The reactive-telegraph equation and a related kinetic model
TThe reactive-telegraph equation and a related kinetic model
Christopher Henderson ∗ and Panagiotis E. Souganidis † October 31, 2017
Abstract
We study the long-range, long-time behavior of the reactive-telegraph equation and a relatedreactive-kinetic model. The two problems are equivalent in one spatial dimension. We point outthat the reactive-telegraph equation, meant to model a population density, does not preservepositivity in higher dimensions. In view of this, in dimensions larger than one, we considera reactive-kinetic model and investigate the long-range, long-time limit of the solutions. Weprovide a general characterization of the speed of propagation and we compute it explicitly inone and two dimensions. We show that a phase transition between parabolic and hyperbolicbehavior takes place only in one dimension. Finally, we investigate the hydrodynamic limit ofthe limiting problem.
The Fisher-KPP equation u t = u xx + u (1 − u )is the classical model used to study the spread of a population in an environment. Being basedon the heat equation, the Fisher-KPP equation exhibits infinite speed of propagation. Indeed thepopulation density is non-zero everywhere for any positive time even if the initial data is compactlysupported.One approach to remove this unphysical behavior is to look at the reactive-telegraph equation τ ρ tt + (1 − τ (1 − ρ )) ρ t = ∆ ρ + ρ (1 − ρ ) . (1.1)This may be stated with more general non-linearities F ( ρ ) in place of ρ (1 − ρ ) and F (cid:48) ( ρ ) in placeof 1 − ρ . In one dimension, this is equivalent, for ρ = p − + p + , to the kinetic system (cid:40) p + t + √ τ p + x = τ ( p − − p + ) + ρ (1 − ρ ) ,p − t − √ τ p − x = τ ( p + − p − ) + ρ (1 − ρ ) . (1.2)The solution ρ is a population density, while p ± represents the density of those individuals movingwith velocity ±
1. The positive parameter τ is related to the relaxation time, which depends on ∗ Corresponding author, Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago,IL 60637, E-mail: [email protected] † Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, E-mail: [email protected] a r X i v : . [ m a t h . A P ] O c t he mean run length between changes in velocity. A small sample of the more applied literatureaddressing these models, including the interpretation of τ can be found in the works of Fedotov[17], Hillen and Othmer [18], Holmes [19], Horsthemke [20], Kac [21], M´endez, Fort, and Farjas[24], and Ortega, Fort, and M´endez [25].We discuss, in particular the work in [17], where the author proposed (1.1) as a model for populationdynamics in more than one dimension, deriving it from a general transport model with the fluxgiven by convolution of the gradient with a kernel, and formally analyzed the long-range, long-timebehavior of the solution with arguments similar to those used to rigorously study the same problemfor the Fisher-KPP equation. However, to even write down this formal analysis it is necessary toknow that the solutions to (1.1) preserve the sign, that is if they start nonnegative, they remainnonnegative.Our first result is to show that, in general, such formal computations cannot be justified when n ≥
2. Indeed, solutions to (1.1) need not remain positive. In other words, it is not possible to ruleout negative population densities, which are unphysical.
Theorem 1.1.
Assume n = 1 and fix ρ ∈ C ( R ) ∩ L ( R ) with ≤ ρ ≤ . If ρ is the solution of (1.1) with initial data ρ ( · ,
0) = ρ and ρ t ( · ,
0) = 0 , then ≤ ρ ( x, t ) ≤ for all ( x, t ) ∈ R × [0 , ∞ ) .If n ≥ , there exists ρ ∈ C ∞ ( R n ) with ≤ ρ ≤ and ( x , t ) ∈ R n × R + such that, if ρ solves (1.1) with initial data ρ ( · ,
0) = ρ and ρ t ( · ,
0) = 0 , then ρ ( x , t ) < . We discuss first why (1.1) preserves positivity when n = 1. In this case, (1.1) is equivalent to (1.2),which preserves positivity because, roughly, ρ is controlled by p from below. Neither the equivalenceof the model (1.1) to a kinetic equation nor the fact that ρ is controlled below by p holds in higherdimensions.As is evident in the proof of Theorem 1.1, (1.1) does not preserve positivity for a wide class ofnon-linearities F . The essential ingredients of the proof are that the equation is well-defined forshort times for data in a “nice” enough Sobolev space and that F is piecewise C near zero with F (0) = 0. In particular, the second claim in Theorem 1.1 is due to the properties of the waveoperator and not the non-linearity ρ (1 − ρ ).In view of the discussion above, the reactive-telegraph equation allows for negative populationdensities, suggesting that it may not be an appropriate biological model in some settings. As such,we restrict our focus to the following generalization of the kinetic system (1.2) p t + a n,τ v · Dp = 1 τ ( ρ − p ) + ρ (1 − ρ ) + in R n × S n − × R + , (1.3)where D is the spatial gradient, p is the density of individuals moving with velocity v ∈ S n − , ρ ( x, t ) := S n − p ( x, v, t ) dv is the population density, a n,τ := (cid:112) n/τ (1.4)is the speed of pure transport, S n − is the unit sphere in R n , R + := (0 , ∞ ) , ffl denotes the normalizedintegral such that ffl S n − dx = 1, and x + := max( x, ρ (1 − ρ ) + so thatthe model preserves positivity and retains the fundamental aspects of the logistic one, ρ (1 − ρ ).In particular, it represents the physical assumption that growth and competition depend only onthe total population at a particular location and do not depend on velocity. Lastly, the √ n factorin a n,τ is to fix the hydrodynamic limit τ → is expected to move at an approximately constant speed, it isstandard to use the hyperbolic long-range, long-time limit, see Barles, Evans and Souganidis [2],Evans and Souganidis [16], and Majda and Souganidis [23] and the large body of literature citingthese works. This limit corresponds to scaling space and time by the same large parameter in orderto capture this linear-in-time propagation while ignoring fluctuations and short-time behavior.As such, we use the hyperbolic scaling ( x, v, t ) (cid:55)→ ( x/(cid:15), v, t/(cid:15) ) and consider the rescaled function p (cid:15) ( x, v, t ) := p ( x/(cid:15), v, t/(cid:15) ) , which solves p (cid:15)t + a n,τ v · Dp (cid:15) = 1 (cid:15)τ ( ρ (cid:15) − p (cid:15) ) + 1 (cid:15) ρ (cid:15) (1 − ρ (cid:15) ) + in R n × S n − × R + , (1.5)with ρ (cid:15) ( x, t ) := S n − p (cid:15) ( x, v, t ) dv. We study the behavior, as (cid:15) →
0, of p (cid:15) and ρ (cid:15) with initial datum p (cid:15) ( · , · ,
0) = p on R n × S n − × { } (1.6)such that (cid:40) ≤ p ≤ , and there exists an open, Lipschitz domain G ⊂ R n so that G := { x ∈ R n : inf v p ( x, v ) > } and G c = { x ∈ R n : sup v p ( x, v ) = 0 } . (1.7)In order to investigate the propagation properties of p (cid:15) and ρ (cid:15) as (cid:15) →
0, we use the classical Hopf-Cole transform p (cid:15) = exp ( J (cid:15) /(cid:15) ), which is a standard tool in the study of front propagation [2, 16, 23],and study first the behavior, as (cid:15) →
0, of the J (cid:15) ’s, which solve J (cid:15)t + a n,τ v · DJ (cid:15) = τ (cid:16) ffl S n − e (cid:15) ( J (cid:15) ( x,v (cid:48) ,t ) − J (cid:15) ( x,v,t )) dv (cid:48) − (cid:17) + ffl S n − e (cid:15) ( J (cid:15) ( x,v (cid:48) ,t ) − J (cid:15) ( x,v,t )) dv (cid:48) (1 − ρ (cid:15) ) + in R n × S n − × R + . (1.8)Note that it is possible to use this transformation since p (cid:15) ≥ R n × S n − × R + . This is shownto be the case in Lemma 2.1 if (1.7) holds.In what follows, when necessary to signify the dependence on τ , we write J (cid:15),τ , J τ and H τ insteadof J (cid:15) , J and H . See Lemma 2.1 for a proof of the preservation of positivity and see Section 6 for a discussion of a related modelwith the logistic non-linearity ρ (1 − ρ ) that does not preserve positivity, which suggests that ρ (1 − ρ ) + is a betterchoice of non-linearity. By “front,” we mean the area between where ρ approximately takes the value 0 and the value 1. heorem 1.2. Assume (1.7) and let J (cid:15) solve (1.8) with initial data J (cid:15) ( · , · ,
0) = (cid:15) log p on R n × S n − . Then, for each τ > , there exists a concave, rotationally invariant function H : R n → R ,defined in (3.4) , such that, as (cid:15) → and uniformly in v and locally uniformly in R n × R + , J (cid:15) converges to J , the unique solution to max ( J t + H ( DJ ) , J ) = 0 in R n × R + J ( · ,
0) = (cid:40) −∞ in G c , in G . (1.9)We make a few brief comments on some technical aspects of Theorem 1.2. First, the limitingfunction J does not depend on v . Second, when p is zero, we define log p to be the extended realvalue −∞ . Lastly, by the locally uniform convergence of J (cid:15) to J , we mean that this convergence isuniform on compact sets K such that either K ⊂ Int { J = −∞} or K ⊂ Int { J < −∞} .Knowing Theorem 1.2 we then infer the following spreading behavior of ρ (cid:15) . Theorem 1.3.
Suppose that (1.7) holds and that J is the solution to (1.9) . Then, as (cid:15) → andlocally uniformly in { J < } , lim (cid:15) → ρ (cid:15) ( x, t ) = 0 . If τ ≤ , then, as (cid:15) → and locally uniformly in Int { J = 0 } , lim (cid:15) → ρ (cid:15) ( x, t ) = 1 . If τ > , then, as (cid:15) → and locally uniformly in Int { J = 0 } , lim inf (cid:15) → ρ (cid:15) ( x, t ) ≥ . Before discussing the proof of Theorem 1.3, we mention the reason that there is a distinction between τ > τ ≤
1. When τ <
1, it is possible to bound p by 1 by using a maximum principle-typeargument. On the other hand, when τ ≥
1, it is shown in Lemma 2.1 that upper bound is order(1 + τ ) / τ . The proof of Theorem 1.3 yields that lim inf (cid:15) → ρ (cid:15) ( x, t ) ≥ { J = 0 } . When τ <
1, the bound above gives us immediately that lim sup (cid:15) → ρ (cid:15) ( x, t ) ≤
1, yielding the exact valueof the limit. When τ ≥
1, this argument does not work.To prove the asymptotic results, we use the classical half-relaxed limits of Barles and Perthame [3]along the lines of [2, 16] and a modification due to Barles and Souganidis [4], which allows us toside-step the technical difficulty that, due to the finite speed of propagation in kinetic equations,the J (cid:15) ’s take the value −∞ . The combination of the half-relaxed limits and the technique of [4]is that, roughly, when the limiting Hamilton-Jacobi equation represents a distance function, no apriori bounds or regularity are needed to pass to the limit (cid:15) → H in Theorem 1.2 is the same one found by Bouin and Calvez [6] and Caillerie [12]in a related context since the linearized equations are the same.Since H in (1.9) is space-time homogeneous, it follows from [16, 23] that J ( x, t ) = min ( I ( x, t ) , , (1.10)where I is the solution to I t + H ( DI ) = 0 in R n × R + , I ( · ,
0) = (cid:40) −∞ in G c , G . Let L be the concave dual of H . Then, see, for example, Lions [22], I ( x, t ) = t sup (cid:26) L (cid:18) x − yt (cid:19) : y ∈ G (cid:27) . (1.11)As mentioned in Theorem 1.2, H is concave and rotationally invarant. From this it follows that H is radially decreasing; this can also be seen from the explicit formula (3.4). As a result, L has the4ame properties. Taking some liberty with notation, we write L ( q ) = L ( | q | ). It then follows that,for t ∈ R + , { x ∈ R n : J ( x, t ) < } = (cid:40) x ∈ R n : sup y ∈ G L (cid:18) x − yt (cid:19) < (cid:41) = (cid:26) x ∈ R n : L (cid:18) d ( x, G ) t (cid:19) < (cid:27) , (1.12)where d ( x, G ) is the usual distance from the point x to the closed set G .In view of Theorem 1.3, it is clear that the front is ∂ { x ∈ R n : J ( x, t ) < } . From the discussionabove, we see that ∂ { x ∈ R n : J ( x, t ) < } = ∂ { x ∈ R n : I ( x, t ) < } = (cid:26) x ∈ R n : L (cid:18) d ( x, G ) t (cid:19) = 0 (cid:27) . (1.13)The next result, which holds for n ≥
2, provides a characterization of the 0-level set of L and,hence, the speed of propagation in terms of a global property of the Hamiltonian.Let c n,τ := − sup q ∈ R n H ( q ) | q | . The claim is:
Proposition 1.4.
Assume (1.7) and n ≥ . Then c n,τ is achieved, c n,τ < a n,τ , and the front isthe set { x ∈ R n : d ( x, G ) = c n,τ t } . Since c n,τ < a n,τ , it follows that, for n ≥
2, the front moves with velocity slower than that of puretransport. Following [8], we call this behavior parabolic to distinguish it from the hyperbolic one,which is observed, as discussed below, for n = 1, when τ ≥ . To heuristically justify the term “parabolic,” we return to the unscaled problem (1.3) and discussthe behavior of ρ when ρ has compact support. Indeed, due to the kinetic nature of (1.3), thesupport of ρ propagates with the speed of the pure transport a n,τ . Proposition 1.4, however, impliesthat the set on which ρ is approximately 1 propagates at speed c n,τ . This difference suggests that ρ has long tails connecting 0 and 1. As a result the profile resembles that of solutions to theFisher-KPP equation, whose long tails are caused by the infinite speed of propagation in the heatequation. See Figure 1 for a cartoon picture of this, and see [8] for an explicit construction wherethe long tails are observed.When n = 1 and n = 2 it is possible to explicitly compute H and, hence, the speed of propagation.This is the contents of the proposition below. We point out that the speed c ,τ is the one givenin [17] and is the speed of the traveling waves constructed by Bouin, Calvez, and Nadin in [8]. Proposition 1.5.
Assume (1.7) . Then c τ, = τ if τ ≤ , √ τ if τ ≥ , and c τ, = (cid:112) τ )1 + τ . When n = 1 and τ ≥ a ,τ = c ,τ . We refer to this behavior as “hyperbolic”. To heuristicallyjustify this term, we again return to the unscaled problem (1.3) and consider initial data ρ withcompact support. Following the discussion above, we see that the support of ρ and the set onwhich ρ approximately takes the value 1 move with the same speed. In particular, we do not5 n,τ t a n,τ t c n,τ t a n,τ tx ρ ρ Hyperbolic behavior Parabolic behaviorFigure 1: Cartoon pictures showing the difference between the hyperbolic behavior (the graph onthe left) seen in one spatial dimension when τ ≥ ρ , which progress with speed a n,τ , with the location ofthe front, which progresses with speed c n,τ , which is strictly less than a n,τ .see the formation of long tails as in the discussion above. Once more, see Figure 1 for a cartoonrepresentation of this behavior and see [8] for an explicit construction where one sees this behavior.It follows from the last two propositions that there is a phase transition, depending on τ , betweenparabolic and hyperbolic behavior only in one dimension. We now heuristically explain the reasonfor this. In order to exhibit hyperbolic behavior, growth due to the reaction must “keep up” withpure transport. That this is possible in only R is related to the fact that p controls ρ from below;that is, for any v ∈ S = {− , } , p ( x, v, t ) / ≤ ρ ( x, t ) for all ( x, t ). In particular, the non-localreaction term ρ (1 − ρ ) + can be bounded below by a local term when ρ is small. This is not possiblein higher dimensions.Formally, taking the limit τ → u t = ∆ u + u (1 − u ).In addition, solutions of (1.3) formally converge, as τ →
0, to the Fisher-KPP equation regardlessof dimension; see, for example, the discussion of Cuesta, Hittmeir, and Schmeiser in [14] and alsoSection 3 of [18].We prove that the limits may be taken in the reverse order to obtain the same limiting object.Indeed, fix u such that G = { u > } and let u (cid:15) be the solution to u (cid:15)t = (cid:15) ∆ u (cid:15) + (cid:15) − u (cid:15) (1 − u (cid:15) ) in R n × R + u (cid:15) ( · ,
0) = u on R n . It is well-known (see [16]) that, as (cid:15) → x, t ), z (cid:15) := (cid:15) log u (cid:15) converges tothe unique solution z ofmax { z t − | Dz | − , z } = 0 in R n × R + z ( · ,
0) = (cid:40) G , −∞ on G c . (1.14)We show that, as τ → J τ → z , implying that lim τ → lim (cid:15) → J τ,(cid:15) = z . This is what one wouldexpect in view of the discussion above, which formally gives that lim (cid:15) → lim τ → J τ,(cid:15) = z . Proposition 1.6.
Assume (1.7) and let J τ and z be respectively the unique solutions to (1.9) and (1.14) . Then, as τ → and locally uniformly in R n × R + , J τ → z . The convergence in Proposition 1.6 essentially follows from the formal convergence of H τ to −|·| − √ n factor in (1.4), there would be a factor n in theequation for lim τ → J τ . This is related to the fact that the variance of any unit vector over S n − is 1 /n . We mention briefly that, in the equation (1.3), the √ n factor can be added or removed bysimply scaling in x and therefore does not affect our analysis.6ext we discuss briefly related works. As mentioned above, the one dimensional problems have beenconsidered from the modeling perspective in [18, 19, 21]. The reactive-telegraph model (1.1) hasonly been rigorously studied in the one dimensional setting. We point out, in particular, the workof Bouin, Calvez, and Nadin [8], in which they prove existence and stability of traveling waves in aweighted space. In addition, there is [17] which was discussed earlier. There has been more workrecently on related kinetic equations in one dimension. Bouin and Calvez [6] obtained convergenceto a limiting equation for the linearization of (1.3) assuming that J is Lipschitz and bounded.Later Bouin, Calvez, and Nadin [9], and Bouin, Calvez, Grenier, and Nadin [7] showed accelerationin a kinetic-reactive equation similar to our setting but where the velocity space is unbounded andthe reaction term is replaced by ρ ( M ( v ) − p ). In higher dimensions, the kinetic model studied herewith ρ (1 − ρ ) replaced by ρ ( M ( v ) − p ) was investigated by Caillerie [12]. In this work Caillerieperforms a limiting procedure with well-prepared initial data and identifies the same Hamiltonianas the one we find above .Finally, during the write-up of this work, we became aware of a parallel work by Bouin and Cail-lerie [5] which has some similarity to the present one. In [5], the authors investigate a related kineticsystem with reaction term ρ ( M ( v ) − p ) and with general dispersal kernel M ( v ). The authors con-sider the long-range, long-time limit of this equation, the question of propagation in the unscaledequation, and the existence of traveling waves. There are many differences between the two papers.First, in the long-range, long-time limit, Bouin and Caillerie consider well-prepared initial data for J (cid:15) , which corresponds to initial conditions that are exponentially decaying (in contrast to thecompactly supported initial conditions considered here and widely considered in the front propa-gation literature ). Second, the equation considered by Bouin and Caillerie enjoys the maximumprinciple. These two facts simplify many technical details in the long-range, long-time limit. Also,the questions considered in the present work about the appropriateness of the reactive-telegraphequation, the difference in behavior with the non-linearity ρ (1 − ρ ) versus ρ (1 − ρ ) + , and the ex-istence and dependence on dimension of the phase transition in τ are not considered in [5]. Onthe other hand, the work of Bouin and Caillerie considers a much broader class of kinetic models,which will be useful in studying a variety of biological models. Organization of the paper
The paper is organized into six main parts. In Section 2 we present a preliminary lemma that givesan upper bound for J (cid:15) that is independent of (cid:15) and prove the convergence part of Theorem 1.3.The Hamiltonian and its properties as well as the proof of Theorem 1.2 are given in Section 3. InSection 4 we prove Proposition 1.4 and Proposition 1.5. In Section 5 we show that we may take thehydrodynamic limit τ → That they are related can be seen by the fact that F ( ρ ) and ρ ( M ( v ) − p ) have the same linearization when M ( v )is 1 for all v ∈ S n − . The authors fix initial data for their analogue of J (cid:15) and apparently assume that this initial data is independentof (cid:15) and finite everywhere. This is in contrast to the present work in which J (cid:15) takes infinite values in G c . Since the study of front propagation is the study of a new population (or other physical quantity) invading apreviously unexplored area, which corresponds to p being zero, this type of assumption is crucial to a full investiga-tion. We note that this is not a mere technical detail – there are many situations where the behavior of compactlysupported initial data and well-prepared initial data are quite different (see, e.g., Bramson [10], Zlatoˇs [27], Cabreand Roquejoffre [11]). he notion of solution We are not making any assumptions for (1.5) and (1.8) to have smooth solutions. We interpretboth equations as well as the limiting Hamilton-Jacobi variational inequalities and equations in theclassical Crandall-Lions viscosity sense.
Acknowledgements
We wish to thank the anonymous referees for a close reading of the manuscript and their veryhelpful comments. CH would like to thank Jacek Jendrej for pointing out the refence [26] regardingregularity of hyperbolic equations. CH was partially supported by the National Science FoundationResearch Training Group grant DMS-1246999. PS was partially supported by the National Sci-ence Foundation grants DMS-1266383 and DMS-1600129 and the Office for Naval Research GrantN00014-17-1-2095.
The a priori bounds
We state as a lemma the fact that (1.5) preserves positivity and yields an upper bound that isindependent of (cid:15) . This is important, since without the positivity of p (cid:15) , J (cid:15) is not well-defined, whilethe upper bound is needed in order to study the limit (cid:15) → Lemma 2.1.
Assume ≤ p ≤ and let p (cid:15) be the solution to (1.5) and (1.6) . Then, for all t ∈ R + , ≤ p (cid:15) ( · , · , t ) ≤ M τ , where M τ = if τ ≤ , (1+ τ ) τ if τ > . Proof.
The positivity of p (cid:15) follows from rewriting (1.5) as p (cid:15)t + a n,τ v · Dp (cid:15) + 1 (cid:15)τ p (cid:15) = 1 (cid:15)τ ρ (cid:15) + 1 (cid:15) ρ (cid:15) (1 − ρ (cid:15) ) + , and observing that the right hand side of the above equation is always nonnegative.For the upper bound, if τ >
1, let q (cid:15) := (1 + τ ) τ − p (cid:15) and ˆ ρ (cid:15) := S n − q (cid:15) ( x, v, t ) dv. It follows that q (cid:15)t + a n,τ v · Dq (cid:15) + 1 (cid:15)τ q (cid:15) = 1 (cid:15)τ ˆ ρ (cid:15) + 1 (cid:15) (cid:18) ˆ ρ (cid:15) − (1 + τ ) τ (cid:19) (cid:18) ˆ ρ (cid:15) − (cid:18) (1 + τ ) τ − (cid:19)(cid:19) + . Straightforward calculations yield that the right hand side above is nonnegative, which, in turn,implies that q (cid:15) ≥ τ ≤ ropagation We present the proof of Theorem 1.2, which is an immediate consequence of Theorem 1.3.
Proof of Theorem 1.2.
Fix any ( x , t ) ∈ { J < } with t >
0. Since J (cid:15) converges locally uniformlyin x, t and uniformly in v to J , then, for all (cid:15) sufficiently small, we havesup v ∈ S n − J (cid:15) ( x , v, t ) ≤ J ( x , t ) / < . It follows that ρ (cid:15) ( x , t ) = S n − p (cid:15) ( x , v, t ) dv ≤ sup v ∈ S n − p (cid:15) ( x , v, t ) = e (cid:15) sup v ∈ Sn − J (cid:15) ( v,x ,t ) ≤ e J ( x ,t (cid:15) , and, hence, lim (cid:15) → ρ (cid:15) ( x , t ) = 0; the local uniformity of the limit is immediate.Now consider any point ( x , t ) ∈ Int { J = 0 } with t >
0. Let φ ( x, t ) = −| t − t | − | x − x | andnotice that J − φ has a strict local minimum at ( x , t ). Hence, there exist ( x (cid:15) , t (cid:15) ) converging to( x , t ) such that min v J (cid:15) ( x, t ) − ψ has a local minimum at ( x (cid:15) , t (cid:15) ). Let v (cid:15) be such that J (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ) =min v J (cid:15) ( x (cid:15) , v, t (cid:15) ).Uaing (1.8), we find φ t ( x (cid:15) , t (cid:15) ) + a n,τ v (cid:15) · Dφ ( x (cid:15) , t (cid:15) ) ≥ τ (cid:18) ρ (cid:15) ( x (cid:15) , t (cid:15) ) p (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ) − (cid:19) + ρ (cid:15) ( x (cid:15) , t (cid:15) ) p (cid:15) ( x e , v (cid:15) , t (cid:15) ) (1 − ρ (cid:15) ( x (cid:15) , t (cid:15) )) + . (2.1)Observe that, since ( x (cid:15) , v (cid:15) , t (cid:15) ) is the location of a global minimum in v of J (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ), ρ (cid:15) ( x (cid:15) , t (cid:15) ) p (cid:15) ( x e , v (cid:15) , t (cid:15) ) ≥ . Moreover, an explicit computation implies that, as (cid:15) → φ t ( x (cid:15) , t (cid:15) ) + a n,τ v (cid:15) · Dψ ( x (cid:15) , t (cid:15) ) → . Then lim (cid:15) → ρ (cid:15) ( x (cid:15) , t (cid:15) ) p (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ) = 1 and lim (cid:15) → (1 − ρ (cid:15) ( x (cid:15) , t (cid:15) )) + = 0 . (2.2)The second limit above implies that lim inf (cid:15) → ρ (cid:15) ( x (cid:15) , t (cid:15) ) ≥ v ∈ S n − , J (cid:15) ( x , v, t ) ≥ J (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ) − φ ( x (cid:15) , t (cid:15) ). Hence ρ (cid:15) ( x , t ) = S n − e J(cid:15) ( x ,v,t (cid:15) dv ≥ S n − e J(cid:15) ( x(cid:15),v(cid:15),t(cid:15) )+ | x(cid:15) − x | | t(cid:15) − t | (cid:15) dv ≥ p (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ) = (cid:18) p (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ) ρ (cid:15) ( x (cid:15) , t (cid:15) ) (cid:19) ρ (cid:15) ( x (cid:15) , t (cid:15) ) . Letting (cid:15) → (cid:15) → ρ (cid:15) ( x , t ) ≥ . (2.3)Since, when τ ≤ p (cid:15) ≤
1, we have ρ (cid:15) ≤
1, (2.3) yieldslim (cid:15) → ρ (cid:15) ( x , t ) = 1 . The Hamiltonian, its properties and the proof of Theorem 1.3
The Hamiltonian H and its properties To motivate the choice of the Hamiltonian H , we first present a formal argument about the limitof the J (cid:15) assuming that J (cid:15) ( x, v, t ) = J ( x, t ) + (cid:15)η ( x, v, t ) + o( (cid:15) ) . Working in { J < } , where we can ignore ρ (cid:15) in (1 − ρ (cid:15) ) + , we get from (1.8) that J t + a n,τ v · DJ = 1 τ (cid:18) S n − e η ( x,v (cid:48) ,t ) − η ( x,v,t ) dv (cid:48) − (cid:19) + S n − e η ( x,v (cid:48) ,t ) − η ( x,v,t ) dv (cid:48) . Since J t is independent of v , there must exist some, independent of v , constant H ( DJ ) so that, forall v ∈ S n − , − H ( DJ ) + 1 τ + a n,τ v · DJ = τ + 1 τ S n − e η ( x,v (cid:48) ,t ) − η ( x,v,t ) dv (cid:48) . The above expresssion leads to the cell (eigenvalue) problem to find, for each p ∈ R n , a uniqueconstant H ( p ) (eigenvalue) and some η = η ( v ; p ) (eigenfunction) such that, for all v ∈ S n − , e η ( v ) ffl S n − e η ( v (cid:48) ) dv (cid:48) = τ + 1 τ − H ( p ) + τ + a n,τ v · p . (3.1)It follows that, if it exists, H ( p ) must be defined implicitly by τ τ = S n − dv − H ( p ) + τ + a n,τ v · p . (3.2)Consider the function Φ : [1 , ∞ ) → R + given byΦ( s ) := S n − dvs + v ; (3.3)here v = ( v , v , . . . , v n ). It is immediate that Φ (cid:48) < s →∞ Φ( s ) = 0. Moreover, as it isshown in the Appendix, Φ(1) = ∞ when n ∈ { , , } , while Φ(1) < ∞ for n > p ∈ R n \ { } . Then, looking at (3.2) and the properties of Φ we assert that H ( p ) := − a n,τ | p | + τ if Φ(1) ≤ τ τ a n,τ | p | ,α if Φ(1) ≥ τ τ a n,τ | p | , (3.4)where, in the latter case, α is the unique negative number such thatΦ (cid:32) − α + τ a n,τ | p | (cid:33) = τ τ a n,τ | p | . (3.5)Note that H is continuous, isotropic, that is depends only on | p | , and H (0) = − . We note thatthis Hamiltonian is the same as found in [5, 12].Next we show that (3.4) is indeed correct for every p , when n ∈ { , , } , and for all p such thatΦ(1) > τ τ a n,τ | p | , if n > . We present the argument in the latter case, since the discussion appliesto the former. 10f Φ(1) > τ τ a n,τ | p | , then η : S n − → R given by e η ( u ) = τ + 1 τ − H ( p ) + τ + a n,τ v · p , (3.6)with H ( p ) given by the second alternative in (3.4), clearly satisfies (3.1). If Φ(1) = τ τ a n,τ | p | , thenthe η given in (3.6) also satisfies (3.1) for all v ∈ S n − except v = − p/ | p | .It is clear that, when Φ(1) < τ τ a n,τ | p | , it is not possible to find such an η . In the proofs, we dealwith this issue by considering “approximate” correctors, which for δ > µ >
0, are given by η µ,δ ( v ) := τ + 1 τ δ − H ( p ) + τ + a n,τ v · p ) µ . (3.7)We now discuss the concavity of H . We present the argument for n >
3, since the other casesfollow similarly without having to deal with the first part of the definition of the Hamiltonian.When | p | > (1+ τ )Φ(1) / ( a n,τ τ ), the concavity is obvious from (3.4). When | p | < (1+ τ )Φ(1) / ( a n,τ τ ),we use (3.2) to obtain S n − D p H ( p ) − a n,τ v (cid:0) − H ( p ) + τ + a n,τ v · p (cid:1) dv = 0 . (3.8)Differentiating again we get S n − D p H ( p ) dv (cid:0) − H ( p ) + τ + a n,τ v · p (cid:1) = − S n − ( D p H ( p ) − a n,τ v ) ⊗ ( D p H ( p ) − a n,τ v ) (cid:0) − H ( p ) + τ + a n,τ v · p (cid:1) dv. Fix any non-zero vector ξ ∈ R n . Then D p H ( p ) ξ · ξ S n − dv (cid:0) − H ( p ) + τ + a n,τ v · p (cid:1) = − S n − | ( D p H ( p ) − a n,τ v ) · ξ | (cid:0) − H ( p ) + τ + a n,τ v · p (cid:1) dv, and concavity follows after noticing that, by its construction, − H ( p ) + τ − a n,τ | p | ≥ | p | = ττa n,τ Φ(1)when Φ(1) < ∞ . Since H is isotropic, this is immediate if we show thatlim sup | p |→ ττan,τ Φ(1) − | D p H ( p ) | ≤ lim | p |→ ττan,τ Φ(1) + | D p H ( p ) | = a n,τ . (3.9)To establish (3.9) we consider two cases depending on whether ffl S n − (1 + v · ˆ p ) − dv is finite or not.Note that this quantity does not depend on ˆ p , since we may simply change variables in the integral.If ffl S n − (1 + v · ˆ p ) − dv < ∞ then, in view of (3.8), we getlim sup | p |→ ττan,τ Φ(1) − | D p H ( p ) | = (cid:12)(cid:12)(cid:12)(cid:12) S n − a n,τ v (1 + v · ˆ p ) dv (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) S n − v · ˆ p ) dv (cid:19) − ≤ a n,τ . If ffl S n − (1 + v · ˆ p ) − dv = ∞ , then, using (3.8), we findlim | p |→ ττan,τ Φ(1) − | D p H ( p ) | = a n,τ , since, otherwise, the left hand side of (3.8) is not finite, and the claim follows.11 he half-relaxed limits and the limiting equation We prove Theorem 1.2. The main tools are the classical (in the theory of viscosity solutions) half-relaxed limits [3] and the methodology of [2]. The problem is that, since J (cid:15) takes infinite values,we do not have uniform bounds. To circumvent this we use an argument introduced in [4] to dealwith this kind of difficulty.We begin with the definition of the half-relaxed limits. Given a family f (cid:15) of uniformly boundedfunctions depending only on x and t , the half-relaxed upper and lower limits f and f are respectively f ( x, t ) := lim sup f (cid:15) ( x, t ) := lim sup y → x,s → t,(cid:15) → f (cid:15) ( y, s ) and f ( x, t ) := lim inf f (cid:15) ( x, t ) := lim inf y → x,s → t,(cid:15) → f (cid:15) ( y, s ) . We also remark that, in view of Lemma 2.1, J (cid:15) ≤ (cid:15) log M τ in R n × S n − × R + . (3.10)Fix A > J (cid:15)A := max { J (cid:15) , − A } . Since J (cid:15)A is bounded, uniformly in (cid:15) , from above and below by (cid:15) log M τ and − A respectively, we takethe half-relaxed upper and lower limits of max v J (cid:15)A and min v J (cid:15)A respectively; that is, we consider J A and J A given by J A ( x, t ) := lim sup max v J (cid:15)A ( x, v, t ) and J A ( x, t ) := lim inf min v J (cid:15)A ( x, v, t ) . (3.11)We point out that max v J (cid:15)A and min v J (cid:15)A are uniformly bounded in (cid:15) and do not depend on v . Hence,their half-relaxed limits are well-defined. Further, we note that we expect the limit of J (cid:15)A to beindependent of v as fluctuations in v will “average out.” This suggests that we lose no informationin taking the maximum and minimum in v .To state the next lemma we introduce some additional notation. First, J A,t refers to the timederivative of J A , and this notation applies similarly to other terms derived from J A . Second, given g : R n → R , g (cid:63) and g (cid:63) are respectively its upper and lower semicontinuous envelopes. Moreover, A := G , − A on ( R n \ G ) . Lemma 3.1.
Assume (1.7) . Then:(i) J A is a (viscosity) sub-solution to max (cid:8) J A,t + H ( DJ A ) , J A , − J A − A (cid:9) ≤ in R n × R + ,J A ( · , ≤ ( A ) (cid:63) on R n ; (3.12) (ii) J A is a (viscosity) super-solution to max (cid:8) J A,t + H ( DJ A ) , J A , − J A − A (cid:9) ≥ in R n × R + ,J A ( · , ≥ ( A ) (cid:63) on R n . (3.13)12he proofs of (3.12) and (3.13) are similar. We separate into cases based on how the Hamiltonianis defined. When H is defined by the second case of (3.4), the proof is based on perturbing thetest function by a small multiple of the corrector. This method is classical, dating back to thework of Evans [15] (see also [23] for another early use and [5, 6] for an application in a very similarcontext). On the other hand, when H is defined by the first case of (3.4), additional care is requiredbecause, as per the discussion in the previous subsection, exact correctors do not exist and a fewmore additional arguments are need in that case. This is where the proofs of (3.12) and (3.13)differ.The proofs of parts (i) and (ii) are quite long and involved, and for the readers convenience weseparate them into two proofs. We first show the proof of (i), which is slightly simpler. Proof of Lemma 3.1(i). (3.12) for positive times:
Since ρ (cid:15) ≥
0, it follows from (1.8) that J (cid:15)t + a n,τ v · DJ (cid:15) ≤ τ (cid:18) S n − e (cid:15) ( J (cid:15) ( · ,v (cid:48) , · ) − J (cid:15) ( · ,v, · )) dv (cid:48) − (cid:19) + S n − e (cid:15) ( J (cid:15) ( · ,v (cid:48) , · ) − J (cid:15) ( · ,v, · )) dv (cid:48) . Noting that − A also solves the inequality above and using that the maximum of two sub-solutionsis itself a sub-solution implies that J (cid:15)A,t + a n,τ v · DJ (cid:15)A ≤ τ (cid:18) S n − e (cid:15) ( J (cid:15)A ( · ,v (cid:48) , · ) − J (cid:15)A ( · ,v, · )) dv (cid:48) − (cid:19) + S n − e (cid:15) ( J (cid:15)A ( · ,v (cid:48) , · ) − J (cid:15)A ( · ,v, · )) dv (cid:48) . (3.14)Let φ be a smooth test function and assume that J A − φ has a strict maximum at ( x , t ) and φ ( x , t ) = J A ( x , t ) . We first assume that t > (cid:8) φ t ( x , t ) + H ( Dφ ( x , t )) , J A ( x , t ) , − J A ( x , t ) − A (cid:9) ≤ . (3.15)That max (cid:8) J A ( x , t ) , − J A ( x , t ) − A (cid:9) ≤ J A and (3.10).Next we show that φ t ( x , t ) + H ( Dφ ( x , t )) ≤ . (3.16)To simplify the presentation and shorten some formulae in what follows we write p := Dφ ( x , t ) and ˆ p := Dφ ( x , t ) | Dφ ( x , t ) | . There are two cases.
Case one:
If Φ(1) > τ τ a n,τ | p | , we consider the perturbed test φ (cid:15) = φ + (cid:15)η ( v ; p ) with η given by (3.6).Let ( x (cid:15) , t (cid:15) ) be a maximum point of max v J (cid:15)A − φ (cid:15) in a neighborhood of ( x , t ). The use of such anapproximating sequence in the theory of viscosity solutions is standard (see, for example, [1, 22]).In short, using the boundedness and continuity of η along with the fact that ( x , t ) is a strict localmaximum of J A − φ , its existence follows directly from the definition of lim sup.13hen there exist v (cid:15) ∈ S n − such that ( x (cid:15) , v (cid:15) , t (cid:15) ) is a maximum point of J (cid:15)A − φ (cid:15) and along asubsequence, which we denote the same way, (cid:15) →
0, there exists v ∈ S n − such that ( x (cid:15) , v (cid:15) , t (cid:15) ) → ( x , v , t ) and J (cid:15)A ( x (cid:15) , v (cid:15) , t (cid:15) ) → J A ( x , t ) . We note that the existence of the limiting vector v ∈ S n − follows from the compactness of S n − .It follows that, for (cid:15) small enough and all v ∈ S n − and all ( x, t ) in a small ball B r ( x , t ) of radius r > φ (cid:15) ( x, v, t ) − φ (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ) ≥ J (cid:15)A ( x, v, t ) − J (cid:15)A ( x (cid:15) , v (cid:15) , t (cid:15) ) . (3.17)Then (3.14) implies φ (cid:15)t ( x (cid:15) , v (cid:15) , t (cid:15) ) + a n,τ v (cid:15) · Dφ (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ) + 1 τ ≤ τ + 1 τ S n − e η ( v (cid:48) ) − η ( v (cid:15) ) dv (cid:48) . (3.18)Using now the definition of η in (3.18) we find φ (cid:15)t ( x (cid:15) , t (cid:15) ) + a n,τ v (cid:15) · Dφ (cid:15) ( x (cid:15) , t (cid:15) ) ≤ − H ( Dφ ( x , t )) + a n,τ v (cid:15) · Dφ ( x , t ) , which, after letting (cid:15) →
0, gives (3.16).
Case two:
If Φ(1) ≤ τ τ a n,τ | Dφ ( x , t ) | , we observe that the previous argument cannot be repeated verbatim, since, as remarked on duringthe previous discussion, we do not have an exact corrector. We follow the same line of proof asabove but with an additional twist to deal with this difficulty.For δ >
0, we consider the “approximate” corrector η δ given by e η δ ( v ) = τ + 1 τ a n,τ | p |
11 + δ + v · ˆ p , and note that S n − e η δ ( v ) dv = τ + 1 τ a n,τ | p | S n −
11 + δ + v · ˆ p dv ≤ τ + 1 τ a n,τ | p | S n −
11 + v · ˆ p = τ + 1 τ a n,τ | p | Φ(1) ≤ . (3.19)Consider the perturbed test function φ δ,(cid:15) ( x, v, t ) = φ ( x, t ) + (cid:15)η δ ( v ). Let ( x δ,(cid:15) , t δ,(cid:15) ) be a maximumpoint of max v J (cid:15)A − φ δ,(cid:15) in a neighborhood of ( x , t ). Then there exists v δ,(cid:15) ∈ S n − such that( x δ,(cid:15) , v δ,(cid:15) , t δ,(cid:15) ) is a maximum point of J (cid:15)A − φ δ,(cid:15) and along a subsequence, which we denote the sameway, (cid:15) →
0, ( x δ,(cid:15) , v δ,(cid:15) , t δ,(cid:15) ) → ( x , v δ, , t ) for some v δ, ∈ S n − and J (cid:15)A ( x δ,(cid:15) , v δ,(cid:15) , t δ,(cid:15) ) → J τA ( x , t ) . Note that the limit of the ( x δ,(cid:15) , t δ,(cid:15) ) is independent of δ since ( x , t ) is a strict local maximum of J A − φ .As before we have φ δ,(cid:15)t ( x δ,(cid:15) , t δ,(cid:15) ) + a n,τ v δ,(cid:15) · Dφ δ,(cid:15) ( x δ,(cid:15) , t δ,(cid:15) ) + 1 τ ≤ τ + 1 τ S n − e η δ ( v (cid:48) ) − η δ ( v δ,(cid:15) ) dv (cid:48) . Using the definition of η δ and (3.19), we find φ t ( x δ,(cid:15) , t δ,(cid:15) ) + a n,τ v δ,(cid:15) · Dφ ( x δ,(cid:15) , t δ,(cid:15) ) + 1 τ ≤ a n,τ | p | (1 + δ + v δ,(cid:15) · ˆ p ) = a n,τ ( | p | (1 + δ ) + v δ,(cid:15) · p ) . (cid:15) →
0, we obtain φ t ( x , t ) + a n,τ v δ, · Dφ ( x , t ) + 1 τ ≤ a n,τ ( | p | (1 + δ ) + v δ, · p ) , and, hence, φ t ( x , t ) − (1 + δ ) a n,τ | Dφ ( x , t ) | + 1 τ ≤ , from which the claim follows after taking δ → (3.12) at t = 0 : To conclude the proof of the sub-solution property, we considerthe case t = 0. Here the first conclusion is that J A satisfies the initial condition in the followingweak sense:min (cid:8) max (cid:8) J A,t + H ( DJ A ) , J A , − J A − A (cid:9) , J A − ( A ) (cid:63) (cid:9) ≤ R n × { } . (3.20)Then an argument as in [2] implies that actually J A ≤ ( A ) (cid:63) on R n × { } , and hence the claim.To prove (3.20) we assume that, for some smooth φ , J A − φ has a strict maximum at ( x , t >
0. If the maximum points at the level (cid:15) are in R n × R + for infinitely many (cid:15) ’s, we argueexactly as before and obtainmax { φ t ( x , t ) + H ( Dφ ( x , t )) , − J A ( x , t ) , − J A ( x , t ) − A } ≤ , otherwise we have J A ( x , t ) ≤ ( A ) (cid:63) , and, hence, the claim.We now proceed with the proof of part (ii). Proof of Lemma 3.1(ii). (3.13) for positive times:
Let φ be a smooth test function and assume that J A − φ has a strict minimum at ( x , t ) and φ ( x , t ) = J A ( x , t ).We claim that max { φ t ( x , t ) + H ( Dφ ( x , t )) , J A ( x , t ) , − J A ( x , t ) − A } ≥ . (3.21)If J A ( x , t ) = 0 or J A ( x , t ) = − A the claim is true. Hence we assume that − A < J A ( x , t ) < φ t ( x , t ) + H ( Dφ ( x , t )) ≥ . (3.22)Again, to simplify the presentation and shorten some formulae, we use as before the notation p and ˆ p for Dφ ( x , t ) and Dφ ( x , t ) / | Dφ ( x , t ) | respectively. There are two cases to consider. Case one:
If Φ(1) > τ τ a n,τ | p | , we use the perturbed test φ (cid:15) = φ + (cid:15)η ( v ; p ) with η given by (3.6).15et ( x (cid:15) , t (cid:15) ) be a minimum point of min v J (cid:15)A − φ (cid:15) in a neighborhood of ( x , t ). Then there exist v (cid:15) ∈ S n − such that ( x (cid:15) , v (cid:15) , t (cid:15) ) is a minimum point of J (cid:15)A − φ (cid:15) and, along a subsequence, whichwe denote the same way, (cid:15) →
0, there exists v ∈ S n − such that ( x (cid:15) , v (cid:15) , t (cid:15) ) → ( x , v , t ) and J (cid:15)A ( x (cid:15) , v (cid:15) , t (cid:15) ) → J A ( x , t ) . We note that, for (cid:15) sufficiently small, − A < J (cid:15)A ( x (cid:15) , v (cid:15) , t (cid:15) ) < J (cid:15)A ( x (cid:15) , v (cid:15) , t (cid:15) ) = J (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ), and, hence, the J (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) )’s are bounded away from 0.Then (1.8) implies φ (cid:15)t ( x (cid:15) , v (cid:15) , t (cid:15) )+ a n,τ v (cid:15) · Dφ (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ) + 1 τ ≥ (cid:18) τ + (1 − ρ (cid:15) ( x (cid:15) , t (cid:15) )) + (cid:19) S n − e J(cid:15) ( x(cid:15),v (cid:48) ,t(cid:15) ) − J ( x(cid:15),v(cid:15),t(cid:15) ) (cid:15) dv (cid:48) . (3.23)Since the left hand side of (3.23) is bounded independently of (cid:15) , so must be the integral term S n − e ( J (cid:15) ( x (cid:15) ,v (cid:48) ,t (cid:15) ) − J ( x (cid:15) ,v (cid:15) ,t (cid:15) )) /(cid:15) dv (cid:48) . Finally, note that ρ (cid:15) ( x (cid:15) , t (cid:15) ) = e J(cid:15) ( x(cid:15),v(cid:15),t(cid:15) ) (cid:15) S n − e J(cid:15) ( x(cid:15),v (cid:48) ,t(cid:15) ) − J(cid:15) ( x(cid:15),v(cid:15),t(cid:15) ) (cid:15) dv (cid:48) . Combining the last two observations and the formula above we conclude thatlim (cid:15) → ρ (cid:15) ( x (cid:15) , t (cid:15) ) = 0 . (3.24)Returning to (3.23), we obtain φ (cid:15)t ( x (cid:15) , v (cid:15) , t (cid:15) )+ a n,τ v (cid:15) · Dφ (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ) + 1 τ ≥ ττ S n − e J(cid:15) ( x(cid:15),v (cid:48) ,t(cid:15) ) − J(cid:15) ( x(cid:15),v(cid:15),t(cid:15) ) (cid:15) dv (cid:48) + o(1) . Using now that ( x (cid:15) , v (cid:15) , t (cid:15) ) is a minimum point of J (cid:15)A − φ (cid:15) , we get φ (cid:15)t ( x (cid:15) , v (cid:15) , t (cid:15) ) + a n,τ v (cid:15) · Dφ (cid:15) ( x (cid:15) , v (cid:15) , t (cid:15) ) + 1 τ ≥ ττ S n − e η ( v (cid:48) ) − η ( v (cid:15) ) dv (cid:48) + o(1) . (3.25)The definition of η in (3.6) then yields φ (cid:15)t ( x (cid:15) , t (cid:15) ) + a n,τ v (cid:15) · Dφ (cid:15) ( x (cid:15) , t (cid:15) ) ≥ − H ( Dφ ( x , t )) + a n,τ v (cid:15) · Dφ ( x , t ) + o(1) , which, after letting (cid:15) →
0, gives the desired inequality.
Case two:
If Φ(1) ≤ τ τ a n,τ | p | , the argument needs to be modified. We recall that this may only occur when n ≥ µ ∈ (0 , µ c ), with µ c := ( n − /
2, and for δ >
0, the “approximate” corrector η µ,δ ( v ) = µ log (cid:18)
11 + δ + v · ˆ p (cid:19) . (3.26)16he perturbed test function is now φ µ,δ,(cid:15) ( x, v, t ) := φ ( x, t ) + (cid:15)η µ,δ ( v ). As in the previous case,let ( x µ,δ,(cid:15) , t µ,δ,(cid:15) ) be local minima of min v J (cid:15)A − φ µ,δ,(cid:15) . Then there exists v µ,δ,(cid:15) ∈ S n − such that( x µ,δ,(cid:15) , v µ,δ,(cid:15) , t µ,δ,(cid:15) ) is a minimum point of J (cid:15)A − φ µ,δ,(cid:15) . Further, along a subsequence, which wedenote the same way, (cid:15) →
0, there exists v µ,δ, ∈ S n − such that ( x µ,δ,(cid:15) , v µ,δ,(cid:15) , t µ,δ,(cid:15) ) convergesto ( x , v µ,δ, , t ) and J (cid:15)A ( x µ,δ,(cid:15) , v µ,δ,(cid:15) , t µ,δ,(cid:15) ) → J A ( x , t ) . That ( x δ,(cid:15) , t δ,(cid:15) ) → ( x , t ) as (cid:15) → x , t ) is a strict minimum of J A − φ. The fact that (3.24) holds is proved as before. Moreover, as in (3.25), we find φ µ,δ,(cid:15)t ( x µ,δ,(cid:15) , v µ,δ,(cid:15) , t µ,δ,(cid:15) ) + a n,τ v µ,δ,(cid:15) · Dφ µ,δ,(cid:15) ( x µ,δ,(cid:15) , v µ,δ,(cid:15) , t µ,δ,(cid:15) ) + 1 τ + o(1) ≥ ττ S n − e η µ,δ ( v (cid:48) ) − η µ,δ ( v µ,δ,(cid:15) ) dv (cid:48) . (3.27)Letting (cid:15) → φ t ( x , t ) + a n,τ v µ,δ, · Dφ ( x , t ) + 1 τ ≥ ττ S n − e η µ,δ ( v (cid:48) ) − η µ,δ ( v µ,δ, ) dv (cid:48) . (3.28)Using now the definition of η µ,δ , we rewrite (3.28) as φ t ( x , t ) + a n,τ v µ,δ, · Dφ ( x , t ) + 1 τ ≥ ττ (1 + δ + v µ,δ, · ˆ p ) µ S n − dv (cid:48) (1 + δ + v (cid:48) · ˆ p ) µ . (3.29)Along a subsequence, which we denote in the same way, δ →
0, it follows that v µ,δ, → v µ, , forsome v µ, , ∈ S n − , and φ t ( x , t ) + a n,τ v µ, , Dφ ( x , t ) + 1 τ ≥ ττ (1 + v µ, , · ˆ p ) µ S n − dv (cid:48) (1 + v (cid:48) · ˆ p ) µ dv (cid:48) . (3.30)It shown in the Appendix that lim µ → µ c S n − dv (cid:48) (1 + v (cid:48) · ˆ p ) µ dv (cid:48) = ∞ , while the left hand side of (3.30) is bounded independently of µ . It follows that v µ, , → − ˆ p as µ → µ c .Hence, φ t ( x , t ) − a n,τ | Dφ ( x , t ) | + 1 τ ≥ lim inf µ → µ c ττ (1 + v µ, , · ˆ p ) µ S n − dv (cid:48) (1 + v (cid:48) · ˆ p ) µ ≥ , and the proof is now complete in the case that t = 0. (3.13) at t = 0 : The case t = 0 is treated very similarly to the work above; seethe proof of the sub-solution property. As such, we omit it. The proof is now complete.From the above, we now easily show that J (cid:15) converges uniformly to the solution of (1.9). In otherwords, we establish Theorem 1.2. Proof of Theorem 1.2.
By construction, J A ≤ J A . Also, since J A is a super-solution to an equationthat J A is a sub-solution to, it follows that J A ≥ J A , by the comparison principle (see Crandall,Lions and Souganidis [13]). Hence, we have that J A = J A . This, in turn, implies that J (cid:15)A convergeslocally uniformly in ( x, t ) and uniformly in v to J τA = J A = J A , which does not depend on v andwhich solves (3.13).On the other hand, it is easily seen that J A = inf {− A, J } and J A = inf {− A, J } . Letting A → ∞ ,we see that J (cid:15) converges locally uniformly to a limit J which solves (1.9).17 The speed of the moving front
Propagation speed when n > Proof of Proposition 1.4.
When n >
3, the fact that H ( p ) / | p | attains its maximum follows fromthe observation that, for large | p | , H ( p ) | p | = 1 τ | p | − a n,τ > − a n,τ = lim | p |→∞ H ( p ) | p | . This last observation also yields that c n,τ < a n,τ . SWhen n = 2 , H is defined by the second alternative in (3.4) and the argument above can not beused. Instead we compute the Hamiltonians in order to conclude that H ( p ) / | p | attains a maximum.We first consider the case n = 2. Using (1.4), (3.4) and the fact that, as shown in the Appendix,for s ≥
1, Φ( s ) = 1 √ s − , (4.1)we find H ( p ) = 1 τ − (cid:115)(cid:18) ττ (cid:19) + 2 τ | p | . (4.2)Next we consider the case n = 3. Using again (1.4), (3.4) and the fact that, as shown in theAppendix, for s >
1, Φ( s ) = 12 log (cid:18) s + 1 s − (cid:19) , (4.3)we obtain H ( p ) = 1 τ − √ | p |√ τ e √ τ τ | p | + 1 e √ τ τ | p | − . (4.4)In either case the claim follows after some straightforward calculations.In view of the discussion in the introduction, to conclude the proof it suffices to show that, for each e ∈ S n − , L ( c n,τ e ) = 0.Let p c ∈ R n \ { } be a maximizing vector in the definition of c n,τ , that is, c n,τ = − H ( p c ) | p c | ;note that, in view of the isotropy of H , any rotation of p c is also a maximizing vector. Thus wemay assume that e · p c = −| p c | .Since, for q ∈ R n , L ( q ) = inf p ∈ R n ( q · p − H ( p )) , it is immediate that L ( c n,τ e ) ≤ ( − c n,τ | p c | − H ( p c )) = 0 . Rewriting the definition of L , we obtain L ( c n,τ e ) = c n,τ inf p ∈ R n | p | (cid:18) e · ˆ p − c n,τ H ( p ) | p | (cid:19) . (4.5)18f L ( c n,τ e ) <
0, then there exists p ∈ R n \ { } such that L ( c n,τ e ) ≤ c n,τ | p | (cid:18) − − c n,τ H ( p ) | p | (cid:19) < . This implies that − c n,τ < H ( p ) | p | , which contradicts the definition of c n,τ . We conclude that L ( c n,τ e ) = 0. Explicit formulas for the speed for n = 1 and n = 2 . Proof of Proposition 1.5.
We begin with the speed when n = 2 since the proof is simple. UsingProposition 1.4 along with (4.2), find c ,τ = − max p ∈ R H ( p ) | p | = − max p ∈ R τ | p | − (cid:115)(cid:18) ττ | p | (cid:19) + 2 τ . Elementary calculus yields that the maximum is attained when | p | = (1 + τ ) √ τ / √
2, giving c ,τ = (cid:112) τ )1 + τ . We now consider the case n = 1. Let α = − H ( p ) + τ − . It follows from (3.4) and the fact that S = {− , } that, for any p ∈ R \ { } , αa n,τ | p | α a ,τ | p | − τ τ a ,τ | p | , and after rearranging this α = 1 + τ τ + (cid:115)(cid:18) τ τ (cid:19) + 1 τ | p | ;that is H ( p ) = 1 − τ τ − (cid:115)(cid:18) τ τ (cid:19) + 1 τ | p | . (4.6)Using elementary calculus, it is easy to see that c ,τ is given by the formula in Proposition 1.5.The definition of the concave dual and elementary calculus now yields L ( q ) = (cid:40) − − τ τ + τ τ (cid:112) − τ | q | , if τ | q | ≤ , −∞ , otherwise . If τ < L ( q ) = 0 if and only if | q | = 2 / (1 + τ ). In view of (1.13), this implies that c ,τ = 2 / (1 + τ ).If τ ≥
1, there is no q ∈ R such that L ( q ) = 0. In this case, we notice that L ( q ) > τ | q | ≤ L ( q ) < ∂ { q ∈ R : L ( q ) < } = { q ∈ R : | q | = 1 / √ τ } . In view of (1.12), thisimplies that c ,τ = 1 / √ τ , finishing the proof. 19 The limit τ → Let J τ be the solution to (1.9), recall that J τ ≤
0, and define J τA := max { J τ , − A } . Consider the half-relaxed limits z A := lim sup τ → J τA and z A := lim inf τ → J τA , (5.1)and note that, by construction, − A ≤ z A ≤ z A ≤ . (5.2)We prove the following lemma. Lemma 5.1.
Let J τ solve (1.9) and define z A and z A by (5.1) . Then(i) z A is a super-solution to max (cid:8) z A,t − | Dz A | − , z A , − A − z A (cid:9) ≥ in R n × R + z A ( · , ≥ ( A ) (cid:63) on R n . (5.3) (ii) z A is a sub-solution to max (cid:8) z A,t − | Dz A | − , z A , − A − z A (cid:9) ≤ in R n × R + , z A ( · , ≤ ( A ) (cid:63) on R n . (5.4)We momentarily postpone the proof of Lemma 5.1 to note that the comparison principle yields z A = z A . Letting then A → ∞ we obtain Proposition 1.6 from Lemma 5.1. The argument followsexactly as in Lemma 3.1, so we omit it. Proof of Lemma 5.1.
We prove only the claim for z A . The argument is similar for z A with similarmodifications as in the proof of Lemma 3.1.Fix any smooth test function φ and any point ( x , t ) such that z A − ψ has a strict local minimumof zero at ( x , t ).Assume that t >
0. If z A ( x , t ) = 0 or − A the claim is immediate from (5.3). Hence, we needonly consider the case where z A ( x , t ) ∈ ( − A,
0) and aim to show that0 ≤ φ t ( x , t ) − | Dφ ( x , t ) | − . It follows from the definition of z A that, as τ →
0, there exists a sequence ( x τ , t τ ) → ( x , t ) suchthat J τA − φ has a local minimum at ( x τ , t τ ) and J τA ( x τ , t τ ) − φ ( x τ , t τ ) → z A ( x , t ) − φ ( x , t ). Inaddition, since z A ( x , t ) ∈ ( − A, J τA ( x τ , t τ ) ∈ ( − A,
0) for all τ sufficiently small.Using the definition of H τ and that J τ satisfies (1.9), when τ is sufficiently small, we obtain, at( x τ , t τ ), τ τ − S n − dv τ + φ t ( x τ , t τ ) + a n,τ v · Dφ ( x τ , t τ ) ≥ . We point out that we need only use this form of the Hamiltonian H (3.4) since, for any fixed p ,Φ(1) > √ nτ | p | / (1 + τ ) when τ is small enough.Dividing both sides by τ and using Taylor’s theorem on the integrand, we see that, at ( x τ , t τ ),0 ≤
11 + τ − (cid:16) − τ φ t ( x τ , t τ ) − √ τ nv · Dφ ( x τ , t τ ) + nτ ( v · Dφ ( x τ , t τ )) + o( τ ) (cid:17) . S n − as it is odd in v , while, acalculation in the Appendix, yields that S n − n ( v · Dφ ( x τ , t τ )) dv = | Dφ ( x τ , t τ ) | . (5.5)Applying Taylor’s theorem again, it is now immediate that, at ( x τ , t τ ),0 ≤ (1 − τ + o( τ )) − τ φ t ( x τ , t τ ) − τ | Dφ ( x τ , t τ ) | + o ( τ ) , which, after dividing by τ and letting τ →
0, yields0 ≤ φ t ( x , t ) − | Dφ ( x , t ) | − . The case when t = 0 may be easily handled by a combination of the methods above and inSection 2. The reactive-telegraph equation: proof of Theorem 1.1 when n = 1 We prove that the reactive-telegraph equation preserves positivity when n = 1. We note that theupper bound in the statement of Theorem 1.1 is clearly not sharp. Due to the lack of maximumprinciple and the structure of the equation (1.1), the upper and lower bounds must be obtainedsimultaneously. Since our main interest is in the lower bound, we do not optimize the proof of theupper bound and use 2 since it is sufficient to obtain the desired lower bound. Proof of Theorem 1.1 when n = 1 . Before beginning we note that it is enough to prove the claimwith nonlinearity ρ (1 − ρ ), since ρ (1 − ρ ) ≤ ρ (1 − ρ ) + . The key tool is the equivalence between (1.1)and (1.3) in one spatial dimension, which may also be written as (1.2). That is, let p ± solve (1.2)with p + ( · ,
0) = p − ( · ,
0) = ρ , then ρ = ( p + + p − ) /
2. Hence, if we show that 0 ≤ p + , p − ≤
2, theresult follows for ρ .Fix (cid:15) >
0. By approximation, we may assume that p ± are smooth and uniformly equal to 2 √ (cid:15) outside a compact set and that the initial data satisfy 2 √ (cid:15) ≤ ρ ≤
1. We define p + (cid:15) ( x, t ) = p + ( x, t ) + (cid:15)t and p − (cid:15) ( x, t ) = p − ( x, t ) + (cid:15)t, and let T (cid:15) = sup { t ∈ (0 , / √ (cid:15) ) : √ (cid:15) < p + (cid:15) ( x, s ) , p − (cid:15) ( x, s ) < − √ (cid:15) for all s ∈ (0 , t ) } . Here, T (cid:15) is the first time that either p + (cid:15) or p − (cid:15) “touches” √ (cid:15) or 2 − √ (cid:15) . It is well-defined and positivedue to the smoothness of ρ that is inherited by p + (cid:15) and p − (cid:15) .Our goal is to show that T (cid:15) = 1 / √ (cid:15) for every (cid:15) >
0. Once we have shown this, the bounds on p + and p − follow by taking (cid:15) →
0. To this end, we proceed by contradiction and assume that T (cid:15) ∈ (0 , / √ (cid:15) ).Let x (cid:15) ∈ R be such that p + (cid:15) ( x (cid:15) , T (cid:15) ) is either √ (cid:15) or 2 − √ (cid:15) . The argument follows similarly if p − (cid:15) ( x (cid:15) , T (cid:15) ) is either √ (cid:15) or 2 − √ (cid:15) .Before continuing with the proof, we briefly justify the existence of x (cid:15) . Let R (cid:15) > ρ ≡ √ (cid:15) on B cR (cid:15) . Since (1.2) has no space or time dependence, the finite speed of21ropagation for kinetic equations implies that p ± x ≡ B cR (cid:15) +( τ(cid:15) ) − / × [0 , / √ (cid:15) ]. As such, (1.2)reduces to an ordinary differential equation in t at every point x ∈ B cR (cid:15) +( τ(cid:15) ) − / . The ODE is thesame for both p + and p − , and, hence, the p + and p − are equal on B cR (cid:15) +( τ(cid:15) ) − / . This, in turn,implies that p + = p − = ρ on B cR (cid:15) +( τ(cid:15) ) − / , which, from (1.2), implies that ρ t = ρ (1 − ρ ) with ρ ( x ) = 2 √ (cid:15) on B cR (cid:15) +( τ(cid:15) ) − / × [0 , / √ (cid:15) ]. At this point, it is clear that p + (cid:15) and p − (cid:15) can achieve neither the value √ (cid:15) nor the value 2 − √ (cid:15) on B cR (cid:15) +( τ(cid:15) ) − / × [0 , / √ (cid:15) ], so long as (cid:15) is sufficiently small. As such, if T (cid:15) < / √ (cid:15) , the smoothness of p + (cid:15) and p − (cid:15) guarantees the existence of x (cid:15) ∈ B R (cid:15) +( τ(cid:15) ) − / .First we consider the case that p + (cid:15) ( x (cid:15) , T (cid:15) ) = √ (cid:15) . Using that p + (cid:15) is smooth and ( x (cid:15) , T (cid:15) ) is the locationof a minimum of p + (cid:15) on R × (0 , T (cid:15) ], we see that, at ( x (cid:15) , T (cid:15) ),( p + (cid:15) ) t + 1 √ τ ( p + (cid:15) ) x ≤ . From this, along with (1.2) and the definition of p + (cid:15) in terms of p + , we get that, at ( x (cid:15) , T (cid:15) ), (cid:15) + 1 τ (cid:0) ρ − p + (cid:1) + ρ (1 − ρ ) = ( p + (cid:15) ) t + 1 √ τ ( p + (cid:15) ) x ≤ . (6.1)The definition of T (cid:15) yields that √ (cid:15) ≤ p − (cid:15) ( x (cid:15) , T (cid:15) ) ≤ − √ (cid:15) . Using these bounds for p − (cid:15) , the factsthat p + (cid:15) ( x (cid:15) , T (cid:15) ) = √ (cid:15) and (cid:15)T (cid:15) < √ (cid:15) , and the relationship between p ± (cid:15) and p ± , we obtain0 ≤ ρ ( x (cid:15) , T (cid:15) ) ≤ . Also, since p + (cid:15) ( x (cid:15) , T (cid:15) ) ≤ p − (cid:15) ( x (cid:15) , T (cid:15) ), it follows that p + ( x (cid:15) , T (cid:15) ) ≤ ρ ( x (cid:15) , T (cid:15) ). Hence, at ( x (cid:15) , T (cid:15) ),1 τ (cid:0) ρ − p + (cid:1) + ρ (1 − ρ ) ≥ . The combination of this inequality with the positivity of (cid:15) contradicts (6.1), finishing this case.Now we consider the case that p + (cid:15) ( x (cid:15) , T (cid:15) ) = 2 − √ (cid:15) . Reasoning as above, we see that, at ( x (cid:15) , T (cid:15) ),0 ≤ ( p + (cid:15) ) t + 1 √ τ ( p + (cid:15) ) x = (cid:15) + 1 τ (cid:0) ρ − p + (cid:1) + ρ (1 − ρ ) . Moving the (cid:15) and p + terms from the right hand side to the left hand side and using that p + = p + (cid:15) − (cid:15)T (cid:15) and T (cid:15) < / √ (cid:15) , we obtain2 τ − √ (cid:15) (2 − τ √ (cid:15) ) τ = 1 τ (cid:0) − √ (cid:15) (cid:1) − (cid:15) ≤ τ (cid:0) p + (cid:15) − (cid:15)T (cid:15) (cid:1) − (cid:15) ≤ τ ρ + ρ (1 − ρ ) . (6.2)On the other hand the definition of T (cid:15) implies that √ (cid:15) ≤ p − (cid:15) ( x (cid:15) , T (cid:15) ) ≤ − √ (cid:15) . From this, we seethat1 − √ (cid:15) ≤ ( p + (cid:15) ( x (cid:15) , T (cid:15) ) − (cid:15)T (cid:15) ) + ( p − (cid:15) ( x (cid:15) , T (cid:15) ) − T (cid:15) )2 = ρ ( x (cid:15) , T (cid:15) ) < p + (cid:15) ( x (cid:15) , T (cid:15) ) + p − (cid:15) ( x (cid:15) , T (cid:15) )2 ≤ − √ (cid:15), or, more succinctly, 1 − √ (cid:15) ≤ ρ ( x (cid:15) , T (cid:15) ) < − √ (cid:15). (6.3)22hen (cid:15) is sufficiently small, depending only on τ , the two inequalities (6.2) and (6.3) are incom-patible. Briefly, this may be seen by considering two cases. First, if ρ ∈ [1 − (cid:15), / τ − √ (cid:15) (2 − τ √ (cid:15) ) τ ≤ τ ρ + ρ (1 − ρ ) ≤ τ + 32 √ (cid:15). from (6.2). This is clearly a contradiction when (cid:15) is sufficiently small. Second, if ρ ∈ [3 / , − √ (cid:15) ],we find from (6.2) 2 τ − √ (cid:15) (2 − τ √ (cid:15) ) τ ≤ τ ρ + ρ (1 − ρ ) ≤ − √ (cid:15)τ + 32 (cid:18) − (cid:19) , which is also a contradiction for (cid:15) is sufficiently small.This finishes the consideration of the second case and thus finishes the proof. The reactive-telegraph equation: proof of Theorem 1.1 when n ≥ We need the following estimate:
Theorem 6.1 (Chapter 1, Theorem 4.1 [26]) . Let F wave : R → R be a smooth function such that F wave (0 ,
0) = 0 . Assume that s ≥ n + 2 , f ∈ H s +1 ( R n ) , and g ∈ H s ( R n ) . Then there exists T > ,depending on F wave , (cid:107) f (cid:107) H s +1 , and (cid:107) g (cid:107) H s , such that the initial value problem (cid:40) u tt = ∆ u + F wave ( u, u t ) in R n × (0 , T ) ,u = f, u t = g on R n × { } , has a unique solution u satisfying, for some universal constant C , (cid:107) u (cid:107) H s +1 ( R n × [0 ,T ]) ≤ C (cid:0) (cid:107) f (cid:107) H s +1 ( R n ) + (cid:107) g (cid:107) H s ( R n ) (cid:1) . With this in hand, we now show that the reactive-telegraph equation (1.1) does not preservepositivity.
Proof of Theorem 1.1 when n ≥ . It is enough to work with n = 2, since any example in thissetting also works for n > ρ ( x ) = (cid:15)e −| x | /δ for (cid:15), δ ∈ (0 ,
1) so that (cid:15) ≤ δ / and let ρ be the solution to (cid:40) τ ρ tt + (1 − τ + 2 τ ρ ) ρ t = ∆ ρ + ρ (1 − ρ ) in R × (0 , T ) ,ρ ( · ,
0) = ρ and ρ t ( · ,
0) = 0 on R , (6.4)where T > u ( x, t ) = e (1 − τ ) t/ τ ρ ( x, t ) satisfies τ u tt − ∆ u = u (cid:18) − (1 − τ ) τ + 2 τ e − (1 − τ ) t/ τ u t − e − (1 − τ ) t/ τ u (cid:19) . v ( x, t ) = u ( x, √ τ t ), we find v tt − ∆ v = v (cid:16) − (1 − τ ) τ + 2 √ τ e − (1 − τ ) t/ τ v t − e − (1 − τ ) t/ τ v (cid:17) on R × (0 , T ) ,v ( · ,
0) = ρ and v t ( · ,
0) = − τ √ τ ρ on R . Using Theorem 6.1 with s = 7 / s = 5 / T > C τ , which isindependent of T and will change from line to line below, such that (cid:107) v t (cid:107) L ∞ ((0 ,T ) × R ) ≤ C Sob (cid:107) v (cid:107) H / ((0 ,T ) × R ) ≤ C τ (cid:107) ρ (cid:107) H / ( R ) ≤ C τ (cid:15)δ − / (6.5)and (cid:107) v (cid:107) L ∞ ((0 ,T ) × R ) ≤ C Sob (cid:107) v (cid:107) H / ((0 ,T ) × R ) ≤ C τ (cid:107) ρ (cid:107) H / ( R )) ≤ C τ (cid:15)δ − / ; (6.6)note that in the second inequality we absorbed C Sob into C τ to simplify the notation.Since (cid:15) ≤ δ / , it follows that, for some C τ depending only on τ , (cid:107) ρ (cid:107) H / ≤ C τ . In view of this and Theorem 6.1, it follows that T does not depend on (cid:15) or δ .Duhamel’s formula for the solution to the wave equation for n = 2 gives v ( x, t ) = 12 B t ( x ) (cid:112) t − | x − y | (cid:20) t − τ √ nτ ρ ( y ) + tρ ( y ) + tDρ ( y ) · ( y − x ) (cid:21) dy + ˆ t s B s ( x ) v (cid:112) s − | x − y | (cid:18) − (1 − τ ) τ + 2 √ τ e − (1 − τ ) t/ τ v t − e − (1 − τ ) t/ τ v (cid:19) dyds. (6.7)At this point we mention that this example does not work in one dimension because the generalform of the solution is different in one dimension. In particular, the crucial term Dρ does notappear.Inserting the bounds (6.5) and (6.6) in (6.7), we find v (0 , t ) ≤ (cid:15) (cid:18) e − t /δ (cid:20) t | − τ | √ τ + 1 − t δ (cid:21) + C τ δ − / t (cid:104) (cid:15)δ − / (cid:105)(cid:19) , which, for δ = t , becomes v (0 , t ) ≤ (cid:15) (cid:18) e − t | − τ | √ τ − e − + C τ t / + (cid:15)C τ t − (cid:19) . The claim follows by choosing first t and then (cid:15) small to obtain v (0 , t ) <
0, and, hence, that ρ (0 , t/ √ τ ) < A kinetic model that does not preserve positivity
In the interest of simplicity, we study here a slightly different model than (1.3) and somewhatsingular initial data. Generalizations to a larger class of equations and initial data are conceptuallystraightforward, though with significantly more involved computations.24e consider solutions to a two-dimensional discrete version of (1.3) given by (cid:40) p t + a ,τ v · Dp = τ ( ρ − p ) + p (1 − p ) in R × S d × R + ,p ( · ,
0) = p on R × S d , (6.8)where S d := { e , e , − e , − e } with e , e the standard basis vectors of R and ρ ( x, t ) := 14 (cid:88) v ∈ S d p ( x, v, t ) . We have:
Proposition 6.2.
Fix τ > . There exist a bounded p with (cid:16)(cid:80) v ∈ S d p (cid:17) / ≤ and ( x , v , t ) ∈ R × S d × R + such that, if p solves (6.8) with initial datum p , then p ( x , v , t ) < . The idea is to choose p having three patches, one moving in the − e direction, one moving in the e direction, and another moving in the e direction, which will eventually collide. Their sum willthen be large enough to force the reaction term to be negative, which will cause the populationdensity of the species moving in the e direction to become negative. Proof of Proposition 6.2.
To choose p , we introduce the (moving) sets X ,t = { ( x , x ) ∈ R : | x | < − x + a ,τ t } , X ,t = { ( x , x ) ∈ R : | x | < x + a ,τ t } , and X ,t = { ( x , x ) ∈ R : | x | < x + a ,τ t } , which, when t = 0, are disjoint cones with a vertex at the origin in the directions − e , e , and e ,respectively, while, for t >
0, they move in time in the directions e , − e , and − e , respectively,with speed a ,τ .Let p be so that p ( x, e ) ≡ , p ( x, e ) = 4 (cid:0) − τ (cid:1) X , ( x ) ,p ( x, − e ) = 4 (cid:0) − τ (cid:1) X , ( x ) , and p ( x, − e ) = 4 (cid:0) − τ (cid:1) X , ( x ) . The constant 4(1 − /τ ) is chosen so that p is in a quasi-equilibrium, that is, for any fixed e ∈ S d ,the right hand side of (6.8) is zero at t = 0.A simple computation shows that, in view of the choice of the sets X i, ,0 ≤ ρ ( · , ≤ R . To simplify the notation, we introduce p ( x, t ) := e − t/τ p ( x + e a ,τ t, e , t ) , which satisfies p t = 1 τ ρ + ρ (1 − ρ ) in R × S d × R + , (6.9)with ρ ( x, t ) = ρ ( x + e a ,τ t, t ). For future reference note that the right hand side of (6.9) isdecreasing in ρ when τ > ρ ≥ ( τ + 1) / τ . 25ix (cid:15) >
0. It is easy to see, from equation (6.8), that there exists T (cid:15) , depending only on (cid:15) , suchthat, for t ∈ (0 , T (cid:15) ),4 (cid:18) − τ (cid:19) X ,t ( x ) ≥ p ( x, e , t ) ≥ − (cid:15) ) (cid:18) − τ (cid:19) X ,t ( x ) , (cid:18) − τ (cid:19) X ,t ( x ) ≥ p ( x, − e ) ≥ − (cid:15) ) (cid:18) − τ (cid:19) X ,t ( x ) , and4 (cid:18) − τ (cid:19) X ,t ( x ) ≥ p ( x, − e , t ) ≥ − (cid:15) ) (cid:18) − τ (cid:19) X ,t ( x ) . (6.10)Fix δ ∈ (0 , T (cid:15) a ,τ ). We point out that the domains of the indicator functions in (6.10) are initiallydisjoint but contain (0 , − δ ) when t > δ/ a ,τ . Then (6.10) yields, for all t ∈ ( δ/ a ,τ , T (cid:15) ), ρ ((0 , − δ ) , t ) ≥ − (cid:15) ) (cid:18) − τ (cid:19) , (6.11)and, hence, for (cid:15) small enough and t > δ/ a ,τ and since τ > ρ ((0 , − δ ) , t ) ≥ ( τ + 1) / τ. (6.12)Using (6.12), the lower bound in (6.11), and the fact that τ > (cid:15) in(6.9), we obtain, for all t ∈ ( δ/ a ,τ , T (cid:15) ), p t ((0 , − δ ) , t ) = 1 τ ρ ((0 , − δ ) , t ) + ρ ((0 , − δ ) , t )(1 − ρ ((0 , − δ ) , t )) ≤ − (cid:15) ) τ (cid:18) − τ (cid:19) + 3(1 − (cid:15) ) (cid:18) − τ (cid:19) (cid:18) − − (cid:15) ) (cid:18) − τ (cid:19)(cid:19) = 3(1 − (cid:15) ) (cid:18) − τ (cid:19) (cid:18) τ − − (cid:15) ) (cid:18) − τ (cid:19)(cid:19) < . (6.13)By a similar, though simpler, computation, p t ((0 , − δ ) , t ) = 0 when t ∈ [0 , δ/ a ,τ ]. Usingthis, (6.13), and that p ((0 , − δ ) ,
0) = 0, we see that, for all t ∈ ( δ/ a ,τ , T (cid:15) ), p (( a ,τ t − δ ) e , e , t ) = p ((0 , − δ ) , t ) < , and the proof is now complete. A Appendix: The integrals (3.3) and (5.5)
We compute the three simple integrals used above.
The integral (3.3)
When n = 1, this computation is simple, since it reduces to a sum. On the other hand, when n >
3, the expression becomes more complicated. As such, we omit these cases. Here, for s > w ∈ S n − , and n = 2 ,
3, we compute the integralΦ( s ) = S n − dvs + v · w . (A.1)26hen n = 2, changing variables we may assume that w = ( | w | , s ) = S dvs + v · w = 1 π ˆ π dθs + cos( θ ) . The substitution r = tan( θ/
2) then impliesΦ( s ) = 2 π ˆ ∞ dr r ) + (1 − r ) = 2 π ˆ ∞ drs + 1 + ( s − r = 1 √ s − . When n = 3, again we change variables so that v · w = v | w | . Then, using spherical coordi-nates, (A.1) becomes Φ( s ) = S dva + v · w = 12 ˆ π sin( θ ) dθa + | w | cos( θ ) . The substitution r = tan( θ/
2) yieldsΦ( s ) = 2 ˆ ∞ r (2 + 1) + 2 sr + ( s − r dr = 2 s − ˆ ∞ r ( r + 1) (cid:16) r + s +1 s − (cid:17) dr = ˆ ∞ (cid:34) rr + 1 − rr + s +1 s − (cid:35) dr = 12 log (cid:18) s + 1 s − (cid:19) . The convergence of the integral in (3.30) to infinity
We prove that the integral term in (3.30) tends to infinity as µ → µ − c where µ c := ( n − / ω n depending only on n , S n − dv (1 + v ) µ = 1 ω n ˆ S n − sin( θ ) n − (1 + cos( θ )) µ dθ, and the change of variables r = tan( θ/
2) yields S n − dv (1 + v ) µ = 1 ω n ˆ ∞ r ) n − (1+ r ) n − (cid:16) − r r (cid:17) µ dr r = 2 n − − µ ω n ˆ ∞ r n − (1 + r ) n − − µ dr. (A.2)This integral is finite only when µ < ( n − /
2. When µ = 1, the integral is finite if and only if n >
3. Hence, Φ(1) < ∞ if and only if n > The integral (5.5)
We show here that, for any vector w ∈ S n − , S n − n ( v · w ) dv = 1 . The general result follows by scaling. Moreover, we consider only the case n >
2, since, for n = 1 , v · w = v gives, S n − ( v · w ) dv = S n − v dv . Using polar coordinates, we obtain S n − ( v · w ) dv = 1 ´ π sin( θ ) n − dθ ˆ π sin( θ ) n − cos( θ ) dθ = 1 ´ π sin( θ ) n − dθ ˆ π (sin( θ ) n − − sin( θ ) n ) dθ = 1 − ´ π sin( θ ) n dθ ´ π sin( θ ) n − dθ . Denoting I n = ´ π sin( θ ) n dθ , which is sometimes referred to as the Wallis integral, we integrate byparts to obtain the recurrence relation I n = I n − − ˆ π sin( θ ) n − cos( θ ) dθ = I n − − n − I n , which implies that I n /I n − = n − n , and, hence, S n − ( v · w ) dv = 1 − I n I n − = 1 n . References [1] G. Barles. Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equa-tions and applications.
J. Differential Equations , 154(1):191–224, 1999.[2] G. Barles, L. C. Evans, and P. E. Souganidis. Wavefront propagation for reaction-diffusionsystems of PDE.
Duke Math. J. , 61(3):835–858, 1990.[3] G. Barles and B. Perthame. Exit time problems in optimal control and vanishing viscositymethod.
SIAM J. Control Optim. , 26(5):1133–1148, 1988.[4] G. Barles and P. E. Souganidis. A remark on the asymptotic behavior of the solution of theKPP equation.
C. R. Acad. Sci. Paris S´er. I Math. , 319(7):679–684, 1994.[5] E. Bouin and N. Caillerie. Spreading in kinetic reaction-transport equations in higher velocitydimensions.
Preprint , 2017. https://arxiv.org/abs/1705.02191.[6] E. Bouin and V. Calvez. A kinetic eikonal equation.
C. R. Math. Acad. Sci. Paris , 350(5-6):243–248, 2012.[7] E. Bouin, V. Calvez, E. Grenier, and G. Nadin. Large deviations for velocity-jump processesand non-local Hamilton-Jacobi equations.
Preprint , 2017. https://arxiv.org/abs/1607.03676.[8] E. Bouin, V. Calvez, and G. Nadin. Hyperbolic traveling waves driven by growth.
Math.Models Methods Appl. Sci. , 24(6):1165–1195, 2014.[9] E. Bouin, V. Calvez, and G. Nadin. Propagation in a kinetic reaction-transport equation:travelling waves and accelerating fronts.
Arch. Ration. Mech. Anal. , 217(2):571–617, 2015.2810] M. Bramson. Convergence of solutions of the Kolmogorov equation to travelling waves.
Mem.Amer. Math. Soc. , 44(285):iv+190, 1983.[11] X. Cabr´e and J.-M. Roquejoffre. The influence of fractional diffusion in Fisher-KPP equations.
Comm. Math. Phys. , 320(3):679–722, 2013.[12] N. Caillerie. Large deviations of a velocity jump process with a Hamilton-Jacobi approach.
C.R. Math. Acad. Sci. Paris , 355(2):170–175, 2017.[13] M. G. Crandall, P.-L. Lions, and P. E. Souganidis. Maximal solutions and universal bounds forsome partial differential equations of evolution.
Arch. Rational Mech. Anal. , 105(2):163–190,1989.[14] C. M. Cuesta, S. Hittmeir, and C. Schmeiser. Traveling waves of a kinetic transport model forthe KPP-Fisher equation.
SIAM J. Math. Anal. , 44(6):4128–4146, 2012.[15] L. C. Evans. The perturbed test function method for viscosity solutions of nonlinear PDE.
Proc. Roy. Soc. Edinburgh Sect. A , 111(3-4):359–375, 1989.[16] L. C. Evans and P. E. Souganidis. A PDE approach to geometric optics for certain semilinearparabolic equations.
Indiana Univ. Math. J. , 38(1):141–172, 1989.[17] S. Fedotov. Wave front for a reaction-diffusion system and relativistic Hamilton-Jacobi dy-namics.
Phys. Rev. E (3) , 59(5, part A):5040–5044, 1999.[18] T. Hillen and H. G. Othmer. The diffusion limit of transport equations derived from velocity-jump processes.
SIAM J. Appl. Math. , 61(3):751–775, 2000.[19] E. E. Holmes. Are diffusion models too simple? a comparison with telegraph models ofinvasion.
The American Naturalist , 142(5):779–795, 1993.[20] W. Horsthemke. Spatial instabilities in reaction random walks with direction-independentkinetics.
Physical Review E , 60(3):2651, 1999.[21] M. Kac. A stochastic model related to the telegrapher’s equation.
Rocky Mountain J. Math. ,4:497–509, 1974. Reprinting of an article published in 1956, Papers arising from a Conferenceon Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972).[22] P.-L. Lions.
Generalized solutions of Hamilton-Jacobi equations , volume 69 of
Research Notesin Mathematics . Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982.[23] A. J. Majda and P. E. Souganidis. Large-scale front dynamics for turbulent reaction-diffusionequations with separated velocity scales.
Nonlinearity , 7(1):1–30, 1994.[24] V. M´endez, J. Fort, and J. Farjas. Speed of wave-front solutions to hyperbolic reaction-diffusionequations.
Physical Review E , 60(5):5231, 1999.[25] V. Ortega-Cejas, J. Fort, and V. M´endez. The role of the delay time in the modeling ofbiological range expansions.
Ecology , 85(1):258–264, 2004.[26] C. D. Sogge.
Lectures on nonlinear wave equations . Monographs in Analysis, II. InternationalPress, Boston, MA, 1995. 2927] A. Zlatoˇs. Sharp transition between extinction and propagation of reaction.