Lipschitz property of bistable or combustion fronts and its applications
aa r X i v : . [ m a t h . A P ] O c t LIPSCHITZ PROPERTY OF BISTABLE OR COMBUSTIONFRONTS AND ITS APPLICATIONS
KELEI WANG † Abstract.
For a class of reaction-diffusion equations describing propagation phe-nomena, we prove that for any entire solution u , the level set { u = λ } is a Lipschitzgraph in the time direction if λ is close to 1. Under a further assumption that u connects 0 and 1, it is shown that all level sets are Lipschitz graphs. By a blowingdown analysis, the large scale motion law for these level sets and a characterizationof the minimal speed for travelling waves are also given. Introduction
Lipschitz property for level sets.
Consider a smooth, entire solution u tothe reaction-diffusion equation ∂ t u − ∆ u = f ( u ) , < u < R n × R . (1.1)In this paper we are mainly interested in the Lipschitz property of the level sets { u = λ } and their geometric motion law at large scales.Our main hypothesis on f are (F1): f ∈ Lip([0 , f (0) = f (1) = 0 and f ∈ C ,α ([0 , γ ) ∪ (1 − γ, α, γ ∈ (0 , (F2): f ′ (1) < (F3): R f ( u ) du > (F4): there exists a θ ∈ (0 ,
1) such that f > θ, f < , θ ) with f ′ (0) <
0, or f ≡ , θ ).Typical examples are the bistable nonlinearity f ( u ) = u ( u − θ )(1 − u ) with θ < / transition fronts , was introduced by Berestycki and Hamel in [6, 7]. Mathematics Subject Classification.
Key words and phrases.
Reaction diffusion equation; front motion; blowing down analysis;Hamilton-Jacobi equations.
The geometry of an entire solution is complicated in general. To study the Lips-chitz property of { u = λ } , we introduce some assumptions on the entire solution u .The first one is (H1): For any t ∈ R , sup x ∈ R n u ( x, t ) = 1.Under this assumption we prove Theorem 1.1 (Half Lipschitz property for entire solutions) . Suppose f satisfies (F1-F3) . There exists a b ∈ (0 , such that, if an entire solution u satisfies (H1) ,then for any λ ∈ [1 − b , , { u = λ } = { t = h λ ( x ) } is a globally Lipschitz graph on R n . In general, if λ is close to 0, { u = λ } does not satisfy this Lipschitz property,see the example given after Theorem 1.3. In order to establish the full Lipschitzproperty, we need more assumptions. A natural one is (H2): u → x, t ) , { u ≥ − b } ) → + ∞ .Here dist denotes the standard Euclidean distance on R n × R . Theorem 1.2 (Full Lipschitz property for entire solutions) . Suppose f satisfies (F1-F4) , and u is an entire solution satisfying (H1-H2) . Then for any λ ∈ (0 , , { u = λ } = { t = h λ ( x ) } is a globally Lipschitz graph on R n . The proof of Theorem 1.1 relies on the propagation phenomena (see Aronson andWeinberger [1]) in (1.1). Roughly speaking, by (F3) , 1 represents a more stablestate than 0, so { u ≈ } will invade { u ≈ } . This gives us a cone of monotonicityat large scales, see Lemma 2.6 for the precise statement. Although this only impliesa Lipschitz property for { u = λ } at large scales, it can be propagated to a realLipschitz property by utilizing some estimates on positive solutions to the linearparabolic equation ∂ t w − ∆ w = f ′ (1) w. Here the nondegeneracy condition f ′ (1) < { u = λ } for λ close to 1, under thehypothesis (H2) , we can apply the maximum principle and sliding method (in thetime direction, cf. Guo and Hamel [14]) to extend this Lipschitz property backwardlyin time, which is Theorem 1.2.1.2. Travelling wave solutions.
The solution u is a travelling wave in the direc-tion − e n and with speed κ >
0, if there exists a function v ∈ C ( R n ) such that u ( x, t ) = v ( x + κte n ) . Here v satisfies the elliptic equation − ∆ v + κ∂ n v = f ( v ) in R n . (1.2)Among the class of travelling wave solutions, the one dimensional travelling waveis of particular importance. By [1, Theorem 4.1], under the hypothesis (F1-F4) ,there exists a unique constant κ ∗ > − g ′′ ( t ) + κ ∗ g ′ ( t ) = f ( g ( t )) , g ( −∞ ) = 0 , g (+ ∞ ) = 1 . (1.3) ISTABLE OR COMBUSTION FRONTS 3
Theorem 1.1 applied to v gives Theorem 1.3 (Half Lipschitz property for travelling waves) . Suppose f satisfies (F1-F3) and v is an entire solution of (1.2) , satisfying sup R n v = 1 . For any λ ∈ [1 − b , , { v = λ } = { x n = h λ ( x ′ ) , x ′ ∈ R n − } is a globally Lipschitz graph on R n − . As in the entire solution case, in general, this property does not hold for levelsets { v = λ } with λ close to 0. For example, in Hamel and Roquejoffre [21], it isshown that when n = 2, there exist solutions v of (1.1), which is monotone in x and satisfies (cid:26) v ( x , x ) → x → + ∞ .v ( x , x ) → ϕ ( x ) locally uniformly as x → −∞ , where ϕ is an L -periodic solution (for some L >
0) of − ϕ ′′ = f ( ϕ ) in R . Hence when λ is close to 0, { v = λ } is the graph of an L -periodic function h λ ,satisfying h λ ( kL ) = −∞ for any k ∈ Z . Clearly it cannot be a globally Lipschitzgraph.As in the entire solution case, in order to get the Lipschitz property for all levelsets, we need more assumptions. A natural one is Theorem 1.4 (Full Lipschitz property for travelling waves) . Suppose f satisfies (F1-F4) , v is an entire solution of (1.2) , satisfying sup R n v = 1 and v ( x ) → uniformly as dist ( x, { v ≥ − b } ) → + ∞ . (1.4) Then for any λ ∈ (0 , , { v = λ } = { x n = h λ ( x ′ ) } is a globally Lipschitz graph on R n − . This theorem also holds for the monostable case, that is, instead of (F4) , weassume (F4 ′ ): f > , f ′ (0) > f to be concave. However, we do not know howto prove the parabolic case, see discussions in Subsection 3.2.Existence, qualitative properties and classification of solutions to (1.2) with Lip-schitz level sets have been studied by many people, see [16, 17, 18, 27, 28, 33, 34,23, 35, 36].1.3. Blowing down limits.
Once we know level sets of u are Lipschitz graphs,we would like to study their large scale structures. Take a b ∈ (0 ,
1) such that { u = b } = { t = h ( x ) } is a globally Lipschitz graph on R n . For any λ >
0, let h λ ( x ) := 1 λ h ( λx ) . K. WANG
They are uniformly Lipschitz. Therefore for any λ i → ∞ , there exist a subsequence(not relabelling) such that h λ i converges to h ∞ in C loc ( R n ). (This limit may dependon the choice of subsequences.)We have the following characterization of h ∞ . Theorem 1.5.
Under the assumptions of Theorem 1.2, the blowing down limit h ∞ is a viscosity solution of |∇ h ∞ | − κ − ∗ = 0 in R n . (1.5) Remark 1.6 (Level set formulation) . Equation (1.5) is the level set formulation ofthe geometric motion equation for the family of hypersurfaces Σ( t ) := { x : h ∞ ( x ) = t } , V Σ( t ) = κ ∗ ν Σ( t ) . (1.6) Here ν Σ( t ) = −∇ h ∞ / |∇ h ∞ | is the unit normal vector of Σ( t ) . See Fife [13, Chapter1] for a formal derivation of this equation.The equation (1.5) also corresponds to the fact that the global mean speed of tran-sition fronts equals κ ∗ , see Hamel [15] . Remark 1.7.
Because h ∞ (0) = 0 , the following representation formula holds for h ∞ (see for example Monneau, Roquejoffre and Roussier-Michon [26, Section 2] ):there exists a closed set Ξ ⊂ S n − such that h ∞ ( x ) = inf ξ ∈ Ξ ξ · x. As a consequence, h ∞ is concave and -homogeneous. The connection between reaction-diffusion equations and motion by mean curva-tures in the framework of viscosity solutions has been explored by many people in1980s and 1990s. In particular, the asymptotic behavior of solutions to the Cauchyproblem for (1.1) has been studied by Barles, Bronsard, Evans, Soner and Souganidisin [11, 3, 2, 5], in the framework of Hamilton-Jacobi equation and level set motions.We use the same idea, but now for the study of entire solutions of (1.1) (in the spiritof [19, 20]), where we are free to perform scalings to study the large scale structureof entire solutions.From this blowing down analysis we also get a characterization of the minimalspeed.
Theorem 1.8.
Suppose f satisfies (F1-F4) , v is an entire solution of (1.2) , satis-fying sup R n v = 1 and (1.4) . Then κ ≥ κ ∗ .Furthermore, if κ = κ ∗ , there exists a constant t ∈ R such that v ( x ) ≡ g ( x n + t ) in R n . Further problems.
To put our results in a wide perspective, here we mentionsome further problems about (1.1) and (1.2). Some of these problems are well knownto experts in this field.
Problem 1.
Extend results in this paper to the monostable case.
Problem 2.
Theorem 1.5 gives only the main order term of the front motionlaw. The next order term has been formally derived in Fife [13]. Using the language
ISTABLE OR COMBUSTION FRONTS 5 of viscosity solutions, the family of hypersurfaces Σ( t ) := { u ( t ) = 1 / } should bean approximate viscosity solution at large scales (in the sense of Savin [31, 32]) ofthe forced mean curvature flow V Σ( t ) = (cid:2) κ ∗ − H Σ( t ) (cid:3) ν Σ( t ) . (1.7)Here H Σ( t ) denotes the mean curvature of Σ( t ). Problem 3.
In [20], Hamel and Nadirashvili proposed a conjecture about theclassification of entire solutions. For travelling wave solutions in the bistable andcombustion case, this conjecture may be broken into two steps:(1) There exists a one to one correspondence between solutions of (1.2) andsolutions of div ∇ h p |∇ h | ! = κ ∗ − κ p |∇ h | . (1.8)This is the travelling wave equation of (1.7), see [26] for a discussion on thisequation.(2) There exists a one to one correspondence between solutions of (1.8) andnonnegative Borel measures on S n − . Problem 4.
In view of the above discussion and Taniguchi’s theorem in [36], anot so ambitious question is if the reverse of Theorem 1.5 is true, that is, given ahomogeneous viscosity solution h ∞ of (1.5), does there exist an entire solution of(1.1) so that its level set { u = 1 / } is asymptotic to { t = h ∞ ( x ) } ?1.5. Notations and organization of the paper.
Throughout the paper we keepthe following conventions. • We use C (large) and c (small) to denote various universal constants, whichcould be different from line to line. • The parabolic boundary of a domain Ω ⊂ R n × R is denoted by ∂ p Ω. • A function u ∈ C , ( R n × R ) if it is C in x -variables and C in t -variable.The remaining part of this paper is organized as follows. In Section 2 we studythe propagation phenomena in (1.1) and use this to prove Theorem 1.1. In Section3 we prove Theorem 1.2 by the sliding method. An elliptic Harnack inequality isestablished in Section 4. In Section 5 we perform the blowing down analysis. InSection 6 we prove Theorem 1.5. In Section 7 we give a representation formula forthe blowing down limits. With these knowledge on blowing down limits, we proveTheorem 1.8 in Section 8 by using the sliding method again.2. Propagation phenomena
Cone of monotonicity at large scales.
Standard parabolic regularity theoryimplies that u , ∇ u , ∇ u and ∂ t u are all bounded in R n × R . By the Lipschitz propertyof u in t , sup x ∈ R n u ( x, t ) is a Lipschitz function of t .We start with the following simple observation, which is related to the hypothesis (H1) . Proposition 2.1.
Either sup x ∈ R n u ( x, t ) ≡ or sup x ∈ R n u ( x, t ) < in ( −∞ , + ∞ ) . K. WANG
Proof.
Denote I := (cid:26) t : sup x ∈ R n u ( x, t ) = 1 (cid:27) . By continuity, I is a closed subset of R .We claim that I is also open. Therefore it is either empty or the entire real line.Indeed, if sup x ∈ R n u ( x, t ) = 1, then there exist a sequence of points x j ∈ R n suchthat u ( x j , t ) →
1. Let u j ( x, t ) := u ( x j + x, t + t ) . By standard parabolic regularity theory and Arzela-Ascolli theorem, u j → u ∞ in C , loc ( R n × R ), where u ∞ is an entire solution of (1.1). Since 0 ≤ u ∞ ≤ u ∞ (0 ,
0) = 1, by (F1) and the strong maximum principle, u ∞ ≡
1. As a conse-quence, for any ε > t ∈ ( − ε, ε ),lim j →∞ u ( x j , t + t ) = 1 . Hence sup x ∈ R n u ( x, t ) = 1 in ( t − ε, t + ε ) and the claim follows. (cid:3) From now on it is always assumed that ( H1 ) holds, i.e. sup x ∈ R n u ( x, t ) ≡ t ∈ R . Lemma 2.2.
For any b > and R > , there exists a constant ε := ε ( b, R ) > suchthat for any ( x, t ) ∈ R n × R , if u ( x, t ) ≥ − ε , then u ≥ − b in B R ( x ) × ( t − R, t + R ) .Proof. This follows from a contradiction argument similar to the proof of Proposition2.1, by applying the strong maximum principle to the limiting solution. (cid:3)
The following result is essentially [1, Lemma 5.1] (see also [30, Lemma 3.5]). Wewill use the notations of forward and backward light cones in space-time: ( C + λ ( x, t ) := { ( y, s ) : s > t, | y − x | < λ ( s − t ) } , C − λ ( x, t ) := { ( y, s ) : s < t, | y − x | < λ ( t − s ) } . Lemma 2.3 (Propagation to state 1) . There exists a constant b ∈ (0 , such thatfor any b ∈ [0 , b ) and δ > , there exists an R := R ( b, δ ) so that the following holds.If w is the solution to the Cauchy problem (cid:26) ∂ t w − ∆ w = f ( w ) in R n × (0 , + ∞ ) ,w (0) = (1 − b ) χ B R , (2.1) where R ≥ R ( b, δ ) , then w ( x, t ) > − b in C + κ ∗ − δ (0 , . By decreasing b further, we may assume f ′ ≤ f ′ (1) / − b , Lemma 2.4.
Given a constant
M > , if w satisfies (cid:26) ∂ t w − ∆ w ≤ − M w in B × ( − , , ≤ w ≤ in B × ( − , , ISTABLE OR COMBUSTION FRONTS 7 then w ≤ CM in B / × ( − / , . This can be proved, for example, by constructing a suitable sup-solution. Thefirst application of this lemma is
Lemma 2.5.
For any entire solution u , if it is not exactly , then inf R n × R u < − b . Proof.
Assume by the contrary, u ≥ − b everywhere. By our choice of b , we get ∂ t (1 − u ) − ∆(1 − u ) ≤ f ′ (1)2 (1 − u ) in R n × R . (2.2)An iteration of Lemma 2.4 gives u ≡ (cid:3) The next lemma is our main technical tool for the proof of Lipschitz property.
Lemma 2.6.
There exist two constants
D > , < b < b so that the followingholds. For any ( x, t ) ∈ { u = 1 − b } , u > − b in C + κ ∗ − δ ( x, t + D ) . Proof.
Take R := R ( b , δ ) according to Lemma 2.3, b := ε ( b , R ) according toLemma 2.2. Then u ( x, t ) = 1 − b implies u ( y, t ) ≥ − b for any y ∈ B R ( x ).Combining Lemma 2.3 and comparison principle, we deduce that u > − b in C + κ ∗ − δ ( x, t ).Now 1 − u satisfies the differential inequality (2.2) in C + κ ∗ − δ ( x, t ). By Lemma 2.4,we find a D >
0, which depends only on b , b and f ′ (1), such that u > − b in C + κ ∗ − δ ( x, t + D ). (cid:3) Three corollaries follow from this lemma. The first two of them are rather directconsequences of this lemma.
Corollary 2.7.
For any ( x, t ) ∈ { u = 1 − b } , u < − b in C − κ ∗ − δ ( x, t − D ) . Corollary 2.8.
For any ( x, t ) ∈ { u = 1 − b } , { u = 1 − b } lies between ∂ C − κ ∗ − δ ( x, t − D ) and ∂ C + κ ∗ − δ ( x, t − D ) . Corollary 2.9. If u is an entire solution of (1.1) satisfying (H1) , then it cannotbe independent of t , unless u ≡ .Proof. Assume by the contrary, ∂ t u ≡ R n × R . By (H1) , there exists a point( x, ∈ { u = 1 − b } . By Lemma 2.6, u > − b in C κ ∗ − δ ( x, D ). Then we get u ≥ − b everywhere. By Lemma 2.5, u ≡ (cid:3) Proposition 2.10.
The level set { u = 1 − b } belongs to the D -neighborhood of aglobally Lipschitz graph { t = h ∗ ( x ) } . K. WANG
Proof.
Let h ∗ ( x ) := inf ( y,s ) ∈{ u =1 − b } (cid:20) s + D + | x − y | κ ∗ − δ (cid:21) . It is a globally Lipschitz function on R n , with its Lipschitz constant at most ( κ ∗ − δ ) − . To check this, we need only to show that h ∗ > −∞ at one point. (This thenimplies that it is finite everywhere.) In fact, take an arbitrary point ( x , t ) ∈ { u =1 − b } . (The existence of such a point is guaranteed by (H1) and Lemma 2.5.) ByCorollary 2.7, we see for any ( y, s ) ∈ { u = 1 − b } , | y − x | > ( κ ∗ − δ ) ( t − D − s ) . In other words, ( x , t ) / ∈ C + κ − δ ( y, s + D ). Then by definition, we get h ∗ ( x ) ≥ t . (cid:3) We modify h ∗ into a smooth function. Take a standard cut-off function η ∈ C ∞ ( R n ), η ≥ R R n η = 1. Define h ∗ ( x ) := Z R n η ( x − y ) [ h ∗ ( y ) + 1] dy. It is directly verified that h ∗ has the same Lipschitz constant with h ∗ . Moreover, bychoosing η suitably, we have h ∗ ≤ h ∗ ≤ h ∗ + 2 . (2.3)Denote Ω ∗ := { ( x, t ) : t > h ∗ ( x ) } . Lemma 2.11.
There exists a universal constant c < such that cb ≤ − u ≤ b on { t = h ∗ ( x ) } . Proof.
The second inequality is a direct consequence of the fact that u > − b inΩ ∗ , thanks to (2.3) and the construction of h ∗ in the proof of Proposition 2.10.The first inequality follows by applying Harnack inequality to the linear parabolicequation ∂ t (1 − u ) − ∆ (1 − u ) = V (1 − u )in the parabolic cylinder B √ D ( x ) × ( t − D, t +9 D ). In the above V := − f ( u ) / (1 − u )is an L ∞ function. (cid:3) Proof of Theorem 1.1.
Before proving Theorem 1.1, first we need to con-struct a comparison function. Consider the problem ( ∂ t w ∗ − ∆ w ∗ = f ′ (1) w ∗ , in Ω ∗ ,w ∗ = 1 on ∂ Ω ∗ . (2.4) Proposition 2.12. (1)
There exists a unique solution of (2.4) in L ∞ (Ω ∗ ) ∩ C ∞ (Ω ∗ ) . (2) There exists a universal constant C such that for any ( x, t ) ∈ Ω ∗ , C e − C [ t − h ∗ ( x )] ≤ w ∗ ( x, t ) ≤ Ce − t − h ∗ ( x ) C . (2.5) ISTABLE OR COMBUSTION FRONTS 9 (3)
There exists a universal constant C such that |∇ w ∗ | w ∗ + | ∂ t w ∗ | w ∗ ≤ C in Ω ∗ . (2.6)(4) There exists a universal constant c such that ∂ t w ∗ w ∗ ≤ − c in Ω ∗ . (2.7)(5) There exists a universal constant C such that |∇ w ∗ || ∂ t w ∗ | ≤ C in Ω ∗ . (2.8) As a consequence, all level sets of w ∗ are Lipschitz graphs in the t -direction.Proof. (i) Existence, uniqueness and regularity of the solution can be proved as inOle˘ınik and Radkeviˇc [29, Chapter 1], see respectively Section 5, Section 6 andSection 8 therein.(ii) The exponential upper bound follows from iteratively applying the estimatein Lemma 2.4. The lower bound follows from iteratively applying the standardHarnack inequality.(iii) By the regularity theory in [29], there exists a universal constant C such that |∇ w ∗ | + | ∂ t w ∗ | ≤ C in Ω ∗ . Hence for any constant vector ( ξ, s ) ∈ R n +1 , ξ · ∇ w ∗ + s∂ t w ∗ is a bounded solutionof (2.4). As in (ii), it converges to 0 as t − h ∗ ( x ) → + ∞ . Then (2.6) follows froman application of the comparison principle.(iv) For any ρ >
0, in the half ball B ρ (0 ,
0) := (cid:8) ( x, t ) : | x | + t < ρ , t < (cid:9) , the function w ρ ( x, t ) := e α ( | x | + t − ρ )is a sup-solution of (2.4), provided that α is small enough (depending only on ρ , thespace dimension n and f ′ (1)).On (cid:8) | x | + t = ρ , t < − ( κ ∗ − δ ) − | x | (cid:9) , there exists a constant c ( ρ ) > ∂ t w ρ ( x, t ) = 2 αtw ρ ( x, t ) ≤ − c ( ρ ) . Since sup R n |∇ h ∗ | ≤ C , for any ( x, h ∗ ( x )), there exists an half ball B /C ( y, s ) tangentto ∂ Ω ∗ at this point. Moreover, because |∇ h ∗ ( x ) | ≤ ( κ ∗ − δ ) − , t − s ≤ − ( κ ∗ − δ ) − | x − y | . By the comparison principle in B /C ( y, s ), w /C ( · − ( y, s )) ≥ w ∗ in B /C ( y, s ). There-fore ∂ t w ∗ ( x, h ∗ ( x )) ≤ − c. Then (2.7) follows from an application of the comparison principle as in (iii).(v) This is a direct consequence of (2.6) and (2.7). (cid:3)
Lemma 2.13.
There exists a universal constant C such that b C ≤ − uw ∗ ≤ Cb in Ω ∗ . (2.9) Proof.
For each k ≥
1, letΩ ∗ k := { ( x, t ) : k − < t − h ∗ ( x ) < k } . Similar to (2.5), for any ( x, t ) ∈ Ω ∗ we have1 C e − C [ t − h ∗ ( x )] ≤ − u ( x, t ) ≤ Ce − t − h ∗ ( x ) C . (2.10)Hence there exists a σ ∈ (0 ,
1) such that σ k f ′ (1) (1 − u ) ≤ [ ∂ t − ∆ − f ′ (1)] (1 − u ) ≤ − σ k f ′ (1) (1 − u ) in Ω ∗ k . For each k , define w ∗ k inductively as the unique solution of ( ∂ t w ∗ k − ∆ w ∗ k = f ′ (1) (cid:0) − σ k (cid:1) w ∗ k in Ω ∗ k ,w ∗ k = w ∗ k − on { t = h ∗ ( x ) + k − } . (2.11)Here w ∗ : ≡ b . As before, the existence and uniqueness of w ∗ k follows from Ole˘ınikand Radkeviˇc [29, Chapter 1].By inductively applying the comparison principle, we get1 − u ≤ w ∗ k in Ω ∗ k . (2.12)Next for each k , denote M k := sup Ω ∗ k w ∗ k w ∗ . A direct calculation shows that ( M k w ∗ ) − σ k is a sup-solution of (2.11) in Ω ∗ k +1 . (Herewe also need an inductive assumption that M k w ∗ < { t = h ∗ ( x ) + k } .) Fromthis we deduce that M k +1 ≤ M − σ k k sup Ω ∗ k +1 ( w ∗ ) − σ k ≤ M − σ k k e Ckσ k , where the last inequality follows by substituting (2.5) to estimate inf Ω ∗ k +1 w ∗ .From this inequality it is readily deduced that there exists a universal, finite upperbound on M k as k → ∞ . Combining this fact with (2.12) we obtain the upper boundin (2.9).The lower bound in (2.9) follows in the same way by considering ( ∂ t w k, ∗ − ∆ w k, ∗ = f ′ (1) (cid:0) σ k (cid:1) w k, ∗ in Ω ∗ k ,w k, ∗ = w k − , ∗ on { t = h ∗ ( x ) + k − } . (cid:3) Corollary 2.14.
There exists a universal constant
C > such that |∇ u | + | ∂ t u | − u ≤ C in Ω ∗ . (2.13) ISTABLE OR COMBUSTION FRONTS 11
Proof.
For any ( x, t ) ∈ Ω ∗ , standard gradient estimate gives |∇ u ( x, t ) | + | ∂ t u ( x, t ) | ≤ C sup B ( x ) × ( t − ,t ) (1 − u ) . (2.14)By the previous lemma, for any ( y, s ) ∈ B ( x ) × ( t − , t ),1 − u ( y, s ) ≤ Cw ∗ ( y, s ) ≤ C w ∗ ( x, t ) ≤ C [1 − u ( x, t )] . (2.15)Here the second inequality follows by integrating (2.6) along the segment from ( y, s )to ( x, t ).Substituting (2.15) into (2.14) we get (2.13). (cid:3) Proposition 2.15.
There exists a b ∈ (0 , b ) such that in { u > − b } , ∂ t u − u ≥ c. (2.16) Proof.
Assume by the contrary, there exists a sequence of points ( x k , t k ) such that u ( x k , t k ) → ∂ t u ( x k , t k )1 − u ( x k , t k ) → . (2.17)Denote R k := dist(( x k , t k ) , ∂ Ω ∗ ). Because u ( x k , t k ) →
1, by (2.10), R k goes toinfinity as k → ∞ .Let u k ( x, t ) := 1 − u ( x k + x, t k + t )1 − u ( x k , t k ) , w k ( x, t ) := w ∗ ( x k + x, t k + t )1 − u ( x k , t k ) . By definition, u k (0 ,
0) = 1, while by (2.9), we have b C ≤ u k w k ≤ Cb in B cR k (0) × ( − cR k , cR k ) . Furthermore, (2.6) and (2.13) are transformed into |∇ u k | + | ∂ t u k | u k ≤ C, |∇ w k | + | ∂ t w k | w k ≤ C in B cR k (0) × ( − cR k , cR k ) . Then by standard parabolic regularity theory, u k and w k are uniformly boundedin C α, α/ loc ( R n × R ). After passing to a subsequence, u k → u ∞ , w k → w ∞ in C , loc ( R n × R ). Both of them are solutions of ∂ t w − ∆ w = f ′ (1) w in R n × R . By [37] or [24], there exists a Borel measure supported on { λ = | ξ | } ⊂ R n +1 suchthat w ∞ ( x, t ) = Z { λ = | ξ | } e [ f ′ (1)+ λ ] t + ξ · x dµ ( ξ, λ ) . Because b C ≤ u ∞ w ∞ ≤ Cb in R n × R , (2.18) there exists a function Θ on { λ = | ξ | } with b /C ≤ Θ ≤ Cb such that u ∞ ( x, t ) = Z { λ = | ξ | } e [ f ′ (1)+ λ ] t + ξ · x Θ( ξ, λ ) dµ ( ξ, λ ) . This follows by writing u ∞ as the same integral representation with another measure e µ , applying Radon-Nikodym theorem to these two measures, and then use (2.18) toestimate the differential d e µdµ .Because w ∞ still satisfies the inequality (2.7), the support of µ is contained in { λ ≤ − f ′ (1) − c } . Hence we also have ∂ t u ∞ = Z { λ = | ξ | } [ f ′ (1) + λ ] e [ f ′ (1)+ λ ] t + ξ · x Θ( ξ, λ ) dµ ( ξ, λ ) ≤ − cu ∞ . This is a contradiction with (2.17). (cid:3)
Theorem 1.1 follows by combining (2.13) and (2.16).3.
Proof of Theorem 1.2
For simplicity, denote h ( x ) := h − b ( x ).3.1. The combustion and bistable case.
In these two cases we need the assump-tion (H2) , that is, u ( x, t ) → x, t ) , { t = h ( x ) } ) → + ∞ .First we use the sliding method to prove Proposition 3.1. u is increasing in t .Proof. The fact that ∂ t u > { u > − b } has been established in Proposition2.15. Now we use the sliding method to show the remaining case.For any λ ∈ R , let u λ ( x, t ) := u ( x, t + λ ) . We want to show that for any λ > u λ ≥ u in R n × R . Step 1. If λ is large enough, u λ ≥ u in R n × R .By (H2) , there exists a constant L > { t
By (3.1) and (F4) , V ≤ { t < h ( x ) − L } . Because (cid:0) u − u λ (cid:1) + = 0 on { t = h ( x ) − L } and (cid:0) u − u λ (cid:1) + → x, t ) , { t = h ( x ) − L } ) → + ∞ , by the maximumprinciple we obtain u ≤ u λ in { t < h ( x ) − L } . Step 2.
Now λ ∗ := inf n λ : ∀ λ ′ > λ, u λ ′ ≥ u in R n × R o is well defined. We claim that λ ∗ = 0.By continuity, u λ ∗ ≥ u in R n × R . By the strong maximum principle, either u λ ∗ > u strictly or u λ ∗ ≡ u . The later is excluded if λ ∗ >
0, because in this case u λ ∗ > u in { t > h ( x ) } . Claim. If λ ∗ >
0, for any
L >
0, there exists a constant ε := ε ( λ ∗ , L ) > u λ ∗ − u ≥ ε in { h ( x ) − L ≤ t ≤ h ( x ) } . We prove this claim by contradiction. Assume there exists a sequence of points( x i , t i ) ∈ { h ( x ) − L ≤ t ≤ h ( x ) } such that u λ ∗ ( x i , t i ) − u ( x i , t i ) →
0. Set u i ( x, t ) := u ( x i + x, t i + t ) . They satisfy the following conditions: • there exists a constant b ( L ) ∈ (0 ,
1) such that b ( L ) ≤ u i (0 , ≤ − b ( L ); • u λ ∗ i ≥ u i in R n × R ; • u λ ∗ i − u i ≥ cλ ∗ (1 − u i ) in { u i ≥ − b } (thanks to Proposition 2.15); • u λ ∗ i (0 , − u i (0 , → i → + ∞ , and the proof of this claim iscomplete.By this claim and Proposition 2.15, for any L >
0, we find another constant ε := ε ( λ ∗ , L ) > λ ≥ λ ∗ − ε , u λ ≥ u in { t ≥ h ( x ) − L } . (3.2)Then as in Step 1, by (3.2) and the comparison principle, for these λ , u λ ≥ u in { t < h ( x ) − L } (hence everywhere in R n × R ). This is a contradiction with thedefinition of λ ∗ . Therefore we must have λ ∗ = 0. (cid:3) Combining this proposition with Corollary 2.9 and strong maximum principle, weobtain
Corollary 3.2. In R n × R , ∂ t u > strictly. Proposition 3.3.
There exists a universal constant c > such that ∂ t u ≥ c |∇ u | in R n × R . Proof. In { u > − b } , this inequality follows by combining Corollary 2.14 andProposition 2.15. In view of Corollary 3.2, following the argument in the second step of the proof ofProposition 3.1, for any
L >
0, we find a positive lower bound for ∂ t u in { h ( x ) − L ≤ t ≤ h ( x ) } . Hence trivially we have |∇ u | ≤ C ≤ Cc ∂ t u in { h ( x ) − L ≤ t ≤ h ( x ) } . Finally, we apply the maximum principle to the linearized equation( ∂ t − ∆) ( ∂ t u − cξ · ∇ u ) = f ′ ( u ) ( ∂ t u − cξ · ∇ u )to show that ∂ t u − cξ · ∇ u ≥ { t < h ( x ) − L } , where ξ is an arbitrary unit vectorin R n and c > (cid:3) Theorem 1.2 is a direct consequence of this proposition.3.2.
A remark on the monostable case.
In this subsection we give a remark onthe monostable case.For this case, we note the following important “hair trigger” phenomena (see [1,Theorem 3.1]).
Lemma 3.4.
For any λ ∈ (0 , , δ > and ( x, t ) ∈ R n × R , there exists a constant D := D ( x, t, λ ) > such that u > λ in C + κ ∗ − δ ( x, t + D ) . Lemma 3.5. If f is monostable, then u → uniformly as dist (( x, t ) , { u ≥ − b } ) → + ∞ .Proof. This follows from the following Liouville type result: suppose u is an entiresolution of (1.1) satisfying 0 ≤ u ≤ − b , then u ≡
0. This Liouville theorem is adirect consequence of Lemma 3.4. (cid:3)
Unfortunately, in this case we need to assume the following assumption: | ∂ t u | + |∇ u | u ≤ C in { t < h ( x ) } . (3.3)Of course, if u is a travelling wave solution, this assumption holds by applyingstandard elliptic Harnack inequality and interior gradient estimates (see Lemma 8.1below), but we do not know how to prove the parabolic case. Lemma 3.6.
Given κ > , assume u is an entire positive solution of ∂ t u − ∆ u = κu. Then |∇ u | ∂ t u ≤ √ κ in R n × R . As a consequence, all level sets of u are Lipschitz graphs in the t direction, with theirLipschitz constants at most √ κ . ISTABLE OR COMBUSTION FRONTS 15
Proof.
By [37] or [24], there exists a Borel measure µ supported on { λ = | ξ | } ⊂ R n +1 such that u ( x, t ) = Z { λ = | ξ | } e [ κ + λ ] t + ξ · x dµ ( ξ, λ ) . Then we have ∂ t u ( x, t ) = Z { λ = | ξ | } [ κ + λ ] e [ κ + λ ] t + ξ · x dµ ( ξ, λ )= Z { λ = | ξ | } (cid:2) κ + | ξ | (cid:3) e [ κ + λ ] t + ξ · x dµ ( ξ, λ ) ≥ √ κ Z { λ = | ξ | } | ξ | e [ κ + λ ] t + ξ · x dµ ( ξ, λ ) ≥ √ κ |∇ u ( x, t ) | . (cid:3) Corollary 3.7.
There exists a constant
L > such that |∇ u | ∂ t u ≤ p f ′ (0) in { t < h ( x ) − L } . Proof.
For any ( x i , t i ) ∈ { t < h ( x ) } with t i − h ( x i ) → −∞ , by Lemma 3.5, u ( x i , t i ) →
0. Set u i ( x, t ) := u ( x i + x, t i + t ) u ( x i , t i ) . By definition, u i > u i (0 ,
0) = 1. Integrating (3.3), we see u i are uniformlybounded in any compact set of R n × R . Then by standard parabolic regularitytheory, we can take a subsequence u i → u ∞ in C , loc ( R n × R ). Here u ∞ is an entiresolution of ∂ t u ∞ − ∆ u ∞ = f ′ (0) u ∞ . The claim then follows from Lemma 3.6. (cid:3)
Theorem 1.2 in the monostable case (under the hypothesis (3.3)) follows from thesame sliding method as in the previous subsection.4.
An elliptic Harnack inequality
From now on, unless otherwise stated, it is always assumed that ( F1 − F4 ) and( H1 − H2 ) hold. In this section we prove an elliptic Harnack inequality for u . Thiswill be used in the blowing down analysis in the next section.In { t > h ( x ) } , what we want has been given in Corollary 2.14, so here we considerthe other part { t < h ( x ) } . Proposition 4.1.
There exists a universal constant
C > such that | ∂ t u | + |∇ u | u ≤ C in { t < h ( x ) } . (4.1)Before proving this proposition, we first notice the following exponential decay of u in { t < h ( x ) } for later use. Proposition 4.2.
Under the hypothesis (H2) , u ( x, t ) ≤ Ce c [ t − h ( x )] in { t < h ( x ) } . (4.2) Proof.
Because f is of combustion type or bistable, by choosing L large enough, wehave ∂ t u − ∆ u ≤ { t < h ( x ) − L } . Take a radially symmetric function ϕ ∈ C ( R n ) such that ϕ ≤ ϕ ( x ) ≡ − κ ∗ | x | outside a large ball, | ∆ ϕ | ≪ |∇ ϕ | ≤ C in R n . By taking a small µ >
0, thefunction w ( x, t ) := e µ [ t − ϕ ( x )] is a super-solution of the heat equation in D := { ( y, t ) : t < ϕ ( x ) } . Moreover, w = 1on ∂ D .For each ( x, t ) ∈ { t < h ( x ) } , by enlarging L further (depending on the Lipschitzconstant of h , but independent of x ), the domain D x := { ( y, s ) : s < h ( x ) − L − ϕ ( y − x ) } ⊂ { s < h ( y ) − L } . A comparison with a suitable translation of w leads to (4.2). (cid:3) Take a large
L > u ≪ { t < h ( x ) − L } . This is possible by (H2) .In { h ( x ) − L ≤ t ≤ h ( x ) } , (4.1) is a direct consequence of the facts that u has apositive lower bound here while both ∂ t u and |∇ u | are bounded. It thus remains toshow that (4.1) holds in { t < h ( x ) − L } .We first prove the combustion case. Proof of Proposition 4.1 in the combustion case. If f is of combustion type, u , ∂ t u and ∂ x i u all satisfy the heat equation in { t < h ( x ) − L } . The estimate (4.1) thenfollows from the comparison principle. For example, because both ∂ t u and u convergeto 0 uniformly as dist(( x, t ) , { t = h ( x ) } ) → + ∞ , if ∂ t u ≤ M u on { t = h ( x ) − L } forsome constant M >
0, then ∂ t u ≤ M u in { t < h ( x ) − L } . (cid:3) Next, we prove the bistable case.
Proof of Proposition 4.1 in the bistable case.
Take a b ∈ (0 ,
1) sufficiently small sothat f ∈ C ,α ([0 , b ]), f ′ ( u ) ≤ f ′ (0) / | f ′ ( u ) − f ′ (0) | ≤ Cu α , for any u ∈ [0 , b ] . (4.3)For any λ ∈ (0 , b ), denote Ω λ := { u < λ } . A direct calculation using (4.3) showsthat for some universal constant C > λ ), ∂ t u − Cλ α − ∆ u − Cλ α ≥ f ′ ( u ) u − Cλ α . On the other hand, ∂ t u is a solution of this linearized equation.Therefore, if we denote M ( λ ) := sup ∂ Ω λ ∂ t uu − Cλ α = sup ∂ Ω λ ∂ t uλ − Cλ α , ISTABLE OR COMBUSTION FRONTS 17 applying the comparison principle as in the proof of the combustion case, we obtain ∂ t u ≤ M ( λ ) u − Cλ α in Ω λ . (4.4)From this inequality and the fact that ∂ Ω λ/ ⊂ Ω λ , we deduce that M (cid:18) λ (cid:19) ≤ M ( λ ) (cid:18) λ (cid:19) − C ( − − α ) λ α . This inequality implies that lim sup λ → M ( λ ) < + ∞ . Substituting this estimate into (4.4), we find a constant C such that for any λ ∈ (0 , b ), in Ω λ \ Ω λ/ , ∂ t u ≤ Cu − C (2 u ) α ≤ Cu. (4.5)Here to deduce the last inequality, we have used the inequality (perhaps after choos-ing a smaller b ) u − C α u α ≤ , if u ≤ b. Finally, the estimate for |∇ u | /u follows by combining (4.5) and Proposition 3.3. (cid:3) Blowing down analysis
Recall that the one dimensional travelling wave g (see (1.3)) is strictly increasing,and it converges to 1 and 0 exponentially as t → ±∞ . In fact, by (F1) and ( F4 ),there exist four positive constants α ± and β ± such that g ( t ) = 1 − α + e − β + t + O (cid:0) e − (1+ α ) β + t (cid:1) as t → + ∞ ,g ( t ) = α − e β − t + O (cid:0) e (1+ α ) β − t (cid:1) as t → −∞ , where β + := − lim t → + ∞ g ′′ ( t ) g ′ ( t ) = − κ ∗ + p κ ∗ − f ′ (1)2 ,β − := lim t →−∞ g ′′ ( t ) g ′ ( t ) = κ ∗ + p κ ∗ − f ′ (0)2 . Because f ′ (0) ≤ β − ≥ κ ∗ .Following [2], set Φ := g − ◦ u . It satisfies ∂ t Φ − ∆Φ = κ ∗ + g ′′ (Φ) g ′ (Φ) (cid:0) |∇ Φ | − (cid:1) . (5.1) Lemma 5.1.
There exists a universal constant
C > such that | ∂ t Φ | + |∇ Φ | ≤ C in R n × R . Proof.
By Proposition 4.1, | ∂ t Φ | ≤ C | ∂ t u | u ≤ C, |∇ Φ | ≤ C |∇ u | u ≤ C, in { u ≤ − b } . By Corollary 2.14, | ∂ t Φ | ≤ C | ∂ t u | − u ≤ C, |∇ Φ | ≤ C |∇ u | − u ≤ C, in { u ≥ − b } . (cid:3) Lemma 5.2 (Semi-concavity) . There exists a universal constant C such that forany ( x, t ) ∈ { Φ > } , ∇ Φ( x, t ) ≤ C Φ( x, t ) , and for any ( x, t ) ∈ { Φ < } , ∇ Φ( x, t ) ≥ C Φ( x, t ) . This follows from a standard maximum principle argument applied to η ∇ Φ( ξ, ξ ),for any ξ ∈ R n and a suitable cut-off function η .For each ε >
0, let Φ ε ( x, t ) := ε Φ( ε − x, ε − t ). It satisfies ∂ t Φ ε − ε ∆Φ ε = κ ∗ + g ′′ ( ε − Φ ε ) g ′ ( ε − Φ ε ) (cid:0) |∇ Φ ε | − (cid:1) . (5.2)By the uniform Lipschitz bound on Φ ε from Lemma 5.1, there exists a subsequenceof ε → ε → Φ ∞ in C loc ( R n × R ).The limit Φ ∞ may depend on the choice of subsequences. But for notationalsimplicity, we will always write ε → ε i → Lemma 5.3.
In the open set { Φ ∞ > } , Φ ∞ is a viscosity solution of ∂ t Φ ∞ + β + |∇ Φ ∞ | − κ ∗ − β + = 0 . (5.3) In the open set { Φ ∞ < } (if non-empty), Φ ∞ is a viscosity solution of ∂ t Φ ∞ − β − |∇ Φ ∞ | − κ ∗ + β − = 0 . (5.4)Since h ( x ) is globally Lipschitz on R n , by taking a further subsequence, we mayalso assume ε i h (cid:0) ε − i x (cid:1) → h ∞ ( x ) in C loc ( R n ) . Lemma 5.4. { Φ ∞ > } = { t > h ∞ ( x ) } .Proof. Because u ≥ − b in { t > h ( x ) } , by Lemma 2.4 we get1 − u ( x ) ≤ Ce − c [ t − h ( x )] in { t > h ( x ) } . Using the expansion of g near infinity, this is rewritten asΦ( x ) ≥ c [ t − h ( x )] − C in { t > h ( x ) } . (5.5)Taking the scaling Ψ ε and letting ε →
0, we obtainΦ ∞ ( x ) ≥ c [ t − h ∞ ( x )] > { t > h ∞ ( x ) } . (5.6) ISTABLE OR COMBUSTION FRONTS 19
Finally, because Φ ≤ g − (1 − b ) in { t < h ( x ) } , Φ ∞ ≤ { t < h ∞ ( x ) } . (cid:3) Lemma 5.5. h ∞ is unbounded from below.Proof. This is a direct consequence of Proposition 2.1. (cid:3)
Lemma 5.6.
The Lipschitz constant of h ∞ is at most κ − ∗ .Proof. For any ( x , t ) ∈ { Φ ∞ > } , for all ε small, Φ ε ( x , t ) ≥ Φ ∞ ( x , t ) /
2. Bydefinition, u ( ε − x , ε − t ) is very close to 1. By Lemma 2.6, for any δ >
0, thereexits a D ( δ ) such that C + κ ∗ − δ ( ε − x , ε − t + D ( δ )) ⊂ { u > − b } . A scaling of thisgives C + κ ∗ − δ ( x , t + D ( δ ) ε ) ⊂ { Φ ε > } . Letting ε → δ →
0, with the helpof (5.6), we deduce that C + κ ∗ ( x , t ) ⊂ { Φ ∞ > } . This implies that the Lipschitzconstant of h ∞ is at most κ − ∗ . (cid:3) Finally, under the assumption of Theorem 1.2, the following non-degeneracy con-dition in Ω −∞ holds. Proposition 5.7. In Ω −∞ , Φ ∞ ( x, t ) ≤ c [ t − h ∞ ( x )] < . (5.7)This can be proved by scaling Proposition 4.2.6. Geometric motion: Proof of Theorem 1.5
In this section we prove Theorem 1.5. This theorem does not follow directly fromthe result on front motion law established in [4], although it can be reduced to thatone by constructing a suitable comparison function. The main reason is, for entiresolutions of (1.1), it is not clear whether |∇ Φ | ≤ Proof of Theorem 1.5.
We divide the proof into two steps, verifying the sub- andsup-solution property respectively.
Step 1.
For any ϕ ∈ C ( R n ) satisfying ϕ ≥ h ∞ and ϕ = h ∞ at one point, saythe origin 0, we want to show that |∇ ϕ (0) | ≤ κ − ∗ .Assume by the contrary, there exists δ > |∇ ϕ (0) | = ( κ ∗ − δ ) − . (6.1)The tangent plane of { t = ϕ ( x ) } at (0 ,
0) is { ( κ ∗ − δ ) t = − x · ξ } , where ξ := ∇ ϕ (0) / |∇ ϕ (0) | . Since h ∞ ≤ ϕ , we find three small constants ρ > t < σ > ∞ ( x, t ) ≥ σ in D := { x · ξ ≥ − ( κ ∗ − δ ) t } ∩ B ρ (0) . For all ε small, consider the Cauchy problem ( ∂ t w − ∆ w = f ( w ) in R n × ( t /ε, + ∞ ) ,w ( x, t /ε ) = (cid:16) − e − cσε − dist ( εx,∂ D ) (cid:17) χ D /ε ( x ) , where χ D /ε denotes the characteristic function of D /ε . As in Lemma 2.3 (or by the motion law for front propagation starting from ∂ ( D ∩ B ρ ), see [2, Main Theorem] or [4, Theorem 9.1]), we get w ( x, ≥ − b in (cid:26) x · ξ ≥ δt ε (cid:27) ∩ B ρ +( κ ∗− δ ) t ε (0) . By comparison principle, u ≥ w in R n × [ t /ε, ∞ > { x · ξ ≥ δt } ∩ B ρ +( κ ∗ − δ ) t (0) . In particular, (0 ,
0) is an interior point of { Φ ∞ > } . This is a contradiction. Step 2.
In the same way, we can show that for any ϕ ∈ C ( R n ) satisfying ϕ ≤ h ∞ and ϕ (0) = h ∞ (0), |∇ ϕ (0) | ≥ κ − ∗ .Assume this is not true, that is, there exists a δ > |∇ ϕ (0) | =( κ ∗ + 3 δ ) − . The only difference with Step 1 is the construction of the compari-son function. Now we need to consider, for all ε small, the Cauchy problem ( ∂ t w − ∆ w = f ( w ) in R n × ( t /ε, + ∞ ) ,w ( x, t ) = 1 − (cid:16) − e − cσε − dist ( εx,∂ D ) (cid:17) χ D /ε ( x ) , where D = { x · ξ ≤ − ( κ ∗ + 2 δ ) t } ∩ B ρ (0).As in Step 1, by the motion law for front propagation starting from ∂ ( D ∩ B ρ )given in [2, Main Theorem] or [4, Theorem 9.1], we deduce that w ( x, ≤ b in (cid:26) x · ξ ≤ − δ t ε (cid:27) ∩ B ρ +( κ ∗− δ ) t ε (0) . This implies that (0 ,
0) is an interior point of { Φ ∞ < } , which is a contradiction. (cid:3) Representation formula for the blowing down limit
In this section, we give an explicit representation formula for Φ ∞ . We first considerthe forward problem (5.3) in Ω + ∞ := { t > h ∞ ( x ) } , and then the backward problem(5.4) in Ω −∞ := { t < h ∞ ( x ) } . The main tool used in this section is the generalizedcharacteristics associated to Φ ∞ . We will follow closely the treatment in Cannarsa,Mazzola and Sinestrari [10].7.1. Forward problem.
This subsection is devoted to the forward problem (5.3)in Ω + ∞ . We first notice the following pointwise monotonicity relation. Lemma 7.1.
For any ( x, t ) , ( y, s ) ∈ Ω + ∞ with t > s , if the segment connecting ( y, s ) and ( x, t ) is contained in Ω + ∞ , then Φ ∞ ( x, t ) ≤ Φ ∞ ( y, s ) + ( κ ∗ + β + ) ( t − s ) + | x − y | β + ( t − s ) . (7.1) Proof.
Since Φ ∞ is Lipschitz, it is differentiable a.e. and satisfies (5.3) a.e. in Ω + ∞ . Byavoiding a zero measure set, we may assume for a.e. τ ∈ [0 , ∞ is differentiableat the point X ( τ ) := ((1 − τ ) y + τ x, (1 − τ ) s + τ t ). (The general case follows by anapproximation using the continuity of Φ ∞ .) ISTABLE OR COMBUSTION FRONTS 21
Then we have ddτ Φ ∞ ( X ( τ )) = ∂ t Φ ( X ( τ )) ( t − s ) + ∇ Φ ( X ( τ )) · ( x − y )= ( κ ∗ + β + ) ( t − s ) − β + |∇ Φ ( X ( τ )) | ( t − s )+ ∇ Φ ( X ( τ )) · ( x − y ) ≤ ( κ ∗ + β + ) ( t − s ) + | x − y | β + ( t − s ) . Integrating this inequality in τ , we obtain (7.1). (cid:3) Next we establish a localized Hopf-Lax formula for Φ ∞ . Lemma 7.2 (Localized Hopf-Lax formula I) . There exists a constant K dependingonly on the Lipschitz constant of Φ ∞ so that the following holds. For any ( x, t ) ∈ Ω + ∞ ,there exists an ε > such that B Kε ( x ) × ( t − ε, t ) ⊂ Ω + ∞ , and Φ ∞ ( x, t ) = min y ∈ B Kε ( x ) (cid:20) Φ ∞ ( y, t − ε ) + ( κ ∗ + β + ) ε + | x − y | β + ε (cid:21) . (7.2) Proof.
Denote Q := B Kε ( x ) × ( t − ε, t ). By Lemma 7.1, we can apply Lions [25,Theorem 10.1 and Theorem 11.1] to deduce thatΦ ∞ ( x, t ) = inf ( y,s ) ∈ ∂ p Q (cid:20) Φ ∞ ( y, s ) + ( κ ∗ + β + ) ( t − s ) + | x − y | β + ( t − s ) (cid:21) . (7.3)If K is large enough (compared to the Lipschitz constant of Φ ∞ ), for any ( y, s ) ∈ ∂B Kε ( x ) × [ t − ε, t ), we haveΦ ∞ ( y, s ) + ( κ ∗ + β + ) ( t − s ) + | x − y | β + ( t − s ) > Φ ∞ ( x, t − ε ) + ( κ ∗ + β + ) ε. Therefore the infimum in (7.3) is attained in the interior of B Kε ( x ) × { t − ε } , andit must be a minimum. (cid:3) Now we use this localized Hopf-Lax formula to study backward characteristiccurves of Φ ∞ . We will restrict our attention to differentiable points. Take a point( x , t ) ∈ Ω + ∞ so that Φ ∞ ( · , t ) is differentiable at x . Denote p := ∇ Φ ∞ ( x , t ).By Lemma 7.2, for any s < t sufficiently close to t , there exists a point ( x ( s ) , s ) ∈C − K ( x , t ) ∩ Ω + ∞ such thatΦ ∞ ( x , t ) = Φ ∞ ( x ( s ) , s ) + ( κ ∗ + β + ) ( t − s ) + | x − x ( s ) | β + ( t − s ) . (7.4) Lemma 7.3.
Under the above setting, we have x ( s ) = x − β + ( t − s ) p . (7.5) Proof.
By Lemma 7.2, for any x close to x ,Φ ∞ ( x, t ) ≥ Φ ∞ ( x ( s ) , s ) + ( κ ∗ + β + ) ( t − s ) + | x − x ( s ) | β + ( t − s ) . (7.6) Subtracting (7.4) from (7.6) leads toΦ ∞ ( x, t ) − Φ ∞ ( x , t ) ≥ x + x − x ( s )4 β + ( t − s ) · ( x − x ) . On the other hand, because Φ ∞ ( · , t ) is differentiable at x , we haveΦ ∞ ( x, t ) − Φ ∞ ( x , t ) = p · ( x − x ) + o ( | x − x | ) . These two relations hold for any x sufficiently close to x , so p = x − x ( s )2 β + ( t − s ) . (cid:3) Corollary 7.4.
The minimum in (7.2) is attained at a unique point.
The curve { ( x ( s ) , s ) : x ( s ) = x − β + ( t − s ) p , s ≤ t } is the backward characteristic curve of Φ ∞ starting from ( x , t ). Lemma 7.5.
Under the above settings, Φ ∞ ( · , s ) is differentiable at x ( s ) . Moreover, ∇ Φ ∞ ( x ( s ) , s ) = p . (7.7) Proof.
Because x ( s ) attains the minimum in (7.3), for any z sufficiently close to x ( s ), we have Φ ∞ ( x ( s ) , s ) + ( κ ∗ + β + ) ( t − s ) + | x − x ( s ) | β + ( t − s ) ≤ Φ ∞ ( z, s ) + ( κ ∗ + β + ) ( t − s ) + | x − z | β + ( t − s ) . After simplification, this isΦ ∞ ( z, s ) ≥ Φ ∞ ( x ( s ) , s ) + x − x ( s )2 β + ( t − s ) · [ z − x ( s )] + O (cid:0) | z − x ( s ) | (cid:1) . Since Φ ∞ ( · , s ) is semi-concave, this inequality implies that Φ ∞ ( · , s ) is differentiableat x ( s ), and its gradient is given by (7.7). (cid:3) By Lemma 7.2 and Lemma 7.5, the characteristic curve can be extended indef-initely in the backward direction, unless it hits the boundary ∂ Ω + ∞ in finite time.Now we show that the later case must happen. Lemma 7.6.
For any ( x , t ) ∈ Ω + ∞ with Φ( · , t ) differentiable at x , there exists an s < t such that ( x − β + ( t − s ) p , s ) ∈ ∂ Ω + ∞ . Proof.
If ( x ( s ) , s ) = ( x − β + ( t − s ) p , s ) ∈ Ω + ∞ , by (7.5), (7.4) can be rewritten asΦ ∞ ( x − β + ( t − s ) p , s ) = Φ ∞ ( x , t ) − (cid:0) κ ∗ + β + + β + | p | (cid:1) ( t − s ) . (7.8)Hence there exists an s such that Φ ∞ ( x − β + ( t − s ) p , s ) = 0 and Φ ∞ ( x − β + ( t − s ) p , s ) > s ∈ ( s , t ]. Because Φ ∞ > + ∞ and Φ ∞ = 0 on ∂ Ω + ∞ , ( x − β + ( t − s ) p , s ) ∈ ∂ Ω + ∞ . (cid:3) ISTABLE OR COMBUSTION FRONTS 23
Lemma 7.7.
For any ( x, t ) ∈ Ω + ∞ , Φ ∞ ( x, t ) = inf y ∈{ h ∞ Choosing ( y, s ) = ( y, h ∞ ( y )) with h ∞ ( y ) < t in (7.1) (here we may assumethe segment connecting this point and ( x, t ) is contained in Ω + ∞ ), and then takinginfimum over y , we obtainΦ ∞ ( x, t ) ≤ inf y ∈{ h ∞ For the backward problem (5.4), we still use backwardcharacteristics to determine the form of Φ −∞ . The proof is similar to the forwardproblem, so most results in this subsection will be stated without proof. Lemma 7.8. For any ( x, t ) , ( y, s ) ∈ Ω −∞ with t > s , Φ ∞ ( x, t ) ≥ Φ ∞ ( y, s ) + ( κ ∗ − β − ) ( t − s ) − | x − y | β − ( t − s ) . (7.10)Because Ω −∞ is convex (see Remark 1.7), the segment connecting ( y, s ) and ( x, t )is always contained in Ω −∞ .In Ω −∞ , e Φ ∞ := − Φ −∞ is a viscosity solution of ∂ t e Φ ∞ + β − |∇ e Φ ∞ | + κ ∗ − β − = 0 . (7.11)Hence we have the following localized Hopf-Lax formula. Lemma 7.9 (Localized Hopf-Lax formula II) . There exists a constant K dependingonly on the Lipschitz constant of Φ ∞ so that the following holds. For any ( x, t ) ∈ Ω −∞ ,there exists an ε > such that B Kε ( x ) × ( t − ε, t ) ⊂ Ω −∞ , and Φ ∞ ( x, t ) = max y ∈ B Kε ( x ) (cid:20) Φ ∞ ( y, t − ε ) + ( κ ∗ − β − ) ε − | x − y | β − ε (cid:21) . (7.12)Take a point ( x , t ) ∈ Ω + ∞ so that Φ ∞ ( · , t ) is differentiable at x . Denote p := ∇ Φ ∞ ( x , t ).By Lemma 7.9, for any s < t sufficiently close to t , there exists a point ( x ( s ) , s ) ∈C − K ( x , t ) ∩ Ω −∞ such thatΦ ∞ ( x , t ) = Φ ∞ ( x ( s ) , s ) + ( κ ∗ − β − ) ( t − s ) − | x − x ( s ) | β − ( t − s ) . (7.13) Lemma 7.10. Under the above setting, we have x ( s ) = x + 2 β − ( t − s ) p . (7.14) Lemma 7.11. Under the above setting, Φ ∞ ( · , s ) is differentiable at x ( s ) . Moreover, ∇ Φ ∞ ( x ( s ) , s ) = p . (7.15) By Lemma 7.9 and Lemma 7.11, the characteristic curve can be extended indef-initely in the backward direction, unless it hits the boundary ∂ Ω −∞ in finite time.Now we show that the latter case must happen. Lemma 7.12. For any ( x , t ) ∈ Ω −∞ with Φ( · , t ) differentiable at x , there existsan s < t such that ( x + 2 β + ( t − s ) p , s ) ∈ ∂ Ω −∞ . Proof. If ( x ( s ) , s ) = ( x + 2 β − ( t − s ) p , s ) ∈ Ω −∞ , by (7.14), (7.13) can be rewrittenas Φ ∞ ( x + 2 β − ( t − s ) p , s ) = Φ ∞ ( x , t ) − (cid:0) κ ∗ − β − − β − | p | (cid:1) ( t − s ) . (7.16)If β − > κ ∗ , there exists an s such that Φ ∞ ( x − β + ( t − s ) p , s ) = 0 andΦ ∞ ( x − β + ( t − s ) p , s ) < s ∈ ( s , t ]. Because Φ ∞ < −∞ andΦ ∞ = 0 on ∂ Ω −∞ , ( x + 2 β − ( t − s ) p , s ) ∈ ∂ Ω −∞ .If β − = κ ∗ , this is still the case, unless p = 0. However, if p = 0, the character-istic curve is ( x , s ), and (7.16) reads asΦ ∞ ( x , s ) ≡ Φ ∞ ( x , t ) , for any s < t . This cannot happen by (5.7). (cid:3) With these lemmas in hand, similar to Lemma 7.7, we get Lemma 7.13. For any ( x, t ) ∈ Ω −∞ , Φ ∞ ( x, t ) = sup y ∈{ h ∞ In this section we consider travelling wave equation (1.2).Denote the constants K + := s κ ∗ β + + κ β , K − := s − κ ∗ β − + κ β − . ISTABLE OR COMBUSTION FRONTS 25 By abusing notations, we will use the following notations about cones in R n : C + λ ( x ) := { y : y n − x n > λ | y ′ − x ′ |} , C − λ ( x ) := { y : y n − x n < − λ | y ′ − x ′ |} . As in Section 5, set Ψ := g − ◦ u . It satisfies − ∆Ψ + κ∂ n Ψ = κ ∗ + g ′′ (Ψ) g ′ (Ψ) (cid:0) |∇ Ψ | − (cid:1) . (8.1)Since this is an elliptic equation, we have the following unconditional, globalLipschitz bound on Ψ. This lemma holds once the nonlinearity satisfies f (0) = f (1) = 0, no matter whether it is monostable, combustion or bistable. Lemma 8.1. There exists a universal constant C such that |∇ Ψ | ≤ C on R n .Proof. By definition, ∇ Ψ = ∇ ug ′ (Ψ) . Since g ′ has a positive lower bound on any compact set of R , by the gradient boundon u , |∇ Ψ | is bounded in { / < u < / } .In { u < / } , g ′ (Ψ) ≥ cg (Ψ) = cu. Hence here we have |∇ Ψ | ≤ C |∇ u | u ≤ C, where the last inequality follows from Harnack inequality and interior gradient esti-mates applied to (1.2).Similarly, in { u > / } , |∇ Ψ | ≤ C |∇ u | − u ≤ C. (cid:3) As before, Ψ is still semi-concave. Lemma 8.2 (Semi-concavity) . There exists a universal constant C such that forany x ∈ { Ψ > } , ∇ Ψ( x ) ≤ C Ψ( x ) , and for any x ∈ { Ψ < } , ∇ Ψ( x ) ≥ C Ψ( x ) . For each ε > 0, let Ψ ε ( x ) := ε Ψ( ε − x ), which satisfies − ε ∆Ψ ε + κ∂ n Ψ ε = κ ∗ + g ′′ ( ε − Ψ ε ) g ′ ( ε − Ψ ε ) (cid:0) |∇ Ψ ε | − (cid:1) . (8.2)By the uniform Lipschitz bound on Ψ ε from Lemma 8.1, for any sequence ε i → 0, there exists a subsequence such that Ψ ε i → Ψ ∞ in C loc ( R n ). Then standardvanishing viscosity method gives Lemma 8.3. In the open set { Ψ ∞ > } , Ψ ∞ is a viscosity solution of κ∂ n Ψ ∞ − κ ∗ + β + (cid:0) |∇ Ψ ∞ | − (cid:1) = 0 . (8.3) In the open set { Ψ ∞ < } (if non-empty), Ψ ∞ is a viscosity solution of κ∂ n Ψ ∞ − κ ∗ − β − (cid:0) |∇ Ψ ∞ | − (cid:1) = 0 . (8.4) Remark 8.4. Equations (8.3) and (8.4) are the corresponding travelling wave equa-tions for the time-dependent Hamilton-Jacobi equations (5.3) and (5.4) . Recall that { v = 1 − b } = { x n = h ( x ′ ) } . As before, we define the blowing down limit h ∞ from h . By Lemma 5.4, we stillhave { Ψ ∞ > } = { x n > h ∞ ( x ′ ) } . Proposition 8.5. The Lipschitz constant of h ∞ is at most p κ /κ ∗ − . In partic-ular, we must have κ ≥ κ ∗ . Remark 8.6. Under the assumptions of Theorem 1.4, the blowing down limit h ∞ is a viscosity solution of |∇ h ∞ | − κ κ ∗ + 1 = 0 in R n − . (8.5) This follows from a reduction of Theorem 1.5.Proof of Proposition 8.5. The blowing down limit of the level set for the entire so-lution v ( x + κte n ) is the graph t = h ∞ ( x ′ ) − x n κ . By Lemma 5.6, its Lipschitz constant is at most κ − ∗ . (cid:3) By Lemma 5.2, Ψ ε are uniformly semi-concave in any compact set of { Ψ ∞ > } .As a consequence, Ψ ∞ is locally semi-concave in this open set. The sup-differentialof Ψ ∞ is then well defined at every point in { Ψ ∞ > } . Recall that ∂ Ψ ∞ ( x ) := (cid:26) ξ ∈ R n : lim sup y → x Ψ ∞ ( y ) − Ψ ∞ ( x ) − ξ · ( y − x ) | y − x | ≤ (cid:27) is a compact convex subset of R n . Because Ψ ε → Ψ ∞ uniformly on any compactset of R n , by the uniform semi-concavity of Ψ ε , we deduce that for any x ε → x ∞ ∈{ Ψ ∞ > } , each limit point of ∇ Ψ ε ( x ε ) as ε → ∈ ∂ Ψ ∞ ( x ∞ ) . (8.6)If Ψ ∞ < { x n < h ∞ ( x ′ ) } , the same result holds for the negative part of Ψ ∞ ,with sup-differentials replaced by sub-differentials.A reduction of Lemma 7.7 and Lemma 7.13 gives Proposition 8.7. • For any x = ( x ′ , x n ) ∈ { Ψ ∞ > } , Ψ ∞ ( x ) = inf y ′ ∈ R n − (cid:20) K + p | x ′ − y ′ | + ( x n − h ∞ ( y ′ )) − κ β + ( x n − h ∞ ( y ′ )) (cid:21) . (8.7) ISTABLE OR COMBUSTION FRONTS 27 • Assume Ψ ∞ < in { x n < h ∞ ( x ′ ) } . Then for any x = ( x ′ , x n ) ∈ { x n The representation formula (8.7) and (8.8) (when β − > κ ∗ ), can beproved directly by rewriting (8.3) and (8.4) as eikonal equations. For example, in thecase of (8.3) , we can define a norm on R n , k · k so that the corresponding unit ball is B K + (cid:16) ′ , − κ β + (cid:17) . (This is because this ball contains the origin as an interior point.)The Hamilton-Jacobi equation (8.3) is equivalent to the eikonal type equation k∇ Ψ( x ) k − . (8.9) Then we can prove that Ψ ∞ ( x ) = inf y ∈ ∂ { Φ ∞ > } k x − y k ∗ . Here k · k ∗ denotes the dual norm of k · k . Now we come to Proof of Theorem 1.8. We have shown that κ ≥ κ ∗ in Proposition 8.5. It remainsto characterize the κ = κ ∗ case.From Proposition 8.5, it is seen that, if κ = κ ∗ , we must have ∇ h ∞ = 0 a.e. in R n − . Since h ∞ (0) = 0, we get h ∞ ≡ R n − . (8.10)This holds for any blowing down limit h ∞ , so the blowing down limit is unique.Substituting (8.10) into (8.7) and (8.8), by noting that κ = κ ∗ implies K + = 1 + κ ∗ β + , K − = 1 − κ ∗ β − , we deduce that Ψ ∞ ( x ) ≡ x n in R n . (8.11) Claim. For any ε > 0, there exists an L ( ε ) > |∇ ′ Ψ | ≤ ε∂ n Ψ in {| Ψ | ≥ L ( ε ) } . By this claim, similar to the proof of Theorem 1.4 (or as in the proof of Gibbonsconjecture in [12, 8]), applying the sliding method we deduce that |∇ ′ Ψ | ≤ ε∂ n Ψ in R n . 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