Maximal and minimal spreading speeds for reaction diffusion equations in nonperiodic slowly varying media
aa r X i v : . [ m a t h . A P ] N ov Maximal and minimal spreading speeds for reactiondiffusion equations in nonperiodic slowly varying media
Jimmy Garnier a,b , Thomas Giletti b , Gregoire Nadin c a UR 546 Biostatistique et Processus Spatiaux, INRA, F-84000 Avignon, Franceb Aix-Marseille Universit´e, LATP UMR 6632, Facult´e des Sciences et TechniquesAvenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, Francec CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris, France Abstract
This paper investigates the asymptotic behavior of the solutions of the Fisher-KPPequation in a heterogeneous medium, ∂ t u = ∂ xx u + f ( x, u ) , associated with a compactly supported initial datum. A typical nonlinearity we con-sider is f ( x, u ) = µ ( φ ( x )) u (1 − u ), where µ is a 1-periodic function and φ is a C increasing function that satisfies lim x → + ∞ φ ( x ) = + ∞ and lim x → + ∞ φ ′ ( x ) = 0. Al-though quite specific, the choice of such a reaction term is motivated by its highlyheterogeneous nature. We exhibit two different behaviors for u for large times, de-pending on the speed of the convergence of φ at infinity. If φ grows sufficiently slowly,then we prove that the spreading speed of u oscillates between two distinct values. If φ grows rapidly, then we compute explicitly a unique and well determined speed ofpropagation w ∞ , arising from the limiting problem of an infinite period. We give aheuristic interpretation for these two behaviors. Key-words: heterogeneous reaction-diffusion equations; spreading speeds; propagation phe-nomena.
AMS classification.
Prefered . We consider the following reaction-diffusion equation in (0 , + ∞ ) × R : ∂ t u = ∂ xx u + f ( x, u ) . (1.1)1e assume that f = f ( x, u ) is locally Lipschitz-continuous in u and of class C in theneighborhood of u = 0 uniformly with respect to x , so that we can define µ ( x ) := f ′ u ( x, . Moreover, f is of the KPP type, that is f ( x,
0) = 0 , f ( x, ≤ , µ ( x ) > f ( x, u ) ≤ µ ( x ) u for all ( x, u ) ∈ R × (0 , . A typical f which satisfies these hypotheses is f ( x, u ) = µ ( x ) u (1 − u ), where µ is a continuous,positive and bounded function.The very specific hypothesis we make on f in this paper is the following: there exist µ ∈ C ( R ) and φ ∈ C ( R ) such that µ ( x ) = µ ( φ ( x )) for all x ∈ R , < min [0 , µ < max [0 , µ and µ is 1-periodic ,φ ′ ( x ) > , lim x → + ∞ φ ( x ) = + ∞ and lim x → + ∞ φ ′ ( x ) = 0 . (1.2)That is, our reaction-diffusion equation is strictly heterogeneous (it is not even almostperiodic or ergodic), which means that it can provide useful information on both efficiencyof recently developed tools and properties of the general heterogeneous problem. But it alsosatisfies some periodicity properties with a growing period near + ∞ . We aim to look at theinfluence of the varying period L ( x ) := x/φ ( x ) on the propagation of the solutions.Note that we do not assume here that there exists a positive stationary solution of (1.1).We require several assumptions that involve the linearization of f near u = 0 but our onlyassumption which is related to the behavior of f = f ( x, u ) with respect to u > f ( x, ≤
0, that is, 1 is a supersolution of (1.1) (it is clear that, up to some change ofvariables, 1 could be replaced by any positive constant in this inequality). It is possible toprove that there exists a minimal and stable positive stationary solution of (1.1) by usingthis hypothesis and the fact that µ is positive [4], but we will not discuss this problem sincethis is not the main topic of this paper. For any compactly supported initial condition u with 0 ≤ u ≤ u
0, we define the minimal and maximal spreading speeds as: w ∗ = sup { c > | lim inf inf x ∈ [0 ,ct ] u ( t, x ) > t → + ∞} ,w ∗ = inf { c > | sup x ∈ [ ct, + ∞ ) u ( t, x ) → t → + ∞} . Note that it is clear, from the strong maximum principle, that for any t > x ∈ R , onehas 0 < u ( t, x ) <
1. One can also easily derive from the homogeneous case [1] that2 p min µ ≤ w ∗ ≤ w ∗ ≤ √ max µ . t → + ∞ u ( t, x + ct ) > w ∗ . This is because we did not assume the existence of a positive stationary solution.Hence, we just require u to “take off” from the unstable steady state 0.The aim of this paper is to determine if some of these inequalities are indeed equalities.The first result on spreading speeds is due to Aronson and Weinberger [1]. They provedthat w ∗ = w ∗ = 2 p f ′ (0) in the case where f does not depend on x . More generally, evenif f does not satisfy f ( u ) ≤ f ′ (0) u for all u ∈ [0 , w ∗ = w ∗ is the minimal speed ofexistence of traveling fronts [1]. However, because of the numerous applications in variousfields of natural sciences, the role of heterogeneity has become an important topic in themathematical analysis.When f is periodic in x , Freidlin and Gartner [7] and Freidlin [6] proved that w ∗ = w ∗ using probabilistic techniques. In this case, the spreading speed is characterized using peri-odic principal eigenvalues. Namely, assume that f is 1-periodic in x , set µ ( x ) := f ′ u ( x, p ∈ R the elliptic operator L p ϕ := ϕ ′′ − pϕ ′ + ( p + µ ( x )) ϕ. (1.3)It is known from the Krein-Rutman theory that this operator admits a unique periodic prin-cipal eigenvalue λ p ( µ ), defined by the existence of a positive 1-periodic function ϕ p ∈ C ( R )so that L p ϕ p = λ p ( µ ) ϕ p . The characterization of the spreading speed [7] reads w ∗ = w ∗ = min p> λ p ( µ ) p . (1.4)Such a formula is very useful to investigate the dependence between the spreading speed andthe growth rate µ . Several alternative proofs of this characterization, based on differenttechniques, have been given in [3, 13]. The spreading speed w ∗ = w ∗ has also been identifiedlater as the minimal speed of existence of pulsating traveling fronts, which is the appropriategeneralization of the notion of traveling fronts to periodic media [2]. Let us mention, with-out getting into details, that the equality w ∗ = w ∗ and the characterization (1.4) have beenextended when the heterogeneity is transverse [10], space-time periodic or compactly sup-ported [3], or random stationary ergodic [7, 12]. In this last case one has to use Lyapounovexponents instead of principal eigenvalues.In all these cases (except in the random one), the operator L p is compact and thus princi-pal eigenvalues are well-defined. When the dependence of f with respect to x is more general,then classical principal eigenvalues are not always defined, which makes the computation ofthe spreading speeds much more difficult. Moreover, in general heterogeneous media, it mayhappen that w ∗ < w ∗ . No example of such phenomenon has been given in space heteroge-neous media, but there exist examples in time heterogeneous media [5] or when the initialdatum is not compactly supported [8].Spreading properties in general heterogeneous media have recently been investigated byBerestycki, Hamel and the third author in [3]. These authors clarified the links betweenthe different notions of spreading speeds and gave some estimates on the spreading speeds.More recently, Berestycki and the third author gave sharper bounds using the notion of3eneralized principal eigenvalues [5]. These estimates are optimal when the nonlinearity isperiodic, almost periodic or random stationary ergodic. In these cases, one gets w ∗ = w ∗ and this spreading speed can be characterized through a formula which is similar to (1.4),involving generalized principal eigenvalues instead of periodic principal eigenvalues. Before enouncing our results, let us first roughly describe the situation. As φ ′ ( x ) → x → + ∞ , the function φ is sublinear at infinity and thus µ ( x ) = µ ( φ ( x )) stays nearits extremal values max µ or min µ on larger and larger intervals. If these intervals aresufficiently large, that is, if φ increases sufficiently slowly, the solution u of (1.1) shouldpropagate alternately at speeds close to 2 √ max µ and 2 √ min µ . Hence, we expect in sucha case that w ∗ = 2 √ max µ and w ∗ = 2 √ min µ .On the other hand, if one writes φ ( x ) = x/L ( x ), then the reaction-term locally looks likean L ( x )-periodic function. Since L ( x ) → + ∞ , as clearly follows from the fact that φ ′ ( x ) → x → + ∞ , one might expect to find a link between the spreading speeds and the limitof the spreading speed w L associated with the L -periodic growth rate µ L ( x ) := µ ( x/L )when L → + ∞ . This limit has recently been computed by Hamel, Roques and the thirdauthor [9]. As µ L is periodic, w L is characterized by (1.4) and one can compute the limitof w L by computing the limit of λ p ( µ L ) for all p . This is how the authors of [9] proved thatlim L → + ∞ w L = min k ≥ M kj ( k ) , (2.5)where M := max x ∈ R µ ( x ) > j : [ M, + ∞ ) → [ j ( M ) , + ∞ ) is defined for all k ≥ M by j ( k ) := Z p k − µ ( x ) dx. (2.6)If φ increases rapidly, that is, the period L ( x ) increases slowly, then we expect to recoverthis type of behavior. More precisely, we expect that w ∗ = w ∗ = min k ≥ M k/j ( k ).We are now in position to state our results. φ We first consider the case when φ converges very slowly to + ∞ as x → + ∞ . As expected,we prove in this case that w ∗ < w ∗ . Theorem 2.1
1. Assume that xφ ′ ( x ) → + ∞ as x → + ∞ . Then w ∗ = 2 p min µ < w ∗ = 2 √ max µ .
2. Assume that xφ ′ ( x ) → C as x → + ∞ . If C is large enough (depending on µ ), then w ∗ < w ∗ . f ( x, u )for which the spreading speeds w ∗ and w ∗ associated with compactly supported initial dataare not equal.In order to prove this Theorem, we will first consider the particular case when µ isdiscontinuous and only takes two values (see Proposition 3.1 below). In this case, we are ableto construct sub- and super-solutions on each interval where µ is constant, and to concludeunder some hypotheses on the length of those intervals. Then, in the general continuouscase, our hypotheses on ( xφ ′ ( x )) − allow us to bound µ from below and above by some twovalues functions, and our results then follow from the preliminary case. Remark 2.2
Note that such a two values case is not continuous, so that our Theorem holdsunder more general hypotheses. In fact, one would only need that µ is continuous on twopoints such that µ attains its maximum and minimum there, so that, from the asymptoticsof φ ( x ) , the function µ ( x ) = µ ( φ ( x )) will be close to its maximum and minimum on verylarge intervals as x → + ∞ . φ We remind the reader that M := max x ∈ R µ ( x ) > j : [ M, + ∞ ) → [ j ( M ) , + ∞ ) isdefined by (2.6). We expect to characterize the spreading speeds w ∗ and w ∗ using thesequantities, as in [9].Note that j ( M ) > µ < M . The function j is clearly a bijection and thusone can define w ∞ := min λ ≥ j ( M ) j − ( λ ) λ = min k ≥ M kj ( k ) . (2.7)We need in this section an additional mild hypothesis on f : ∃ C > , γ > f ( x, u ) ≥ f ′ u ( x, u − Cu γ for all ( x, u ) ∈ R × (0 , + ∞ ) . (2.8) Theorem 2.3
Under the additional assumptions ( ) , φ ∈ C ( R ) and φ ′′ ( x ) /φ ′ ( x ) → , and φ ′′′ ( x ) /φ ′ ( x ) → , as x → + ∞ , (2.9) one has w ∗ = w ∗ = w ∞ . Note that (2.9) implies ( xφ ′ ( x )) − → x → + ∞ . Hence, this result is somehow comple-mentary to Theorem 2.1. However, this is not optimal as this does not cover all cases. Aninteresting and open question would be to refine those results to get more precise necessaryand sufficient conditions for the equality w ∗ = w ∗ to be satisfied. This could provide someinsight on the general heterogeneous case, where the establishment of such criteria is animportant issue.This result will mainly be derived from Theorem 2.1 of [5]. We first construct someappropriate test-functions using the asymptotic problem associated with µ L ( x ) = µ ( x/L )as L → + ∞ . This will enable us to compute the generalized principal eigenvalues and thecomputation of the spreading speeds will follow from Theorem 2.1 in [5].5 .3 Examples
We end the statement of our results with some examples which illustrate the different possiblebehaviors.
Example 1: φ ( x ) = β (ln x ) α , with α, β >
0. This function clearly satisfies the hypothesesin (1.2). • If α ∈ (0 , / ( xφ ′ ( x )) = (ln x ) − α / ( βα ) → + ∞ as x → + ∞ . Hence, theassumptions of case 1 in Theorem 2.1 are satisfied and one has w ∗ = 2 √ min µ and w ∗ = 2 √ max µ . • If α = 1, then xφ ′ ( x ) = β for all x and thus we are in the framework of case 2 inTheorem 2.1, which means that we can conclude that w ∗ < w ∗ provided that β is smallenough. • Lastly, if α >
1, then straightforward computations give φ ′′ ( x ) /φ ′ ( x ) ∼ − βα (ln x ) − α → x → + ∞ ,φ ′′′ ( x ) /φ ′ ( x ) ∼ βαx (ln x ) − α → x → + ∞ . Hence, the assumptions of Theorem 2.3 are satisfied and there exists a unique spreadingspeed: w ∗ = w ∗ = w ∞ . Example 2: φ ( x ) = x α , α ∈ (0 , . This function clearly satisfies the hypotheses in (1.2)since α <
1. One has φ ′′ ( x ) /φ ′ ( x ) = α − αx α → φ ′′′ ( x ) /φ ′ ( x ) = ( α − α − αx α → x → + ∞ . Thus, the assumptions of Theorem 2.3 are satisfied and w ∗ = w ∗ = w ∞ . Example 3: φ ( x ) = x/ (ln x ) α , α > . This function satisfies (1.2) and one has φ ′ ( x ) = 1(ln x ) α − α (ln x ) α +1 ,φ ′′ ( x ) = − αx (ln x ) α + α ( α + 1) x (ln x ) α +2 ,φ ′′′ ( x ) = αx (ln x ) α − α ( α + 1)( α + 2) x (ln x ) α +3 . It follows that φ ′′ ( x ) /φ ′ ( x ) → φ ′′′ ( x ) /φ ′ ( x ) → x → + ∞ since the terms in x willdecrease faster than the terms in ln x . Thus, the assumptions of Theorem 2.3 are satisfiedand w ∗ = w ∗ = w ∞ . Organization of the paper:
Theorem 2.1 will be proved in Section 4. As a first step toprove this Theorem, we will investigate in Section 3 the case where µ is not continuous6nymore but only takes two values µ + and µ − . Lastly, Section 5 is dedicated to the proof ofTheorem 2.3. Acknowledgements:
The authors would like to thank Fran¸cois Hamel and Lionel Roquesfor having drawn their attention to the problems investigated in this paper. This article wascompleted while the third author was visiting the Department of mathematical sciences ofBath whose hospitality is gratefully acknowledged.
We assume first that µ is discontinuous and only takes two distinct values µ − , µ + ∈ (0 , + ∞ ).Moreover, we assume that there exist two increasing sequences ( x n ) n and ( y n ) n such that x n +1 ≥ y n ≥ x n for all n , lim n → + ∞ x n = + ∞ and µ ( x ) = ( µ + if x ∈ ( x n , y n ) ,µ − if x ∈ ( y n , x n +1 ) . (3.10) Proposition 3.1
We have:1. If y n /x n → + ∞ , then w ∗ = 2 √ µ + .2. If x n +1 /y n → + ∞ , then w ∗ = 2 √ µ − .3. If y n /x n → K > , then w ∗ ≥ √ µ + K ( K −
1) + p µ + /µ − .4. If x n +1 /y n → K > , then w ∗ ≤ √ µ − K + p µ + /µ − K + p µ − /µ + . It is clear in part 3 (resp. 4) of Proposition 3.1 that the lower bound on w ∗ (resp. upperbound on w ∗ ) goes to 2 √ µ + (resp. 2 √ µ − ) as K → + ∞ . Hence, for K large enough, we getthe wanted result w ∗ < w ∗ .
1. We first look for a subsolution of equation (1.1) going at some speed c close to 2 √ µ − .Let φ R be a solution of the principal eigenvalue problem: ∂ xx φ R = λ R φ R in B R ,φ R = 0 on ∂B R ,φ R > B R . (3.11)We normalize φ R by k φ R k ∞ = 1. We know that λ R → R → + ∞ . Let c < √ µ − and R large enough so that − λ R < µ − − c /
4. Then v ( x ) = e − cx φ R ( x ) satisfies: ∂ xx v + c∂ x v + µ − v = (cid:16) µ − − c λ R (cid:17) v > B R .
7y extending φ R by 0 outside B R , by regularity of f and since f ′ u ( x, ≥ µ − for any x ∈ R ,for some small κ , we also have in (0 , + ∞ ) × R : ∂ xx κv + c∂ x κv + f ( x + ct, κv ) ≥ . Hence, w ( t, x ) := κv ( x − ct ) is a subsolution of (1.1). Without loss of generality, we canassume that u (1 , x ) ≥ w (1 , x ), thus for any t ≥ u ( t, x ) ≥ w ( t, x ). That is, for any speed c < √ µ − , we have bounded u from below by a subsolution of (1.1) with speed c. Inparticular, let t n := x n + Rc , then u ( t n , x ) ≥ w ( t n , x ) for all x ∈ R , which is positive on a ball of radius R around x n + R .2. Take an arbitrary c ′ < √ µ + and let φ R ′ a solution of the principal eigenvalue prob-lem (3.11) with R ′ such that − λ R ′ < µ + − c ′ /
4. As above, there exists e v ( x ) = κ ′ e − c ′ x φ R ′ ( x )compactly supported such that ∂ xx e v + c ′ ∂ x e v + f ( x + x n + R + c ′ t, e v ) ≥ , (3.12)as long as e v = 0 where f ′ u ( x + x n + R + c ′ t, = µ + , that is( − R ′ + x n + R + c ′ t, R ′ + x n + R + c ′ t ) ⊂ ( x n , y n ) , which is true for R > R ′ and 0 ≤ t ≤ y n − x n − R − R ′ c ′ . As R could be chosen arbitrarily large, we can assume that the condition R > R ′ is indeedsatisfied. Moreover, as lim inf n → + ∞ y n /x n > n → + ∞ x n = + ∞ , we can assume that n is large enough so that y n − x n > R and thus the second condition is also satisfied. Hence, e w ( t n + t, x ) := e v ( x − x n − R − c ′ t ) is a subsolution of (1.1) for t ∈ (0 , y n − x n − R − R ′ c ′ ) and x ∈ R . We can take κ ′ small enough so that κ min y ∈ B (0 ,R ′ ) φ R ( y ) > κ ′ e | c − c ′| R ′ . (3.13)For all x ∈ B ( ct n , R ′ ), one has: w ( t n , x ) = κe − c ( x − ctn )2 φ R ( x − ct n ) ≥ κ (cid:16) min y ∈ B (0 ,R ′ ) φ R ( y ) (cid:17) e − ( c − c ′ )2 ( x − ct n ) e − c ′ ( x − ct n ) ≥ κ (cid:16) min y ∈ B (0 ,R ′ ) φ R ( y ) (cid:17) e −| c − c ′| R ′ e − c ′ ( x − ct n ) ≥ κ ′ e − c ′ ( x − ct n ) ≥ κ ′ e − c ′ ( x − ctn )2 φ R ′ ( x − ct n ) = e w ( t n , x ) , (3.14)8ince ct n = x n + R by definition. Moreover, u ( t n , x ) ≥ w ( t n , x ) for all x ∈ R . The parabolicmaximum principle thus gives u ( t n + t, x ) ≥ e w ( t n + t, x ) for all t ∈ (cid:16) , y n − x n − R − R ′ c ′ (cid:17) and x ∈ R .
3. We can now conclude. Indeed, for n large enough one has: u (cid:18) t n + y n − x n − R − R ′ c ′ , y n − R ′ (cid:19) ≥ e w (cid:18) t n + y n − x n − R − R ′ c ′ , y n − R ′ (cid:19) = e v (0) . Since the construction of e v did not depend on n, the above inequality holds independentlyof n, which implies that:inf n u (cid:18) t n + y n − x n − R − R ′ c ′ , y n − R ′ (cid:19) > . If y n /x n → + ∞ , we have y n − R ′ t n + y n − x n − R − R ′ c ′ = y n − R ′ x n c + y n − x n − R − R ′ c ′ → c ′ as n → + ∞ . It follows that w ∗ ≥ c ′ for any c ′ < √ µ + . The proof of part 1 of Proposition 3.1 is completed.If y n /x n → K , we have y n − R ′ t n + y n − x n − R − R ′ c ′ → K c + K − c ′ as n → + ∞ . As this is true for any c ′ < √ µ + and c < √ µ − , this concludes the proof of part 3 ofProposition 3.1. (cid:3) Let λ + = √ µ + be the solution of λ − √ µ + λ + = − µ + . One can then easily check, fromthe KPP hypothesis, that the function v ( t, x ) := min (cid:0) , κe − λ + ( x − √ µ + t ) (cid:1) is a supersolution of equation (1.1) going at the speed 2 √ µ + , for any κ >
0. Since u iscompactly supported, we can choose κ such that v (0 , · ) ≥ u in R . Thus, for any t ≥ x ∈ R , u ( t, x ) ≤ v ( t, x ). In particular, the inequality holds for t = t n the smallest time suchthat v ( t, y n ) = 1. Note that t n = y n / (2 √ µ + ) + C where C is a constant independent of n .Then for all x ∈ R , u ( t n , y n + x ) ≤ v ( t n , y n + x ) = min (cid:0) , e − λ + x (cid:1) . We now look for a supersolution moving with speed 2 √ µ − locally in time around t n . Let usdefine w ( t n + t, y n + x ) := min (cid:0) v ( t n + t, y n + x ) , e − λ − ( x − √ µ − t ) (cid:1) λ − = √ µ − . Note that λ − < λ + , thus u ( t n , y n + x ) ≤ v ( t n , y n + x ) = w ( t n , y n + x ).We now check that w is indeed a supersolution of equation (1.1). We already knowthat v is a supersolution and it can easily be seen as above from the KPP hypothesis that( t, x ) e − λ − ( x − √ µ − t ) is a supersolution only where f ′ u ( · ,
0) = µ − . Thus, we want theinequality v ( t n + t, y n + x ) ≤ e − λ − ( x − √ µ − t ) to be satisfied if y n + x ( y n , x n +1 ) . Recall that v ( t n + t, y n + x ) = min (cid:0) , e − λ + ( x − √ µ + t ) (cid:1) for all t > x ∈ R . Thus, the inequality issatisfied if t ≥ x ≤ x ≥ λ + √ µ + − λ − √ µ − λ + − λ − t = 2( λ + + λ − ) t. It follows that w ( t n + t, y n + x ) is indeed a supersolution of equation (1.1) in R as long as0 ≤ λ + + λ − ) t ≤ x n +1 − y n , (3.15)and that u ( t n + t, y n + x ) ≤ w ( t n + t, y n + x ) for any t verifying the above inequality.To conclude, let now 2 √ µ + > c > √ µ − , and t ′ n the largest t satisfying (3.15), i.e. t ′ n = x n +1 − y n λ + + λ − ) . The sequence ( t ′ n ) n tends to + ∞ as n → + ∞ since lim inf n → + ∞ x n +1 /y n > n → + ∞ y n = + ∞ . Moreover, one has u ( t n + t ′ n , y n + ct ′ n ) ≤ w ( t n + t ′ n , y n + ct ′ n ) → n → + ∞ since c > √ µ − .If x n +1 /y n → + ∞ as n → + ∞ , as t n = y n / (2 √ µ + ) + C , one gets t ′ n /t n → + ∞ as n → + ∞ . Hence, y n + ct ′ n t n + t ′ n → c as n → + ∞ . It follows that w ∗ ≤ c for any c > √ µ − . This proves part 2 of Proposition 3.1.If x n +1 /y n → K as n → + ∞ , we compute y n + ct ′ n t n + t ′ n → c λ + + λ − ) ( K − √ µ + + K − λ + + λ − ) as n → + ∞ . Hence, w ∗ is smaller than the right hand-side. As c ∈ (2 √ µ − , √ µ + ) is arbitrary, λ − = √ µ − and λ + = √ µ + , we eventually get w ∗ ≤ √ µ − K + p µ + /µ − K + p µ − /µ + , which concludes the proof of part 4 of Proposition 3.1. (cid:3) The continuous case
We assume that µ is a continuous and 1-periodic function. Let now ε be a small positiveconstant and define µ − < µ + by: ( µ + := max µ − ε,µ − := min µ . We want to bound µ from below by a function taking only the values µ − and µ + , in orderto apply Theorem 3.1. Note first that there exist x − ∈ (0 ,
1) and δ ∈ (0 ,
1) such that µ ( x ) > µ + for any x ∈ ( x − , x − + δ ). We now let the two sequences ( x n ) n ∈ N and ( y n ) n ∈ N defined for any n by: ( φ ( x n ) = x − + n,φ ( y n ) = x − + n + δ. Note that since φ is strictly increasing and φ (+ ∞ ) = + ∞ , then those sequences indeed exist,tend to + ∞ as n → + ∞ , and satisfy for any n , x n < y n < x n +1 . It also immediately followsfrom their definition that for all x ∈ R ,µ ( x ) ≥ e µ ( x ) where e µ ( x ) := ( µ + if x ∈ ( x n , y n ) ,µ − if x ∈ ( y n , x n +1 ) . We now have to estimate the ratio y n /x n in order to apply Proposition 3.1. Note that: δ = φ ( y n ) − φ ( x n ) = Z y n x n φ ′ ( x ) dx. (4.16)Moreover, under the hypothesis xφ ′ ( x ) → x → + ∞ , and since ( x n ) n , ( y n ) n tend to + ∞ as n → + ∞ : Z y n x n φ ′ ( x ) dx = Z y n x n (cid:18) xφ ′ ( x ) × x (cid:19) dx = o (cid:18) ln (cid:18) y n x n (cid:19)(cid:19) as n → + ∞ . (4.17)From (4.16) and (4.17), we have that y n x n → + ∞ . To conclude, we use the parabolic maximumprinciple and part 1 of Proposition 3.1 applied to problem (1.1) with a reaction term e f ≤ f such that e f ′ u ( x,
0) = e µ ( x ) for all x ∈ R . It immediately follows that w ∗ ≥ √ max µ − ε . Since this inequality holds for any ε > w ∗ = 2 √ max µ .We omit the details of the proof of w ∗ = 2 √ min µ since it follows from the same method.Indeed, one only have to choose y ′− and δ ′ in (0 ,
1) such that µ ( x ) < min µ + ε for any x ∈ ( y ′− , y ′− + δ ′ ) and let two sequences such that ( φ ( y ′ n ) = y ′− + n,φ ( x ′ n +1 ) = y ′− + n + δ ′ . One can then easily conclude as above using part 2 of Proposition 3.1. (cid:3) .2 Proof of part 2 of Theorem 2.1 As before, we bound µ from below by a two values function, that is, for all x ∈ R , µ ( x ) ≥ e µ ( x ) where e µ ( x ) := ( µ + = max µ − ε if x ∈ ( x n , y n ) ,µ − = min µ if x ∈ ( y n , x n +1 ) , where ε a small positive constant and the two sequences ( x n ) n and ( y n ) n satisfy for any n : x n < y n < x n +1 ,φ ( x n ) = x − + n,φ ( y n ) = x − + n + δ ( ε ) for some δ ( ε ) > ,x n → + ∞ and y n → + ∞ . Here, under the assumption that xφ ′ ( x ) → /C , we get δ ( ε ) = φ ( y n ) − φ ( x n ) = Z y n x n φ ′ ( x ) dx = Z y n x n (cid:18) xφ ′ ( x ) × x (cid:19) dx = 1 C ln (cid:18) y n x n (cid:19) + o (cid:18) ln (cid:18) y n x n (cid:19)(cid:19) as n → + ∞ . Hence, y n x n → e δ ( ε ) C as n → + ∞ . We can now apply the parabolic maximum principle and part 3 of Proposition 3.1 to get w ∗ ≥ √ max µ − ε e δ ( ε ) C ( e δ ( ε ) C −
1) + p (max µ − ε ) / min µ . (4.18)Notice that the dependence of δ on ε prevents us from passing to the limit as ε → ε >
0, one can easily check thatthe right-hand side in the inequation (4.18) converges as C → + ∞ to 2 √ max µ − ε .One can proceed similarly to get an upper bound on w ∗ , that is: w ∗ ≤ p min µ + ε e δ ′ ( ε ) C + p max µ / (min µ + ε ) e δ ′ ( ε ) C + p (min µ + ε ) / max µ , (4.19)where ε can be chosen arbitrary small and δ ′ ( ε ) is such that µ ( x ) ≤ min µ + ε on someinterval of length δ ′ ( ε ). It is clear that the right-hand side of (4.19) converges to 2 √ min µ + ε as C → + ∞ .Therefore, by choosing ε < (max µ − min µ ) /
2, one easily gets from (4.18) and (4.19)that for C large enough, w ∗ < w ∗ . This concludes the proof of part 2 of Theorem 2.1. More-over, note that the choice of C to get this strict inequality depends only on the function µ ,by the intermediate of the functions δ ( ε ) and δ ′ ( ε ). (cid:3) The unique spreading speed case
We begin with some preliminary work that will be needed to estimate the spreading speeds.The proof of Theorem 2.3 is then separated into two parts: the first part (Section 5.2) isdevoted to the proof that w ∗ ≤ w ∞ , while in the second part (Section 5.3) we prove that w ∗ ≥ w ∞ . For all p ∈ R , we define H ( p ) := (cid:26) j − ( | p | ) if | p | ≥ j ( M ) ,M if | p | < j ( M ) . (5.20)The fundamental property of this function is given by the following result. Proposition 5.1 (Propositions . and . in [9]) For all p ∈ R , H ( p ) is the unique realnumber such that there exists a continuous -periodic viscosity solution v of ( v ′ ( y ) − p ) + µ ( y ) = H ( p ) over R . (5.21)Next, we will need, as a first step of our proof, the function v given by Proposition 5.1 tobe piecewise C . This is true under some non-degeneracy hypothesis on µ . We will checkbelow in the second part of the proof of Theorem 2.3 that it is always possible to assumethat this hypothesis is satisfied by approximation. Lemma 5.2
Assume that µ ∈ C ( R ) and thatif µ ( x ) = max R µ , then µ ′′ ( x ) < . (5.22) Then for all p ∈ R , equation (5.21) admits a -periodic solution v p ∈ W , ∞ ( R ) which ispiecewise C ( R ) . Proof.
The proof relies on the explicit formulation of v p . Assume first that p > j ( M ) = j (cid:0) k µ k ∞ (cid:1) .Then it is easy to check (see [9]) that v p ( x ) := px − Z x p H ( p ) − µ ( y ) dy (5.23)satisfies (5.21). Then, the definition of j implies that v p is 1-periodic and, as µ ∈ C ( R )and H ( p ) > µ ( y ) for all y ∈ R , the function v p is C ( R ). The case p < − j ( M ) is treatedsimilarly.Next, if | p | ≤ j ( M ), let F define for all Y ∈ [0 ,
1] by: F ( Y ) := p + Z Y p M − µ ( y ) dy − Z Y p M − µ ( y ) dy. F is continuous and, as | p | ≤ j ( M ), F (0) = p + Z p M − µ ( y ) dy = p + j ( M ) ≥ . Similarly, F (1) = p − j ( M ) ≤
0. Thus, there exists X ∈ [0 ,
1] so that F ( X ) = 0. We nowdefine: v p ( x ) = px − Z x p M − µ ( y ) dy for all x ∈ [0 , X ] ,px − Z X p M − µ ( y ) dy + Z xX p M − µ ( y ) dy for all x ∈ [ X, . (5.24)From the definition of X , the function v p is 1-periodic. It is continuous and derivable at anypoint x ∈ [0 , \{ X } with v ′ p ( x ) = (cid:26) p − p M − µ ( x ) for all x ∈ [0 , X ) ,p + p M − µ ( x ) for all x ∈ ( X, . Hence, it satisfies (5.21) in the sense of viscosity solutions. Lastly, for all x ∈ (0 , X ) so that µ ( x ) = M , one has v ′′ p ( x ) = µ ′ ( x )2 p M − µ ( x ) . If µ ( x M ) = M , then (5.22) implies that µ ( x ) < M for all x = x M close to x M and a Taylorexpansion gives lim x → x M ,x = x M v ′′ p ( x ) = p − µ ′′ ( x M ) / . Hence, v ′′ p can be extended to a continuous function over (0 , X ). Similarly, it can be extendedover ( X, v ′′ p is bounded over [0 ,
1] and that it is piecewise C ( R ). (cid:3) For any p ∈ R , define the elliptic operator: L p ϕ := ϕ ′′ − pϕ ′ + ( p + µ ( φ ( x ))) ϕ. Lemma 5.3
For all p ∈ R , let ϕ p ( x ) := exp (cid:16) v p ( φ ( x )) φ ′ ( x ) (cid:17) . (5.25) If (5.22) holds and µ ∈ C ( R ) , then ϕ p is piecewise C ( R ) and one has L p ϕ p ( x ) − H ( p ) ϕ p ( x ) ϕ p ( x ) → as x → + ∞ . (5.26)14 roof. The function ϕ p is piecewise C ( R ) since v p is piecewise C ( R ). For all x so that v p is C in x , we can compute ϕ ′ p ( x ) = (cid:18) v ′ p ( φ ( x )) − φ ′′ ( x )( φ ′ ( x )) v p ( φ ( x )) (cid:19) ϕ p ( x ) ,ϕ ′′ p ( x ) = (cid:18) φ ′ ( x ) v ′′ p ( φ ( x )) − φ ′′ ( x ) φ ′ ( x ) v ′ p ( φ ( x )) + (cid:18) φ ′′ ( x )) ( φ ′ ( x )) − φ ′′′ ( x )( φ ′ ( x )) (cid:19) v p ( φ ( x )) (cid:19) ϕ p ( x )+ (cid:18) v ′ p ( φ ( x )) − φ ′′ ( x )( φ ′ ( x )) v p ( φ ( x )) (cid:19) ϕ p ( x ) . This gives L p ϕ p ( x ) − H ( p ) ϕ p ( x ) ϕ p ( x ) = φ ′ ( x ) v ′′ p ( φ ( x )) − φ ′′ ( x ) φ ′ ( x ) v ′ p ( φ ( x )) + (cid:18) φ ′′ ( x )) ( φ ′ ( x )) − φ ′′′ ( x )( φ ′ ( x )) (cid:19) v p ( φ ( x )) − φ ′′ ( x )( φ ′ ( x )) v p ( φ ( x )) v ′ p ( φ ( x )) + (cid:18) φ ′′ ( x )( φ ′ ( x )) v p ( φ ( x )) (cid:19) + v ′ p ( φ ( x )) − p (cid:18) v ′ p ( φ ( x )) − φ ′′ ( x )( φ ′ ( x )) v p ( φ ( x )) (cid:19) + p + µ ( φ ( x )) − H ( p ) , = φ ′ ( x ) v ′′ p ( φ ( x )) − φ ′′ ( x ) φ ′ ( x ) v ′ p ( φ ( x )) + (cid:18) φ ′′ ( x )) ( φ ′ ( x )) − φ ′′′ ( x )( φ ′ ( x )) (cid:19) v p ( φ ( x )) − φ ′′ ( x )( φ ′ ( x )) v p ( φ ( x )) v ′ p ( φ ( x )) + (cid:18) φ ′′ ( x )( φ ′ ( x )) v p ( φ ( x )) (cid:19) +2 p φ ′′ ( x )( φ ′ ( x )) v p ( φ ( x )) . As v p is periodic and W , ∞ , v ′′ p is bounded. It follows from (2.9) that L p ϕ p ( x ) − H ( p ) ϕ p ( x ) ϕ p ( x ) → x → + ∞ . (cid:3) Lemma 5.4
Define ϕ p as in Lemma 5.3. Then ln ϕ p ( x ) x → as x → + ∞ . (5.27) Proof.
One has ln ϕ p ( x ) x = v p ( φ ( x )) φ ′ ( x ) x for all x ∈ R . (5.28)The function x v p ( φ ( x )) is clearly bounded since v p is periodic. Hence, (2.9) gives theconclusion. (cid:3) .2 Upper bound for the spreading speed Proof of part 1 of Theorem 2.3.
We first assume that µ ∈ C ( R ). Let us now show that w ∗ ≤ w ∞ . Let c > w ∞ and c ∈ ( w ∞ , c ). We know that there exists p ≥ j ( M ) > w ∞ = min p ′ ≥ j ( M ) H ( p ′ ) /p ′ = H ( p ) /p. Let k ≥ M so that p = j ( k ) > , and ϕ p defined as in Lemma 5.3. We know from Lemma 5.3that there exists X > | L p ϕ p ( x ) − kϕ p ( x ) | ≤ ( c − w ∞ ) j ( k ) ϕ p ( x ) for all x > X. (5.29)Let u be defined for all ( t, x ) ∈ [0 , + ∞ ) × R by: u ( t, x ) := min { , ϕ p ( x ) e − j ( k )( x − h − c t ) } , where h ∈ R is large enough so that u ( x ) ≤ u (0 , x ) for all x ∈ R (this is always possiblesince u is compactly supported). Moreover, u ( t, x ) < ϕ p ( x ) < e j ( k )( x − h − c t ) ,which is equivalent to x − v p ( φ ( x )) / ( j ( k ) φ ′ ( x )) > h + c t ≥ h . Lemma 5.4 yields that theleft hand-side of this inequality goes to + ∞ as x → + ∞ . Hence, we can always take h largeenough so that u ( t, x ) < x > X . It follows that for all ( t, x ) ∈ [0 , + ∞ ) × R suchthat u ( t, x ) <
1, one has ∂ t u − ∂ xx u − f ( x, u ) ≥ ∂ t u − ∂ xx u − µ ( x ) u ≥ j ( k ) c u − L j ( k ) (cid:0) ϕ p (cid:1) ( x ) e − j ( k )( x − h − c t ) ≥ j ( k ) c u − ku − ( c − w ∞ ) j ( k ) u ≥ (cid:0) j ( k ) w ∞ − k (cid:1) u = 0 . It follows from the parabolic maximum principle that u ( t, x ) ≥ u ( t, x ) for all ( t, x ) ∈ [0 , + ∞ ) × R .Hence, for all given x ∈ R , u ( t, x ) ≤ ϕ p ( x ) e − j ( k )( x − h − c t ) for all t > . Let ε > ε < j ( k )( c − c ) /c . Lemma 5.4 yields that there exists R > x > R, ln (cid:0) ϕ p ( x ) (cid:1) ≤ εx . Let T = R/c and take t ≥ T and x ≥ ct . One hasln (cid:16) ϕ p ( x ) e − j ( k )( x − h − c t ) (cid:17) = ln (cid:0) ϕ p ( x ) (cid:1) − j ( k )( x − h − c t ) ≤ ( ε − j ( k )) x + j ( k )( h + c t ) ≤ (cid:0) εc + j ( k )( c − c ) (cid:1) t + j ( k ) h → −∞ as t → + ∞ since εc < j ( k )( c − c ). Hence,lim t → + ∞ max x ≥ ct (cid:16) ϕ p ( x ) e − j ( k )( x − h − c t ) (cid:17) = 0 as t → + ∞ , µ ∈ C ( R ).Lastly, if µ ∈ C ( R ) is an arbitrary 1-periodic function, then one easily concludes bysmoothing µ from above. Indeed, one can find a sequence ( µ n ) n ∈ C ( R ) N converginguniformly to µ , and such that for all n ∈ N and x ∈ R , µ ( x ) ≤ µ n ( x ).It follows from the maximum principle thatlim t → + ∞ max x ≥ ct u ( t, x ) = 0 for all w > min k ≥ M k/j n ( k ) , where j n ( k ) = Z p k − µ n ( x ) dx ≥ j ( k ) > . Letting n → + ∞ , one getslim t → + ∞ max x ≥ ct u ( t, x ) = 0 for all w > w ∞ , which concludes the proof. (cid:3) Proof of part 2 of Theorem 2.3.
First, assume that µ ∈ C ( R ) satisfies (5.22). Let ϕ p as in Lemma 5.3. For all δ >
0, take R large enough so that L p ϕ p ≥ ( H ( p ) − δ ) ϕ p at anypoint of ( R, + ∞ ) where ϕ p is piecewise C . It is easy to derive from the proof of Lemma 5.3that ϕ ′ p /ϕ p is bounded and uniformly continuous. Take C > | ϕ ′ p ( x ) | ≤ Cϕ p ( x ) forall x ∈ R .We need more regularity in order to apply the results of [5]. Consider a compactlysupported nonnegative mollifier χ ∈ C ∞ ( R ) so that R R χ = 1 and define the convoled function ψ p := exp (cid:0) χ ⋆ ln ϕ p (cid:1) ∈ C ( R ). One has ψ ′ p /ψ p = χ ⋆ (cid:16) ϕ ′ p /ϕ p (cid:17) . Hence, | ψ ′ p ( x ) | ≤ Cψ p ( x ) forall x ∈ R and, as ϕ ′ p /ϕ p and (cid:0) ϕ ′ p /ϕ p (cid:1) are uniformly continuous, up to some rescaling of χ ,we can assume that (cid:13)(cid:13)(cid:13)(cid:12)(cid:12) χ ⋆ (cid:16) ϕ ′ p ϕ p (cid:17)(cid:12)(cid:12) − (cid:12)(cid:12) ϕ ′ p ϕ p (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) ∞ ≤ δ, (cid:13)(cid:13)(cid:13) χ ⋆ (cid:16)(cid:12)(cid:12) ϕ ′ p ϕ p (cid:12)(cid:12) (cid:17) − (cid:12)(cid:12) ϕ ′ p ϕ p (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) ∞ ≤ δ and k χ ⋆ µ − µ k ∞ ≤ δ. We now compute ψ ′′ p ψ p = (cid:12)(cid:12)(cid:12) ψ ′ p ψ p (cid:12)(cid:12)(cid:12) + χ ⋆ (cid:16) ϕ ′′ p ϕ p − (cid:12)(cid:12)(cid:12) ϕ ′ p ϕ p (cid:12)(cid:12)(cid:12) (cid:17) = (cid:12)(cid:12)(cid:12) χ ⋆ (cid:16) ϕ ′ p ϕ p (cid:17)(cid:12)(cid:12)(cid:12) − χ ⋆ (cid:16)(cid:12)(cid:12)(cid:12) ϕ ′ p ϕ p (cid:12)(cid:12)(cid:12) (cid:17) + χ ⋆ (cid:16) ϕ ′′ p ϕ p (cid:17) ≥ − δ + χ ⋆ (cid:16) ϕ ′′ p ϕ p (cid:17) . It follows that L p ψ p ψ p = ψ ′′ p − pψ ′ p + µ ( x ) ψ p ψ p ≥ − δ + χ ⋆ (cid:16) ϕ ′′ p ϕ p − p ϕ ′ p ϕ p (cid:17) + µ ( x ) ≥ − δ + χ ⋆ (cid:16) H ( p ) − µ − δ (cid:17) + µ ( x ) ≥ − δ + H ( p ) + µ ( x ) − χ ⋆ µ ( x ) ≥ − δ + H ( p )17n ( R, + ∞ ). On the other hand, Lemma 5.4 yields ψ p ∈ A R , where A R is the set of admissibletest-functions (in the sense of [5]) over ( R, ∞ ): A R := (cid:8) ψ ∈ C ([ R, ∞ )) ∩ C (( R, ∞ )) ,ψ ′ /ψ ∈ L ∞ (( R, ∞ )) , ψ > R, ∞ ) , lim x → + ∞ x ln ψ ( x ) = 0 (cid:9) . (5.30)Thus, one has λ ( L p , ( R, + ∞ )) ≥ H ( p ) − δ , where the principal eigenvalue λ is defined by λ ( L p , ( R, ∞ )) := sup { λ | ∃ φ ∈ A R such that L p φ ≥ λφ in ( R, ∞ ) } , (5.31)Hence, lim R → + ∞ λ ( L p , ( R, + ∞ )) ≥ H ( p ) for all p > f from below by such a nonlinearity. As min R µ > f is of class C in the neighborhoodof u = 0, we know that there exists θ ∈ (0 ,
1) so that f ( x, u ) > x ∈ R and u ∈ (0 , θ ) . Let ζ = ζ ( u ) a smooth function so that0 < ζ ( u ) ≤ u ∈ (0 , θ ) , ζ ( u ) = 0 for all u ≥ θ and ζ ( u ) = 1 for all u ∈ (0 , θ . Define f ( x, u ) := ζ ( u ) f ( x, u ) for all ( x, u ) ∈ R × [0 , f ≤ f in R × [0 ,
1] and f ′ u ( x,
0) = f ′ u ( x,
0) = µ ( φ ( x )) for all x ∈ R . Let u the solution of (1.1) with nonlinearity f instead of f and initial datum u . Theparabolic maximum principle yields u ≥ u .Since the function f satisfies the hypotheses of Theorem 2.1 in [5], we conclude thatlim t → + ∞ min x ∈ [0 ,wt ] u ( t, x ) = 1 for all w ∈ (cid:16) , min p> H ( p ) p (cid:17) . It follows that w ∗ ≥ min p> H ( p ) p = min k ≥ M kj ( k ) . Next, assume that µ ∈ C ( R ) does not satisfy (5.22). Let y ∈ R so that µ ( y ) = max y ∈ R µ ( y ). Take a 1-periodic function χ ∈ C ( R ) so that χ (0) = 0, χ ( y ) > y = 0 and χ ′′ (0) >
0. Define for all n ∈ N and x ∈ R : µ n ( y ) := µ ( y ) − n χ ( y − y ) . This 1-periodic function satisfies (5.22) for all n and one has 0 < µ n ≤ µ for n large enough.It follows from the maximum principle thatlim inf t → + ∞ min ≤ x ≤ wt u ( t, x ) > w ∈ (cid:16) , min k ≥ M kj n ( k ) (cid:17) , j n ( k ) = Z p k − µ n ( x ) dx ≥ j ( k ) > k ≥ M . Letting n → + ∞ , one has µ n ( y ) → µ ( y ) uniformly in y ∈ R and thuslim inf t → + ∞ min ≤ x ≤ wt u ( t, x ) > w ∈ (0 , w ∞ ) , which concludes the proof in this case.Lastly, if µ ∈ C ( R ) is an arbitrary 1-periodic function, then one easily concludes bysmoothing µ as in the previous step. (cid:3) References [1] D.G. Aronson, and H.F. Weinberger. Multidimensional nonlinear diffusions arising inpopulation genetics.
Adv. Math. , 30:33–76, 1978.[2] H. Berestycki, and F. Hamel. Front propagation in periodic excitable media.
Comm.Pure Appl. Math. , 55:949–1032, 2002.[3] H. Berestycki, F. Hamel, and G. Nadin. Asymptotic spreading in heterogeneous diffusiveexcitable media.
J. Func. Anal. , 255(9):2146–2189, 2008.[4] H. Berestycki, F. Hamel, and L. Rossi. Liouville type results for semilinear ellipticequations in unbounded domains.
Annali Mat. Pura Appli. , 186:469-507, 2007.[5] H. Berestycki, and G. Nadin. Spreading speeds for one-dimensional monostable reaction-diffusion equations. preprint .[6] M. Freidlin. On wave front propagation in periodic media.
In: Stochastic analysis andapplications, ed. M. Pinsky, Advances in Probability and related topics , 7:147–166, 1984.[7] M. Freidlin, and J. G¨artner. On the propagation of concentration waves in periodic andrandom media.
Sov. Math. Dokl. , 20:1282–1286, 1979.[8] F. Hamel, and G. Nadin. Spreading properties and complex dynamics for monostablereaction-diffusion equations. preprint .[9] F. Hamel, G. Nadin and L. Roques. A viscosity solution method for the spreading speedformula in slowly varying media.
Indiana Univ. Math. J , to appear.[10] J.-F. Mallordy, and J.-M. Roquejoffre. A parabolic equation of the KPP type in higherdimensions.
SIAM J. Math. Anal. , 26(1): 1–20, 1995.[11] J. Nolen, M. Rudd, and J. Xin. Existence of KPP fronts in spatially-temporally periodicadvection and variational principle for propagation speeds.
Dynamics of PDE , 2(1):1–24,2005. 1912] J. Nolen, and J. Xin. Asymptotic Spreading of KPP Reactive Fronts in IncompressibleSpace-Time Random Flows.
Ann. de l’Inst. Henri Poincare – Analyse Non Lineaire ,26(3):815–839, 2008.[13] H. Weinberger. On spreading speed and travelling waves for growth and migrationmodels in a periodic habitat.