g0ζ τ−1g1ζ. 3.8
cLetα∈0,1. Suppose thatg ≥0 on0,1 andgζ>0 for allζ∈α−ρ/2α,1. Then given anyc >0 there existsτ >0 such that for allζ∈0,1
gζ≥cτ−1/αg0ζ ατ−1g1ζ. 3.9 dSuppose thatg0<0 andgζ 0 forζ∈1−,1 for some. Then there existc∈R
andτ >0 such that for allζ∈0,1
gζ≤cτ−1/αg0ζ ατ−1g1ζ. 3.10 Proof. Consider first statementa. Let us writeF Fαρandf fαρ. Letα >1 and letM >0 be large enough such that for somec1, c2>0 and allx≤ −M
Fx≥c1|x|−α, fx≤c2|x|−1−α. 3.11
This is possible by2.6. Letc >0 be given, then we may increaseMfurther such that also c0cγ1Mr ≥c2
α−1
α cα/α−1, 3.12
wherer α/α−1−γ >0. Now setδF−M. Then for 0< ζFx≤δ, that is,x <−M, and for allτ≥M/cα/α−1
cτ−1/αg0Fx ατ−1g1Fx
cτ−1/α ατ−1x fx
≤ α−1
α cα/α−1|x|1/1−αfx
≤ α−1
α cα/α−1|x|1/1−αc2|x|−1−α α−1
α cα/α−1c2|x|−α2/α−1,
3.13
where a standard calculus argument has been used to see that the expression cτ−1/α ατ−1xis maximal forτ |x|/cα/α−1. We estimate further, using the choice ofM,
cτ−1/αg0Fx ατ−1g1Fx≤c0cγ1Mr|x|−α2/α−1
≤c0cγ1|x|r|x|−α2/α−1c0 c1|x|αγ
≤c0Fxγ≤gFx.
3.14
Therefore, for allτ≥M/cα/α−1and allζ < δF−M,
cτ−1/αg0ζ ατ−1g1ζ≤gζ. 3.15
Sinceg >0 onδ,1 by assumption, this inequality can be achieved also onδ,1 by increasing τeven further. This proves parta.
The proof of partbis straight forward: givenc >0, note thatcg0ζ g1ζ≤0 on the interval0, F−c . Thencg0ζ g1ζ/τ≤gζis true ifτis sufficiently large.
To prove partc, let againc > 0 be given. Letu∗ inf{u ∈ 0,1 | gu > 0}. Then u∗<α−ρ/2αF0. Pickτlarge enough such thatF−cατ1−1/α> u∗. This is possible since α <1. Then on0, u∗ ,
cτ−1/αg0ζ ατ−1g1ζ≤gζ 3.16
since the left-hand side is nonpositive there byProposition 2.1. By increasingτ further, we can obtain this estimate also forζ ∈u∗,1 , using again thatgis assumed to be positive on 1−β/2,1 .
To prove partd, note first that
cτ−1/αg0ζ ατ−1g1ζ≥gζ 3.17
on an interval0, δ as soon asτ−1 g0 > 0, that is, for sufficiently smallτ. Increasingc sufficiently and noting thatg 0 nearζ 1 extends this inequality to the entire interval 0,1 .
Proof ofTheorem 3.1. The proof uses the same argument for all three parts; so we give details only in part a. Let > 0. We replace u with u 1u and g with g, where gζ 1g1−1ζ. Then
∂tuAug u. 3.18
Thengζ≥c0ζγforζ∈0,1/2 , possibly with a changedc0, and additionallyg > 0 on0,1 . Letc >0 be given, then there existsτ >0 such that
gζ≥c1τ−1/αg0ζ ατ−1g1ζ 3.19
for allζ∈0,1 byLemma 3.3. By extendingg0andg1to be zero on1,1 , this inequality is true on0,1 . Now find a constantdsuch thatv0x Hx−d≤ux, 0for allx. This is possible since limx→ ∞ux, 0 1. ByProposition 4.3, we see that
ux, t≥F
x−dt−1/α
3.20
for allx∈R, t >0. This is in particular true fortτ. Now useProposition 4.1and3.19to infer that
ux, t≥F
x−d c1tτ−1/α
3.21
for allx∈R, t≥τ. Therefore fort≥τandx≥ −ct,
ux, t≥F
x−d c1tτ−1/α
≥F
−ct−d c1tτ−1/α F
t−dτ−1/α .
3.22
Ast → ∞, the right-hand side goes to 1. Rewriting this in terms ofu, we see that lim inf
t→ ∞ inf
x≥−ctux, t ≥1−1. 3.23
Since >0 was arbitrary, the desired result follows.
In case of partb, the same argument can be used without changes, appealing to part bofLemma 3.3.
In case of partc, we have to restrict such that g > 0 on α−ρ/2α,1 , that is, g >0 onα−ρ/2α1,1 . The rest of the proof is again unchanged, using partcof Lemma 3.3.
Proof ofTheorem 3.2. We replaceuwithu 1/2uandgwithg, where gζ 1/2g 2ζfor 0≤ζ≤1/2 andgζ 0 forζ∈1/2,1 . Theng0 g0<0 and
∂tuAug u. 3.24
ByLemma 3.3, partd, there existc >0 andτ >0 such that
gζ≤c−1τ−1/αg0ζ ατ−1g1ζ. 3.25 Since limx→ ∞ux, 0 1/2 andux,0 0 for x < 0, we can findd > 0 such thatFx dτ−1/α≥ux, 0for allx∈R. UsingProposition 4.1, one sees that
F
xd c−1tτ−1/α
≥ux, t 3.26
for allx∈R, t >0. Therefore fort >0 andx≤ −ct,
ux, t≤F
xd c−1tτ−1/α
≤F
−ctd c−1tτ−1/α F
−tdτ−1/α .
3.27
The right-hand side tends to 0 ast → ∞. In terms ofu, this implies lim sup
t→ ∞ sup
x≤−ctux, t 0. 3.28
This concludes the proof.
4. Facts about Fractional Reaction-Diffusion Equations
In this section we summarize some basic theory about1.7that is needed in this note. A broader and deeper discussion may be found in10 .
We work in the Banach space Clim
w∈CR| lim
x→ ∞wxand lim
x→ −∞wxexist
4.1
equipped with the supremum norm · . Let A Aαρ be the pseudodifferential operator with symbol1.8and parametersα, ρ, with 0 < α < 2 and|ρ| < min{α,2−α}. Subscripts α, ρwill again frequently be suppressed. Solutions of the free equation2.1with initial data u·,0 ϕ ∈ Clim then can be written in terms of the fundamental solution given in2.2, namely,
ux, t
RWαρ
x−y, t ϕ
y
dy. 4.2
For fixedαandρandϕ∈ Clim, we therefore define
Stϕx ux, t, 4.3
where u is given by 4.2. Then Stt≥0 is a positive C0 semigroup on Clim and a Feller semigroup on the subspace of functions inClimthat vanish at±∞. Ifψis a continuous function from0, T toClim, then solutions of the inhomogeneous equation
∂tux, t Aux, t ψx, t, u·,0 ϕ 4.4
can be written with the variation-of-constants formula:
u·, t Stϕ t
0
St−sψ·, sds. 4.5
A continuous curveu:0, T → Clim that satisfies4.5is commonly called a mild solution of4.4. Next letg :0,∞×R → Rbe locally Lipschitz continuous in both variables and let ϕ∈ Clim. Then the equation∂tux, t Aux, t gt, ux, t for which1.7is a special case has a unique mild solutionu∈C0, T,Clim, where 0< T ≤ ∞is maximal. EitherT ∞, or
u·, t → ∞ast↑T. The solution can be obtained as the locally in time uniform limit of the iteration scheme:
un1·, t Stϕ t
0
St−sgs, un·, sds n0,1, . . . 4.6
withu0 being arbitrary, for example,u0·, t Stϕ. 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 in22 , following the seminal work on this topic in23 . For the sake of completeness, a comparison result is stated here, and its proof is sketched.
Proposition 4.1. Letu, v∈C0, T ,Climbe mild solutions of the equations
∂tuAugu, ∂tvAvhv, 4.7
whereg, h:R → Rare locally Lipschitz continuous. If gζ≤hζ, ∀ζ∈R,
u·,0≤v·,0, 4.8
then
ux, t≤vx, t ∀x, t∈R×0, T . 4.9
Proof. LetMmax0,T u·, tv·, t1. Letλ >|gζ||hζ|for all|ζ| ≤M. Without loss of generality we may assume thatgandhare constant outside−M, M . Set
Ux, t eλtux, t, Vx, t eλtvx, t, 4.10
and observe thatUandV satisfy
∂tUAUgt, U,
∂tVAV ht, V 4.11
withgt, ζ λζeλtge−λtζandht, ζdefined similarly. Clearly,gt, ζ ≤ ht, ζfor allζ.
The functiongis nondecreasing in its second argument, since for almost allζ
∂ζgt, ζ λg e−λζ
≥0. 4.12
Consider the iteration scheme:
Un1·, t Stu·,0 t
0
St−sgs, Un·, sds 4.13
and similarly forVnandh. The scheme for theUnconverges to the limitU, and the scheme for theVnconverges to the limitV.
We now employ a standard induction argument to show thatUn≤VnonR×0, T for alln. LetU0·, t Stu·,0andV0·, t Stv·,0, thenU0 ≤V0onR×0, T sinceSis a positive semigroup andu·,0≤v·,0. SupposeUn ≤VnonR×0, T , then
Un1·, t Stu·,0 t
0
St−sgs, Un·, sds
≤Stv·,0 t
0
St−sgs, Vn·, sds
≤Stv·,0 t
0
St−shs, Vn·, sds Vn1·, t,
4.14
which completes the induction step. Taking the limit, this implies thatU ≤V and therefore alsou≤vonR×0, T . This proves the proposition.
Corollary 4.2. Consider a mild solutionu ∈ C0, T,Climof 1.7and assume thatg is locally Lipschitz continuous. Ifgγ ≥ 0 for someγ andu·,0 ≥ γ, thenu·, t ≥ γ for allt. If gγ ≥ 0 ≥gδfor someγ < δandγ ≤ u·,0≤ δ, thenγ ≤u·, t≤ δfor allt, and the solution can be continued toR×0,∞.
The proof consists in observing that the constant functionsvx, t γand wx, t δ solve 1.7 with right-hand sides 0 and therefore must be pointwise bounds for the solution, byProposition 4.1. If the solution remains bounded between two constants, then its supremum norm remains bounded and it can be continued toR×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. Let u ∈ C0, T ,Clim be a mild solution of 1.7, with locally Lipschitz continuousg. Assume that
ux,0≥v0x a0N
j1
ajH x−cj
∀x∈R, 4.15
where aj ∈ R, c1 < c2· · · < cN, and H is the Heaviside function. Letγ minRv0xand δ maxRv0x. Assume also thatg≥0 onγ, δ . Then
ux, t≥a0N
j1
ajVαρ
x−cj, t
∀x∈R,0< t≤T, 4.16
whereVαρis defined in2.4.
Proof. We know thatux, t≥γand thus may assume thatgζ≥0 also forζ < γ. For arbitrary , σ >0, we set
vσx 1 σ
σ 0
v0x−zdz−. 4.17
Thenvσis piecewise linear and constant outside the intervalc1, cNσ ; in particular,vσ ∈ Clim. Solving2.1with initial datav,σgives the solution
Gσx, t 1 σ
σ 0
N j1
ajVαρ
x−z−cj, t
a0−. 4.18
Given > 0, it is possible to findσ > 0 such that vσ < u0xonR, sinceu0 is uniformly continuous. Clearly,γ−≤Gσ ≤δ. We may therefore viewGσ as a solution of1.7with a right-hand sidehthat satisfieshζ 0≤gζforζ≤δandhζ≤gζalso forζ > δ. Then byProposition 4.1
ux, t≥G,σx, t ∀x∈R,0< t≤T. 4.19
Sendδto 0, then sinceVαρ is uniformly continuous,4.16is obtained witha0 replaced by a0−on the right-hand side. Now sendto 0 and4.16follows.
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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