Exponential propagation for fractional reaction-diffusion cooperative systems with fast decaying initial conditions
aa r X i v : . [ m a t h . A P ] O c t Exponential propagation for fractional reaction-diffusioncooperative systems with fast decaying initial conditions.
Anne-Charline COULON a and Miguel YANGARI a,ba Institut de Math´ematiques, Universit´e Paul Sabatier118 Route de Narbonne, F-31062 Toulouse Cedex 4, France. b Departamento de Ingenier´ıa Matem´atica, Universidad de ChileBlanco Encalada 2120, Santiago, Chile.
Abstract
We study the time asymptotic propagation of sectorial solutions to the fractionalreaction-diffusion cooperative systems. We prove that the propagation speed is expo-nential in time, and we find the precise exponent of propagation. This exponent dependson the smallest index of the fractional laplacians and on the principal eigenvalue of thematrix DF (0) where F is the reaction term. We also note that this speed does notdepend on the space direction. The reaction diffusion equation with Fisher-KPP nonlinearity ∂ t u + ( −△ ) α u = f ( u ) (1.1)with α = 1, has been the subject of intense research since the seminal work by Kolmogorov,Petrovskii, and Piskunov [1]. Of particular interest are the results of Aronson and Weinberger[2] which describe the evolution of compactly supported data. They showed that there existsa critical threshold c ∗ = 2 p f ′ (0) such that, for any compactly supported initial value u in [0 , c > c ∗ then u ( t, x ) → {| x | ≥ ct } as t → + ∞ and if c < c ∗ then u ( t, x ) → {| x | ≤ ct } as t → + ∞ . This corresponds to a linear propagation ofthe fronts. In addition, (1.1) admits planar travelling wave solutions connecting 0 and 1. Email addresses: [email protected], [email protected] Acknowledgements:
The research leading to these results has received funding from the European Re-search Council under the European Unions Seventh Framework Programme (FP/2007-2013) / ERC GrantAgreement n.321186 - ReaDi -Reaction-Diffusion Equations, Propagation and Modelling. M. Y. was sup-ported by Becas de Doctorado, Conicyt-Chile and Senescyt-Ecuador. The authors thank Professor J.-M.Roquejoffre for fruitful discussions. α ∈ (0 ,
1) in (1.1),appear in physical models when the diffusive phenomena are better described by L´evy pro-cesses allowing long jumps, than by Brownian processes - obtained when α = 1. The L´evyprocesses occur widely in physics, chemistry and biology. Recently these models have at-tracted much interest. In connection with the discussion given above, in the recent paper[3], Cabr´e and Roquejoffre showed that for any compactly supported initial condition, ormore generally for initial values decaying faster than | x | − d − α , where d is the dimensionof the spatial variable, the speed of propagation becomes exponential in time. They alsoshowed that no travelling wave exist. Their result was sharpened and extended in [4], whoproposed a new (and more flexible) argument to treat models of the form (1.2). In the casein which the initial condition decay slowly, [5] states that the solution spreads exponentiallyfaster with a larger index than in the previous case. All these results are in great contrastwith the case α = 1. They indeed notice that diffusion only plays a role for small times,the large time dynamics being given by a simple transport equation. The scheme of theirproof will be reproduced here, but some steps - and this is why it makes system (1.2) worthstudying - become less easy. The small time study will require the manipulation of somePolya integrals, and the transport equation will also become more complex.The work on the single equation (1.1) can be extended to reaction-diffusion systems.The first definitions of spreading speeds for cooperative systems in population ecology andepidemic theory are due to Lui in [6]. In a series of papers, Lewis, Li and Weinberger[7],[8], [9] studied spreading speeds and travelling waves for a particular class of cooperativereaction-diffusion systems, with standard diffusion. Results on single equations in the singu-lar perturbation framework proved by Evans and Souganidis in [10] have also been extendedby Barles, Evans and Souganidis in [11]. The viscosity solutions framework is studied in [12],with a precise study of the Harnack inequality. In these papers, the system under study isof the following form ∂ t u i − d i ∆ u i = f i ( u ) , where, for m ∈ N ∗ , u = ( u i ) mi =1 is the unknown.For all i ∈ J , m K := { , ..., m } , the constants d i are assumed to be positive as well asthe bounded, smooth and Lipschitz initial conditions, defined from R d to R + . The essentialassumptions concern the reaction term F = ( f i ) mi =1 . This term is assumed to be smooth, tohave only two zeroes 0 and a ∈ R m in [0 , a ], and for all i ∈ J , m K , each f i is nondecreasingin all its components, with the possible exception of the ith one. The last assumption meansthat the system is cooperative. Under additional hypotheses, which imply that the point 0is unstable, the limiting behaviour of the solution u = ( u i ) mi =1 is understood.Here, we focus on similar systems, keeping the same assumptions on f , but consideringthat at least one diffusive term is given by a fractional Laplacian. More precisely, we focuson the large time behaviour of the solution u = ( u i ) mi =1 , for m ∈ N ∗ , to the fractionalreaction-diffusion system: (cid:26) ∂ t u i + ( −△ ) α i u i = f i ( u ) , t > , x ∈ R d ,u i (0 , x ) = u i ( x ) , x ∈ R d , (1.2)2here α i ∈ (0 ,
1] and α := min J ,m K α i < . Note that when α i = 1, then ( −△ ) α i = −△ . As general assumptions, we impose, for all i ∈ J , m K , the initial condition u i to be nonnegative, non identically equal to 0, continuousand to satisfy u i ( x ) = O( | x | − ( d +2 α i ) ) as | x | → + ∞ . (1.3)We also assume that for all i ∈ J , m K , the function f i satisfies f i (0) = 0 and that system(1.2) is cooperative, which means: f i ∈ C ( R m ) and ∂ j f i > , on R m , for all j ∈ J , m K , j = i. (1.4)We will make additional assumptions on the reaction term F = ( f i ) mi =1 that are notgeneral but enable us to understand the long time behaviour of a class of monotone systems.The aim of this paper is to understand the time asymptotic location of the level setsof solutions to (1.2). We show that the speed of propagation is exponential in time, witha precise exponent depending on the smallest index α := min i ∈ J ,m K α i and on the principaleigenvalue of the matrix DF (0) where F = ( f i ) mi =1 . Also we note that this speed does notdepend on the space direction.For what follows and without loss of generality, we suppose that α i +1 ≤ α i for all i ∈ J , m − K so that α = α m <
1. Before stating the main results, we need some additionalhypotheses on the nonlinearities f i , for all i ∈ J , m K .(H1) The principal eigenvalue λ of the matrix DF (0) is positive,(H2) There exists Λ > s = ( s i ) mi =1 ∈ R m + satisfying | s | ≥ Λ, we have f i ( s ) ≤ s = ( s i ) mi =1 ∈ R m + satisfying | s | ≤ Λ, Df i (0) s − f i ( s ) ≥ c δ s i δ ,(H4) For all s = ( s i ) mi =1 ∈ R m + satisfying | s | ≤ Λ, Df i (0) s − f i ( s ) ≤ c δ | s | δ , (H5) F = ( f i ) mi =1 is globally Lipschitz on R m ,where the constants c δ and c δ are positive and independent of i ∈ J , m K , and for all j ∈ { , } δ j ≥ d + 2 α . This lower bound on δ and δ is a technical assumption to make the supersolution andsubsolution to (1.2), we construct, to be regular enough. Note that one may easily produceexamples of functions F satisfying (H1) to (H5).We are now in a position to state our main theorem, which show that the solution to(1.2) move exponentially fast in time. 3 heorem 1.1 Let d ≥ and assume that F satisfies (1.4) and (H1) to (H5). Let u bethe solution to (1.2) with a non negative, non identically equal to 0 and continuous initialcondition u satisfying (1.3). Then there exists τ > large enough such that for all i ∈ J , m K ,the following two facts are satisfied: a) For every µ i > , there exists a constant c > such that, u i ( t, x ) < µ i , f or all t ≥ τ and | x | > ce λ d +2 α t . b) There exist constants ε i > and C > such that, u i ( t, x ) > ε i , f or all t ≥ τ and | x | < Ce λ d +2 α t . The plan to set Theorem 1.1 is organized as follows. First, in the short section 2, we statea local existence result of solutions for cooperative systems involving fractional diffusion andwe state a comparison principle for this type of solutions which, although standard, is crucialfor the sequel. In Section 3 we deal with finite time and large x decay estimates. The endof this paper, Section 4 is devoted to the proof of Theorem 1.1. Recall that the operator A = − diag(( − ∆) α , . . . , ( − ∆) α m ) is sectorial (see [13]) in ( L ( IR d )) m ,with domain D ( A ) = H α ( IR d ) × . . . × H α m ( IR d ). If now u satisfies the assumptions ofTheorem 1.1, it is in ( L ( IR d )) m , so that the Cauchy Problem (1.2) has a unique maximalsolution, defined on an interval of the form [0 , t max ); moreover the L -norm of u blows up as t → t max if t max < + ∞ . Finally, we have u ∈ C ((0 , t max ) , D ( A )) ∩ C ([0 , t max ) , ( L ( IR d )) m )and dudt ∈ C ((0 , t max ) , ( L ( IR d )) m ). A standard iteration argument and Sobolev embeddingsthen yield u ∈ C p ((0 , t max ) , ( H q ( R d )) m )for every integer p and q . Theorem 2.1
Consider
T > , and let u = ( u i ) mi =1 and v = ( v i ) mi =1 such that: u ∈ C ((0 , T ] , D ( A )) ∩ C ([0 , T ] , ( L ( R d )) m ) ∩ C ((0 , T ) , ( L ( R d )) m ) ; and v ∈ C ([0 , T ] × IR d ) ∩ C ((0 , T ) × IR d ) . Assume that, for all i ∈ J , m K , we have ∂ t u i + ( −△ ) α i u i ≤ f i ( u ) , ∂ t v i + ( −△ ) α i v i ≥ f i ( v ) , where f i satisfies (1.4). If for all i ∈ J , m K and x ∈ R d , u i (0 , x ) ≤ v i (0 , x ) we have u ( t, x ) ≤ v ( t, x ) for all ( t, x ) ∈ [0 , T ] × R d . Proof.
Let us define for all i ∈ J , m K , w i = u i − v i . Then w i satisfies w i (0 , x ) ≤ ∂ t w i + ( −△ ) α i w i ≤ f i ( u ) − f i ( v ) = Z ∇ f i ( σu + (1 − σ ) v ) dσ. ( u − v )= Z ∇ f i ( ζ σ ) dσ.w, (2.1)4here ζ σ = σu + (1 − σ ) v . Notice now that w + i ∈ C ((0 , T ) , H α i ( R d )) ∪ W , ∞ ((0 , T ) , L ( R d )).So, taking the scalar product of (2.1) with the vector function ( w + i ) mi =1 and integrating over R d , we have Z R d w + i ∂ t w i dx + Z R d w + i ( −△ ) α i w i dx ≤ Z R d w + i Z ∇ f i ( ζ σ ) dσ.w dx. (2.2)Recall that Z R d w + i ( −△ ) α i w i dx ≥
0. So we have, since ∂ j f i ( ζ σ ) ≥ ddt (cid:20)Z R d ( w + i ) dx (cid:21) ≤ Z R d Z ∂ i f i ( ζ σ ) dσ ( w + i ) dx + m X j =1 ,j = i Z R d Z ∂ j f i ( ζ σ ) dσw + i w + j dx ≤ C m X j =1 Z R d ( w + j ) dx, where C is a constant that depends on m . Doing this procedure for each i ∈ J , m K andadding, we get for t ∈ [0 , T ] ddt " m X i =1 Z R d ( w + i ) dx ≤ C m X i =1 Z R d ( w + i ) dx. So, by Gronwall’s inequality, we have w i ≤ , T ] × IR d . (cid:3) From hypothesis (H2), we deduce that the positive vector M = Λ , where is the vectorof size m with all entries equal to 1, is a supersolution to (1.2), if the initial condition u = ( u i ) mi =1 is smaller than M . So, from Theorem 2.1, we have 0 ≤ u ( t, x ) ≤ M . Toprove global existence, it remains to prove a locally finite L bound; this is done in the nextsubsection. Now, we are in position to establish an algebraic upper bound for the solutions of (1.2).From (H5), we know that, for i ∈ J , m K and j ∈ J , m K | ∂ j f i ( s ) | ≤ Lip ( f i ) , for all s ∈ R m , where Lip ( f i ) is the Lipschitz constant of f i . Taking l = max i ∈ J ,m K Lip ( f i ), we have for all s = ( s i ) mi =1 ≥ f i ( s ) = Z Df i ( σs ) dσ · s ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X j =1 s j Z ∂f i ∂s j ( σs ) dσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ l m X j =1 s j . (3.1)5et us consider v = ( v i ) mi =1 the solution of the following system (cid:26) ∂ t v + Lv = Bv, t > , x ∈ R m v (0 , · ) = u , R m , (3.2)where L = diag(( −△ ) α , ..., ( −△ ) α m ), B = ( b ij ) mi,j =1 is a matrix with b ij = l for all i, j ∈ J , m K . By (3.1) and Theorem 2.1, we conclude that u ≤ v in [0 , + ∞ ) × R d . A finite timeupper bound for u is given by the following lemma. Lemma 3.1
Let d ≥ and let u = ( u i ) mi =1 be the solution of system (1.2), with a nonnegative, non identically equal to 0 and continuous initial condition u satisfying (1.3), andreaction term F = ( f i ) mi =1 satisfying (1.4) and (H1) to (H5). Then, for all i ∈ J , m K , thereexists a locally bounded functions C : (0 , + ∞ ) → R + such that for all t > and | x | largeenough, we have u i ( t, x ) ≤ C ( t ) | x | d +2 α . Taking Fourier transforms in each term of system (3.2), we have (cid:26) ∂ t F ( v ) = ( A ( | ξ | ) + B ) F ( v ) , ξ ∈ R d , t > F ( v )(0 , · ) = F ( u ) , R d , where A ( | ξ | ) = diag( −| ξ | α , ..., −| ξ | α m ). Thus, we have that F ( v )( t, ξ ) = e ( A ( |·| )+ B ) t F ( u )( ξ )and then, for all x ∈ R d and t ≥ u ( t, x ) ≤ v ( t, x ) = F − ( e ( A ( |·| )+ B ) t ) ∗ u ( x ) . (3.3)In what follows, we prove that for each time t >
0, the solution of (1.2) decays as | x | − d − α forlarge values of | x | . Due to the decay of u at infinity, we only need to prove that the entriesof F − ( e ( A ( |·| )+ B ) t ) have the desired decay. The following lemma is needed to prove that wecan rotate the integration line of a small angle ε > F − ( e ( A ( |·| )+ B ) t ). Lemma 3.2
For all z ∈ { z ∈ C | ≤ arg( z ) < π α } and t ≥ , we have (cid:13)(cid:13)(cid:13) e ( A ( z )+ B ) t (cid:13)(cid:13)(cid:13) ≤ e ( k B k−| z | α cos(2 α arg( z ))) t + e ( k B k−| z | α cos(2 α arg( z ))) t , (3.4) and if I t ( z ) := Z t e ( t − s )( A ( z )+ B ) [ e sB , A ( z )] e sA ( z ) ds, (3.5) then there exists C : (0 , ∞ ) → R + a locally bounded function such that k I t ( z ) k ≤ C ( t )( | z | α e −| z | α cos(2 α arg( z )) t + | z | α e −| z | α cos(2 α arg( z )) t ) . (3.6)6 roof. Let z be in { z ∈ C | ≤ arg( z ) < π α } . There exist j ∈ J , m K , and k ∈ J , m K suchthat (cid:13)(cid:13) e ( A ( z )+ B ) t (cid:13)(cid:13) = ( e ( A ( z )+ B ) t ) jk . Consider the system (cid:26) ∂ t w = ( A ( z ) + B ) w, z ∈ C , t > ,w (0 , z ) = e j z ∈ C , where e j is the j th vector of the canonical basis of R m . Thus, we have w ( t, z ) = e ( A ( z )+ B ) t .e j and k w k = (cid:13)(cid:13)(cid:13) e ( A ( z )+ B ) t (cid:13)(cid:13)(cid:13) . Multiply (3.1) by the conjugate transpose w and take the real part to get12 ∂ t k w k + m X l =1 cos(2 α l arg( z )) | z | α l | w l | = Re ( Bw.w ) ≤ k B k k w k . The choice of arg( z ) and Gronwall’s Lemma end the proof.To prove (3.6), it is sufficient to notice that, for s ∈ [0 , t ], we have (cid:13)(cid:13)(cid:13) e sA ( | z | e i arg( z ) ) (cid:13)(cid:13)(cid:13) ≤ e −| z | α cos(2 α arg( z )) s + e −| z | α cos(2 α arg( z )) s , (cid:13)(cid:13)(cid:13) [ e sB , A ( | z | e i arg( z ) )] (cid:13)(cid:13)(cid:13) ≤ C ( t )( | z | α + | z | α ) , where C : (0 , + ∞ ) → R + is a locally bounded function, and due to (3.4), we also have (cid:13)(cid:13)(cid:13) e ( A ( | z | e i arg( z ) )+ B )( t − s ) (cid:13)(cid:13)(cid:13) ≤ e ( k B k−| z | α cos(2 α arg( z )))( t − s ) + e ( k B k−| z | α cos(2 α arg( z )))( t − s ) . (cid:3) Proof for d = 1 . In this proof, we denote by C : (0 , + ∞ ) → R + a locally bounded function.From (3.3), we only have to find an upper bound to F − ( e ( A ( |·| )+ B ) t ). First, we consider for t ≥ z ∈ C , w ( t, z ) := e tB e tA ( z ) . Thus, w satisfies the Cauchy problem (cid:26) ∂ t w = ( A ( z ) + B ) w + [ e tB , A ( z )] e tA ( z ) , t > , z ∈ C w (0 , z ) = Id, z ∈ C , where [ e tB , A ( z )] = e tB A ( z ) − A ( z ) e tB . By Duhamel’s formula, we get for all z ∈ C and t ≥ e t ( A ( z )+ B ) = e tB e tA ( z ) − Z t e ( t − s )( A ( z )+ B ) [ e sB , A ( z )] e sA ( z ) ds. (3.7)Thus, for all t > x ∈ R , we have F − ( e ( A ( |·| )+ B ) t )( x ) = Z R e ixξ e ( A ( | ξ | )+ B ) t dξ (3.8)= Z R e ixξ e tB e tA ( | ξ | ) dξ − Z R e ixξ I t ( | ξ | ) dξ = e tB diag( p α ( t, x ) , ..., p α m ( t, x )) − Z R e ixξ I t ( | ξ | ) dξ, i ∈ J , m K , p α i is the heat kernel of the operator ( − ∆) α i in R , [3]. Since α =min i ∈ J ,m K α i ∈ (0 , | x | , we clearly have (cid:13)(cid:13) e tB diag( p α ( t, x ) , ..., p α m ( t, x )) (cid:13)(cid:13) ≤ C ( t ) | x | α . (3.9)It remains to bound from above the following quantity: Z R e ixξ I t ( | ξ | ) dξ = 2 Z ∞ cos( xr ) I t ( r ) dr = 2 ℜ e (cid:18)Z ∞ e ixr I t ( r ) dr (cid:19) . We use the following two facts. First, for all t ≥
0, the function z e ixz I t ( z ) is holomor-phic on C \ { } . Second, for δ > R > {± δe iθ , θ ∈ [0 , ε ] } (respec-tively {± Re iθ , θ ∈ [0 , ε ] } ), the entries of I t tends to 0 as δ tends to 0 (respectively R tends to+ ∞ , due to Lemma 3.2). Consequently, we can rotate the integration line of a small angle ε ∈ (0 , π α ) and the quantity we have to bound from above becomes R ∞ e ixre iε I t ( re iε ) dr ,with I t ( re iε ) = Z t e ( t − s )( A ( re iε )+ B ) [ e sB , A ( re iε ))] e sA ( re iε )) ds. From Lemma 3.2, taking η t = (cid:13)(cid:13)(cid:13)(cid:13)Z ∞ e ixre iε I t ( re iε ) dr (cid:13)(cid:13)(cid:13)(cid:13) we get, for large values of | x | η t ≤ C ( t ) Z ∞ e − xr sin( ε ) ( r α e − r α cos(2 α ε ) t + r α e − r α cos(2 α ε ) t ) dr ≤ C ( t ) | x | α Z ∞ e − ˜ r sin( ε ) (˜ r α e − ˜ r α | x | α cos(2 α ε ) t + ˜ r α e − ˜ r α | x | α cos(2 α ε ) t ) d ˜ r ≤ C ( t ) | x | α . (3.10)With (3.8), (3.9) and (3.10), we conclude that for large values of | x | and for all t ≥ (cid:13)(cid:13)(cid:13) F − ( e ( A ( |·| )+ B ) t )( x ) (cid:13)(cid:13)(cid:13) ≤ C ( t ) | x | α , which concludes the proof. (cid:3) Now, we state the proof of Lemma 3.1 in the higher space dimension case, i.e. when d > Proof.
As previously, from (3.3), we only need to bound from above the function F − ( e ( A ( |·| )+ B ) t ).Let t > | x | >
1, using the spherical coordinates system in dimension d >
1, the defi-nition of Bessel Function of first kind (see [14] and [15]), we have F − ( e ( A ( |·| )+ B ) t )( x ) = C d Z ∞ Z − e ( A ( r )+ B ) t cos( | x | rs ) r d − (1 − s ) d − dsdr = C d | x | d − Z ∞ e ( A ( r )+ B ) t J d − ( | x | r ) r d dr, C d is a positive constant depending on d .The matrix e ( A ( r )+ B ) t is split into two pieces as done in (3.7), to get F − ( e ( A ( |·| )+ B ) t )( x ) = e tB diag( p α ( t, x ) , ..., p α m ( t, x )) − C d | x | d − Z ∞ I t ( r ) J d − ( | x | r ) r d dr, where I t has been defined in (3.5). From [3], the first piece of the right hand side has thecorrect algebraic decay. It remains to bound from above the second piece. In fact, using theWhittaker function (defined in [15] for example), we have for all x ∈ R d and all t > C d | x | d − Z ∞ I t ( r ) J d − ( | x | r ) r d dr = C d | x | d − √ π ℜ e (cid:18)Z ∞ I t ( r ) e d − iπ W , d − (2 i | x | r ) r d − dr (cid:19) = C d | x | d √ π ℜ e (cid:18)Z ∞ I t (˜ r | x | − ) e d − iπ W , d − (2 i ˜ r )˜ r d − d ˜ r (cid:19) As done in the one dimension case, since the Whittaker function is bounded, we canrotate the integration line of a small angle ε ∈ (0 , π α ). Thus, using (3.6), we have the resultif we prove that the following integral Z ∞ (cid:12)(cid:12)(cid:12) W , d − (2 i ˜ re iε ) (cid:12)(cid:12)(cid:12) ˜ r d − (˜ r α + ˜ r α ) d ˜ r is convergent. From [14], W , d − has the following asymptotic expressions, thus W , d − ( z ) ∼ | z |→ + ∞ e − z and W , d − ( z ) ∼ | z |→ − Γ( d − ) − (cid:18) ln( z ) + Γ ′ ( d − )Γ( d − ) (cid:19) z d − , if d = 2 Γ( d − d − ) z − d , if d ≥ . (cid:3) The following result is important and needed to prove Theorem 1.1. It sets an algebraicallylower bound for the solutions of the cooperative system (1.2). This result is valid for anydimension d ∈ N ∗ . Moreover, since for all i ∈ J , m K , f i (0) = 0, we have for all s = ( s i ) mi =1 ∈ R m with 0 ≤ s ≤ M f i ( s ) = Z Df i ( σs ) dσ · s = m X j =1 s j Z ∂f i ∂s j ( ζ σ ) dσ where ζ σ = σs ∈ [0 , M ] and ∂f i ∂s j : [0 , M ] → R is continuous for all i, j ∈ J , m K , since thesystem is cooperative, there exist constants γ ij > i ∈ J , m K and j ∈ J , m K : | ∂ i f i ( ζ σ ) | ≤ γ ii and γ ij ≤ ∂ j f i ( ζ σ ) . (3.1)9 emma 3.3 Let u = ( u i ) mi =1 be the solution of the system (1.2), with non negative, nonidentically equal to 0 and continuous initial condition u satisfying (1.3) and with reactionterm F = ( f i ) mi =1 satisfying (1.4), (H1), (H2) and (H5). Then, for all i ∈ J , m K , x ∈ R d and t ≥ , we have: u i ( t, x ) ≥ c t e − γ mm t t d α +1 + | x | d +2 α , (3.2) where c is a positive constant and γ mm is defined in (3.1) . Proof.
We split the proof into three steps: first, we prove the result for i = m , which servesas an initiation of the process. In an intermediate step, for all i ∈ J , m − K , t ≥ s ∈ [0 , t − p α i ( · , t − s ) ⋆ ( s d α +1 + |·| d +2 α ) − , that decays like | x | − ( d +2 α ) for large values of | x | . In a third step, for all i ∈ J , m − K , t ≥ s ∈ [0 , t − u i ( t, · ) can be bounded from below by an expression that only depends on theintegral Z t p α i ( · , t − s ) ⋆ ( s d α +1 + |·| d +2 α ) − ds . Step 1.
We have for all x ∈ R d and t > ∂ t u m + ( −△ ) α m u m = f m ( u ) ≥ Z ∂ m f m ( ζ σ ) dσu m ≥ − γ mm u m , where γ mm is defined in (3.1). By the maximum principle of reaction diffusion equations, wehave for all t ≥ u m ( t, x ) ≥ e − γ mm t ( p α m ( t, · ) ∗ u m )( x ) , Since u m ( · ) ξ ∈ R d such that u m ( y ) ≥ C for all y ∈ B R ( ξ ) for some R >
C >
0. If | x | > R , t ≥ α := α m < p α m ( t, · ) ∗ u m )( x ) ≥ C Z | y − ξ |≤ R B − tt d α +1 + | x − y | d +2 α dy = C Z | z |≤ R B − tt d α +1 + | x − ξ − z | d +2 α dz. We also have | x − ξ − z | ≤ (2 + ξR ) | x | . Thus t d α +1 + | x − ξ − z | d +2 α ≤ (cid:18) ξR (cid:19) d +2 α t d α +1 + (cid:18) ξR (cid:19) d +2 α | x | d +2 α . Then( p α m ( t, · ) ∗ u m )( x ) ≥ CB − (2 + ξR ) d +2 α Z | z |≤ R tt d α +1 + | x | d +2 α dz = e Ctt d α +1 + | x | d +2 α , C is a positive constant. If | x | ≤ R and t ≥ p α m ( t, · ) ∗ u m )( x ) ≥ Z B (0) B − tt d α +1 + | x − y | d +2 α u m ( y ) dy ≥ B − tt d α +1 + ( R + 1) d +2 α Z B (0) u m ( y ) dy ≥ Ctt d α +1 ≥ Ctt d α +1 + | x | d +2 α , for some small constant C >
0. Then, there exist C m > x ∈ R d and t ≥ u m ( t, x ) ≥ C m te − γ mm t t d α +1 + | x | d +2 α . (3.3) Step 2.
By similar computations as done in Step 1, it is possible to find a constant
C > x ∈ R d , t ≥ s ∈ [0 , t − α i = 1 then p α i ( · , t − s ) ⋆ ( s d α +1 + |·| d +2 α ) − ( x ) ≥ π ( t − s )) d Z R d e − | y | t − s ) s d α +1 + | x − y | d +2 α dy ≥ π ( t − s )) d ( s d α +1 + | x | d +2 α ) , - if α i ∈ (0 ,
1) then p α i ( · , t − s ) ⋆ ( s d α +1 + |·| d +2 α ) − ( x ) ≥ Z R d t − s ) d αi +1 + | y | d +2 α i )( s d α +1 + | x − y | d +2 α ) dy ≥ ( t − s ) − d αi s d α +1 + | x | d +2 α . Step 3.
For i ∈ J , m − K , we have for all x ∈ R d and t ≥ ∂ t u i + ( −△ ) α i u i ≥ Z ∂ m f i ( ζ σ ) dσu m + Z ∂ i f i ( ζ σ ) dσu i ≥ γ im u m − δ i u i , where ζ σ = σu and δ i ≥ max( γ ii , γ m + 1). Then, by the maximum principle of reactiondiffusion equations and Duhamel’s formula, we have for all ( t, x ) ∈ R + × R d u i ( t, x ) ≥ e − δ i t ( p α i ( t, · ) ∗ u i )( x ) + γ im e − δ i t Z t Z R d p α i ( t − s, y ) u m ( s, x − y ) e δ i s dyds. t ≥
1, and using (3.3), we get u i ( t, x ) ≥ C m γ im e − δ i t Z t − Z R d p α i ( t − s, y ) se ( δ i − γ mm ) s s d α +1 + | x − y | d +2 α dyds Using Step 2, we get the following lower bound, for all x ∈ R d , t ≥
1, taking C i smaller ifnecessary: u i ( t, x ) ≥ C i e − δ i t t d α Z t − se ( δ i − γ mm ) s − ss d α +1 + | x | d +2 α ds ≥ C i te − γ mm t t d α +1 + | x | d +2 α . (cid:3) Inspired by the formal analysis done in [4], we construct an explicit supersolution (respec-tively subsolution) of the form v ( t, x ) = a (cid:16) b ( t ) | x | δ ( d +2 α ) (cid:17) − δ φ , (4.1)where b ( t ) is a time continuous function asymptotically proportional to e − δλ t , φ = ( φ ,i ) mi =1 ∈ R m is the normalised (positive) principal eigenvector of DF (0) associated to the principaleigenvalue λ , and δ is equal to δ (respectively δ ) defined in (H3) (respectively (H4)). Lemma 4.1
Let v be defined as in (4.1). Then, there exist a constant D > such that forall i ∈ J , m K , t > and x ∈ R d | ( −△ ) α i v i ( t, x ) |≤ Db ( t ) αiδ ( d +2 α ) v i ( t, x ) , where α i ∈ (0 , . Proof.
The case α i = 1 is trivial. For α i ∈ (0 ,
1) and δ ≥ d + 2 α , since ( − ∆) α i is 2 α i -homogeneous, we only need to prove | ( −△ ) α i w ( x ) |≤ Dw ( x )where w ( x ) = (1 + | x | δ ( d +2 α ) ) − δ .We consider the following decomposition, which is the central part of the proof:( −△ ) α i w ( x ) = Z | y | > | x | / w ( x ) − w ( y ) | x − y | d +2 α i dy + Z B | x | / ( x ) w ( x ) − w ( y ) | x − y | d +2 α i dy + Z {| x |≤ | y |≤ | x |}\ B | x | / ( x ) w ( x ) − w ( y ) | x − y | d +2 α i dy + Z | y |≤| x | / w ( x ) − w ( y ) | x − y | d +2 α i dy. (cid:3) In what follows, we will use the results of previous sections to obtain appropriate suband super solutions to (1.2) of the form (4.1). We divide the proof of Theorem 1.1 in twolemmas.
Lemma 4.2
Assume that F satisfies (1.4), (H1), (H2), (H3) and (H5). Let u be the solutionto (1.2) with u satisfying the assumptions of Theorem 1.1. Then, for every µ = ( µ i ) mi =1 > ,there exists c > such that, for all t > τ , with τ > large enough n x ∈ R d | | x | > ce λ d +2 α t o ⊂ n x ∈ R d | u ( t, x ) < µ o . Proof:
We consider the function u given by (4.1) with δ = δ as in (H3). The idea is toadjust a > b ( t ) so that the function u serves as supersolution of (1.2).In the sequel, a is any positive constant satisfying a ≥ (cid:18) D + λ c h (cid:19) δ max i ∈ J ,m K (cid:18) φ ,i (cid:19) , where c h is defined in (H2). For any constant B ∈ (0 , (1 + Dλ − ) − δ d +2 α )2 α ), where D > b ′ ( t ) + δ Db ( t ) αδ d +2 α ) +1 + δ λ b ( t ) = 0 , b (0) = ( − Dλ − + B − αδ d +2 α ) ) − δ d +2 α )2 α (4.2)whose solution is given by b ( t ) = ( − Dλ − + B − αδ d +2 α ) e αλ d +2 α t ) − δ d +2 α )2 α . For all t ≥
0, we have 0 ≤ b ( t ) ≤ b (0) ≤
1. Using Lemma 4.1, we have for all i ∈ J , m K ∂ t u i + ( −△ ) α i u i − f i ( u ) = ∂ t u i + ( −△ ) α i u i − Df i (0) u + [ Df i (0) u − f i ( u )] ≥ aφ ,i δ (1 + b ( t ) | x | δ ( d +2 α ) ) δ +1 n − b ′ ( t ) − δ Db ( t ) αδ d +2 α ) +1 − δ λ b ( t ) o | x | δ ( d +2 α ) + aφ ,i (1 + b ( t ) | x | δ ( d +2 α ) ) δ +1 n − Db ( t ) αδ d +2 α ) − λ + c h φ δ ,i a δ o ≥ . Finally, due to Lemma 3.1, for a fixed t >
0, there exists t ≥ x ∈ R d andall i ∈ J , m K , we have u i ( t , x ) ≥ u i ( t , x ) . Thus, by Theorem 2.1 we have, for all t ≥ t , all x ∈ R d and all i ∈ J , m K : u i ( t + t − t , x ) ≥ u i ( t, x ) . For any ( µ i ) mi =1 >
0, we define for i ∈ J , m K the constants c d +2 αi := aφ ,i e λ ( t − t ) [ µ i B δ ] − . c = max i ∈ J ,m K c i , if | x | > ce λ d +2 α t , then, for all t > τ := t and all i ∈ J , m K u i ( t, x ) ≤ u i ( t + t − t , x ) = aφ ,i (1 + b ( t + t − t ) | x | δ ( d +2 α ) ) δ < µ i . (cid:3) Lemma 4.3
Let d ≥ and assume that F satisfies (1.4) , (H1), (H2), (H4) and (H5). Let u be the solution to (1.2) with a non negative, non identically equal to 0 and continuousinitial condition u satisfying (1.3). Then, for all i ∈ J , m K , there exist constants ε i > and C > such that, u i ( t, x ) > ε i , for all t ≥ t and | x | < Ce λ d +2 α t , with t > large enough. Proof:
As in the previous proof, we consider the function u given by (4.1) with δ = δ defined in (H4). Since, u i (0 , · ) ≤ u i may not hold for all i ∈ J , m K , we look for a time t > u i (0 , · ) ≤ u i ( t , · ) for all i ∈ J , m K . Indeed, let L be a constant greater thanmax { D, λ } , where D is given by Lemma 4.1. We choose t ≥ max(1 , Dλ − ) large enough,so that if we set a = min i ∈ J ,m K C i e − γ mm t i ∈ J ,m K φ ,i t d α and B = (cid:18) t (cid:19) ( d +2 α )2 α δ , (4.3)then a ≤ min i ∈ J ,m K φ ,i λ c δ δ and B ≤ ( Dλ − ) − ( d +2 α )2 α δ , where c δ is defined in (H4). Then we set b ( t ) = ( Dλ − + B − αδ d +2 α ) e αλ d +2 α t ) − ( d +2 α )2 α δ . Using Lemma 4.1 and (H3), similarly to the previous proof, we can state that, for all i ∈ J , m K , ∂ t u i + ( −△ ) α i u i − f i ( u ) ≤ , in (0 , + ∞ ) × R d . From Lemma 3.3, we know that for all i ∈ J , m K and all x ∈ R d u i ( t , x ) ≥ c t e − γ mm t t d α +11 + | x | d +2 α .
14y (4.3), we deduce ct e − γ mm t (1 + b (0) | x | δ ( d +2 α ) ) δ ≥ c t e − γ mm t (1 + b (0) δ | x | d +2 α ) ≥ aφ i ( t d α +11 + | x | d +2 α ) . Therefore, we get, for all i ∈ J , m K , u i ( t , · ) ≥ u i (0 , · ) in R d , and by Theorem 2.1, we havefor all t ≥ t u i ( t, · ) ≥ u i ( t − t , · ) , in R d Finally we choose ε i = aφ ,i δ and C d +2 α = e − λ t B − δ . If t ≥ τ := t and | x | ≤ Ce λ d +2 α t , we have u i ( t, x ) ≥ u i ( t − t , x ) = aφ ,i (1 + b ( t − t ) | x | δ ( d +2 α ) ) δ ≥ aφ ,i δ = ε i . References [1] A.N. Kolmogorov, I.G. Petrovsky and N.S. Piskunov, ´Etude de l’´equation de la dif-fusion avec croissance de la quantit´e de mati`ere et son application `a un probl`emebiologique , Bull. Univ. ´Etat Moscou S´er. Inter. A 1 (1937) 1-26.[2] D.G. Aronson and H.F. Weinberger,
Multidimensional nonlinear diffusions arisingin population genetics , Adv. Math. 30 (1978), 33-76.[3] X. Cabr´e and J. Roquejoffre.
The influence of fractional diffusion in Fisher-KPPequation . Preprint, arXiv:1202.6072v1, (2012).[4] X. Cabr´e, A. C. Coulon, and J. M. Roquejoffre.
Propagation in Fisher-KPP typeequations with fractional diffusion in periodic media . C. R. Math. Acad. Sci. Paris,350 (2012), no. 19-20, 885-890.[5] P. Felmer and M. Yangari.
Fast Propagation for Fractional KPP Equations withSlowly Decaying Initial Conditions . SIAM J. Math. Anal., 45(2), 662-678.[6] R. Lui.
Biological growth and spread modeled by systems of recursions. I. Mathemat-ical theory . Math. Biosci. 93(2), 269-295 (1989).[7] M. Lewis, B. Li and H. Weinberger.
Spreading speed and linear determinacy for two-species competition models . J. Math. Biol. 45, 219-233 (2002).[8] H.F. Weinberger, M. Lewis and B. Li.
Anomalous spreading speeds of cooperativerecursion systems . J. Math. Biol. 55, 207-222 (2007).[9] H.F. Weinberger, M. Lewis and B. Li.
Analysis of linear determinacy for spread incooperative models . J. Math. Biol. 45, 183-218 (2002).1510] L. C. Evans and P. E. Souganidis,
A PDE approach to geometric optics for certainsemilinear parabolic equations , Indiana Univ. Math. J. 45(2) (1989), 141–172.[11] G. Barles and L. C. Evans and P. E. Souganidis,
Wavefront propagation for reaction-diffusion systems of PDE . Duke Math. J. 61 (1990), 835-858.[12] J. Busca and B. Sirakov,
Harnack type estimates for nonlinear elliptic systems andapplications , Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 21 (2004), 543–590.[13] D. Henry,
Geometric Theory of Semilinear Parabolic Equations , Springer-Verlag,New York (1981).[14] M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions With For-mulas, Graphs, and Mathematical Tables . Dover Publications, New York. 1972.[15] A. Erd´elyi,
Higher Transcendental Functions. Vol. I , New York-Toronto-London,McGraw-Hill Book Company, Inc.,1953.[16] M. Bonforte and J. Vazquez.