Nonlocal Front Propagation Problems in BoundedDomains with Neumann-type Boundary Conditions

and Applications

Francesca Da Lio (1)

Christina Inwon Kim (2)

Dejan Slepÿcev (3)

Key-words: Front propagation,nonlocal reaction-diffusion equations, asymptotic be-havior, geometrical approach, level-set approach, Neumann boundary condition, angleboundary condition, viscosity solutions.

Abstract

This paper is concerned with the asymptotic behavior as ε! 0 of the solutionsof nonlocal reaction-diffusion equations of the form ut¡∆u+ε−2f(u, ε

R0 u) = 0

in O£ (0, T ) associated with nonlinear oblique derivative boundary conditions.We show that such behavior is described in terms of an interface evolving withnormal velocity depending not only on its curvature but also on the measureof the set it encloses. To this purpose we introduce a weak notion of motionof hypersurfaces with nonlocal normal velocities depending on the volume theyenclose, which extends the geometric deÞnition of generalized motion of hy-persurfaces in bounded domains introduced by G. Barles and the Þrst authorin [BDL] to solve a similar problem with local normal velocities depending onthe normal direction and the curvature of the front. We also establish com-parison and existence theorems of viscosity solutions to initial-boundary valueproblems for some singular degenerate nonlocal parabolic pdes with nonlinearNeumann-type boundary conditions

(1)Dipartimento di Matematica. Universita di Torino. Via Carlo Alberto 10. 10123 Torino, Italy(2)Department of Mathematics, MIT, Cambridge, MA 02139, U.S.A.(3)Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, CanadaThe Þrst author was partially supported by M.U.R.S.T, project: Analisi e controllo di equazioni dievoluzione deterministiche e stocastiche.

1

Introduction

In this paper we study the limiting behaviour, as ε! 0, of the solution of the followingequation

uε,t ¡∆uε + b(x) ¢Duε + ε−2f(uε, εZO

uε) = 0 in O £ (0, T ), (1)

where O is a smooth bounded domain of IRN , T > 0, uε:O£[0, T ]! IR is the solution,the nonlinearity f(u, v) is smooth and f0(u) := f(u, 0) is the derivative of a double-well potentialW . A typical example is f0(u) := 2u(u

2¡1) and f(u, v) = f0(u)+vh(u).The equation (1) is obtained as a limit as a system of reaction-diffusion equationsoften referred to as a Belousov-Zhabotinskii model (see, e.g. [CHL]). We consider (1)together with a nonlinear boundary condition of the form

G(x, t,Du) = 0 on ∂O £ (0, T ), (2)

where G : ∂O£ (0, T )£ IRN ! IR is a continuous function satisfying : for any T > 0,there exists a constant ν(T ) > 0 such that, for all λ > 0, x 2 ∂O, t 2 [0, T ], p 2 IRN ,one has

G(x, t, p+ λn(x))¡G(x, t, p) ¸ ν(T )λ .Typical examples of such boundary conditions, besides the homogeneous Neumannboundary condition, are the oblique derivative boundary condition

∂uε∂γ

= 0 on ∂O £ (0, T ) , (3)

where γ : ∂O £ (0, T )! IRN is a Lipschitz continuous vector Þeld such that γ(x, t) ¢n(x) > 0 on ∂O £ (0, T ), n(x) being the unit exterior normal vector to ∂O at x andthe capillarity type boundary condition

∂u

∂n= θ(x, t)jDuj on ∂O £ (0, T ) , (4)

where θ : ∂O £ (0, T )! IR is, say, a locally Lipschitz continuous function such thatjθ(x, t)j < 1 on ∂O £ (0, T ). Finally we impose an initial data

uε(x, 0) = g(x) on O £ f0g , (5)

where g 2 C(O).When f(u, v) is independent of v, i.e, f(u, v) = f0(u) the equation (1) reduces

to the Allen-Cahn equation [AC] modelling the motion of the sharp interface -the

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antiphase boundary- between regions of different phases of a material. The mainfeature of the solution of the Allen-Cahn equation is that the zero level set of thesolution approximates (as ε ! 0) the motion by mean curvature. Formal derivationof this connection was carried out by Allen-Cahn [AC], Keller, Rubinstein, Stenberg[KRS], Fife [F] and many others.A Þrst rigorous, but partial, proof of this result was proposed by Chen [C] in the

case when the motion by mean curvature is classical i.e. when the fronts are smoothhypersurfaces evolving smoothly. This means in fact a small time result since it iswell-known that, for motion by mean curvature, singularities develop in Þnite time.In order to rigorously prove and even formulate the result for all time, a suitable

notion of generalized motion by mean curvature is needed in order to deÞne it past thedevelopment of singularities. This question was solved in a rather general way by thelevel-set approach (see Osher and Sethian [OS] Evans and Spruck [ES] and Chen,Giga and Goto [CGG].) Then a different but related approach using the propertiesof the (signed) distance to the front was introduced (see Soner [S] and Barles, Sonerand Souganidis [BSS].) For a general review of these theories, their relationship aswell as other related facts we refer to Souganidis [Sou1, Sou2].In order to treat the case of more complicated reaction-diffusion equations, Barles

and Souganidis introduced in [BS] a more geometrical approach. Recently Barlesand the Þrst author in [BDL] apply this approach to the the asymptotics of reaction-diffusion equations in bounded domains and with Neumann-type boundary condition,thus extending the result obtained by Katsoulakis, Kossioris, and Reitich [KKR] forconvex domains and with homogeneous Neumann boundary condition.All the results mentioned above concern with the asymptotics of local reaction

diffusion equation. The Þrst asymptotic result for nonlocal equations of the form(1) was provided by Chen, Hilhorst, Logak [CHL]. In [CHL] the authors proved thatthe limiting behaviour of the solution uε of

uε,t ¡∆uε + ε−2f(uε, εROuε) = 0 in O £ (0, T ),

∂u

∂n(x, t) = 0 on ∂O £ (0, T ),

uε(x, 0) = g(x) on O,

(6)

is governed by the motion of an interface (Γt)t with the following normal velocity Vν

Vν = ¡trDν + c0(λ(Ω+t )¡ λ(Ω−t )) (7)

where ν and Dν denote the exterior normal vector to Γt and its derivative , Ω+t is the

region enclosed by Γt, Ω−t = O n (Γt [Ω+t ), λ(Ω±t ) is the Lebesgue measure of Ω±t and

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c0 is a certain constant depending only on the velocity of the traveling wave solutionassociated with f. Yet their asymptotic result is proved under the assumptions that(Γt)t is a family of smooth hypersurfaces evolving smoothly according to the law (7)and never touches the boundary ∂O, therefore their convergence result holds as longas a smooth evolution of the motion exists.The aim of this paper is to rigorously establish for all time the connection between

the equations (1) set in bounded domains with nonlinear Neumann boundary condi-tions and the motion of fronts with normal velocity (7) and satisfying a contact angleboundary condition on the boundary. In order to do that the Þrst step is to Þnd a wayto interpret the evolution with nonlocal normal velocities past the development ofthe singularities, moreover the weak deÞnition has to be sufficiently ßexible in orderto be able to justify the appearance of an interface in the asymptotic analysis of thereaction-diffusion equations.To this purpose we consider the motion of hypersurfaces Γt with general normal

velocityVν = v(x, t, ν, Dν,Ωt) (8)

under the assumption that v is decreasing with respect to Dν and increasing withrespect to Ωt (which is the region enclosed by Γt) in order to guarantee that the result-ing evolutions satisfy the geometric maximum-type principle (the so called avoidance-inclusion property). A typical example is the evolution law (7) with c0 ¸ 0. When vis not monotone with respect to either arguments such property mail fail (see e.g. thecounter-example in [Ca]). In [Ca] Cardaliaguet proposed a weak deÞnition of motionof compact hypersurfaces in IRN by nonlocal velocities of the form (8) including alsothe case when v is not increasing with respect to Ωt. In this last case he showed theexistence of approximate solutions to the problem which converge to the generalizedsolution of the problem only under suitable regularity assumptions on v.In this paper we follow an approach which is very close to the one developed for

local motions by Barles and Souganidis in [BS] and later modiÞed by Barles and theÞrst author [BDL] to treat problems with Neumann-type boundary conditions. Werecall that both approaches in [BS] and [BDL] consist in considering the evolutionof open sets instead of hypersurfaces (which is quite natural from the point of viewof the applications) and rely on the monotonicity property of the front propaga-tions which, roughly speaking, can be expressed in the following way : if (Ω1s)s∈(a,b),(Ω2s)s∈(a,b) are two families of open subsets evolving with the same normal velocitythen, if Ω1t ½ Ω2t for some t 2 (a, b), one has

Ω1s ½ Ω2s for any s 2 [t, b).

The key points used in [BS] and in [BDL] are that (i) it is enough to test againstfamilies of smooth open subsets evolving smoothly, (ii) this has to be done only on

4

small time interval and (iii) one can use families whose normal velocities are smalleror bigger than the considered normal velocity.At this level of generality these basic ideas apply more or less readily in our

framework.Indeed we extend the notion of viscosity solution to initial-boundary value prob-

lems for nonlocal parabolic pdes with nonlinear Neumann boundary conditions (al-ready introduced by one of the authors in [Sl] in the case of homogeoneous Neumannboundary condition). Then we provide a general comparison result between semicon-tinuous viscosity sub- and supersolutions to Neumann-type problems for a large classof nonlocal degenerate parabolic (possibly singular) pdes which includes the case ofnonlocal geometric equations such as for example ut ¡ tr[(I ¡ Du− Du)D2u]¡ c0jDujµ(x, t, u(¢, t)) = 0 in O £ (0, T ),

G(x, t,Du) = 0 in ∂O £ (0, T ),(9)

where c0 > 0, µ(x, t, u(¢, t)) := λ(Ω+t,x(u))¡λ((Ω+t,x(u))c), Ω+t,x(u) = fu(¢, t) > u(x, t)g,for all p6= 0 p = jpj−1p and p−p denotes the symmetric matrix deÞned by (p−p)ij =pipj, for all 1 · i, j · N. As a consequence of the comparison result we get bymeans of the Perron Method (which can be applied with minor changes to the case ofnonlocal equations) the existence of a unique continuous viscosity solution. Moreoverunder the additional assumption that G is homogeneous of degree 1 in p, the level-setapproach can be extended to geometric pdes (9), namely we can deÞne the frontΓt moving by (8) as the zero level set of a the unique solution u of (9) (with theconvention that the set Ωt enclosed by Γt is given by fu(¢, t) > 0g).In analogy with what was done in [BS] and [BDL] the geometric approach we

want to apply to our problem is based on the idea that given the geometric deÞnitionit suffices to justify the assumptions when everything is smooth. Even though wecan still consider small-time smooth approximations of the fronts in our case, local-in-space approximations by smooth functions as in [BS] and [BDL] does not applydirectly to problems like (1). Indeed the main difficulty in our analysis lies in Þndinga proper approximation of the nonlocal term in the equation, both in the deÞnitionof the generalized fronts and in the asymptotic analysis.In this paper we focus on nonlocal velocities depending on the Lebesque measure of

the region Ωt (or on any Þnite measure which is absolutely continuous with respect tothe Lebesgue one) and in this particular case we can Þnd appropriate approximationsof the volume of the set Ωt (see Theorem 2.2). This fact motivates our deÞnition,namely we say, roughly speaking, that a family of open sets (Ωt)t is a generalizedsolution for motions with the nonlocal velocity Vν if and only if it is a generalizedsolution, according to the geometric deÞnition introduced in [BDL], for motion with

5

local velocities obtained by replacing the volume of the set Ωt with smooth functionsof time which are less (resp. bigger) than the volume of Ωt. We also show that, as in[BS] and [BDL], our approach is essentially equivalent to the level-set approach (seeTheorem 2.3).We then extend the abstract method introduced in [BS] and [BDL] to justify the

appearance of moving interfaces in the asymptotic limits of problems like (1). Oneof the main assumptions which allows us to apply readily this method and to havea rather simple proof of the asymptotics is that the nonlinearity f is supposed tobe nonincreasing with respect to v. Indeed under this condition we can replace thenonlocal term in the reaction-diffusion equations with some suitable smooth functionsof the time and to use in this way the asymptotic result obtained in [BDL] in the caseof equations with x, t and ε- dependent f 0s.We remark that under this monotonicityassumption we are able to prove a comparison result for viscosity solutions of (1) whichautomatically yields, by means of the Perron Method, the existence of a continuoussolution. Finally we show how it would be possible, by using the same approach,to extend the asymptotics under a suitable relaxation of the monotonicity conditionof f with respect to v. In particular we have in mind the case when this conditionholds only in the interval between the two stable equilibria of W, and one of the mainexample is f(u, v) = 2(u+ v)(u2 ¡ 1).Two main issues that we would like to understand and investigate in the future

is the asymptotics when fv ¸ 0 (namely f it is still monotone with respect to v butin the opposite way) and to see if our approach can be applied to reaction-diffusionsystems like the one studied in [SS] and [HLS].This paper is organized as follows. In Section 1 we extend the deÞnition of viscosity

solutions to nonlocal second order degenerate parabolic pdes introduced by one of theauthors in [Sl]. We also prove the comparison between discontinuous viscosity sub-and supersolutions of initial-boundary value problems for a large class of nonlocaldegenerate, possibly singular, parabolic pdes (including the geometric ones) withnonlinear Neumann-type boundary conditions, thus extending the comparison resultobtained in [Sl] in the case of homogeneous Neumann boundary conditions. In Section2 we introduce the new deÞnition for motions with nonlocal velocities and angleboundary condition and show its connection with the level set approach. Section 3is devoted to the application of the new deÞnition to the study of the asymptotics ofreaction-diffusion equations: we Þrst present a general abstract method and then weapply it to the model case of the nonlocal Allen-Cahn Equation (1) with a nonlinearNeumann boundary condition.

Acknowledgement : The authors would like to thank Prof. P. Souganidis forproposing the problem and the Þrst author wishes to thank also Prof. G. Barles forthe helpful discussions.

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1 Viscosity solutions for nonlocal equations, exis-

tence and uniqueness

Let O ½ IRN be a bounded open set with at least C1 boundary and B the set of allmeasurable subsets of O. The topology on B is generated by the distance d(A,B) =λ(A4B), where λ is the Lebesgue measure and ∆ is the symmetric difference.Let F be a real-valued, locally bounded function on O£ [0,1)£ IRN £S(N)£B,

which is continuous in O £ [0,1)£ IRN n f0g £ S(N)£B, where S(N) is the set ofreal symmetric N £ N matrices and let G be a real-valued, continuous function on∂O£(0,1)£IRN . We consider nonlocal degenerate (and possibly singular) parabolicequations, with nonlinear Neumann-type boundary conditions, of the following form: (i) ut + F (x, t,Du,D

2u, fy 2 O : u(y, t) ¸ u(x, t)g) = 0 in O £ (0, T ),(ii) G(x, t,Du) = 0 in ∂O £ (0, T ),(iii) u(x, 0) = u0(x) in O,

(10)with T > 0.Some examples of such operators F are:

F (x, t, p,X,K) := ¡Trace[(I ¡ p− p)A(x, t, p)X] + b(x, t) ¢ p¡ c0jpj(λ(K)¡ λ(Kc))

with c0 > 0, where for all p6= 0, p = jpj−1p and p− p denotes the symmetric matrixdeÞned by (p− p)ij = pipj , for all 1 · i, j · N, and

F (x, t, p,X,K) := ¡Trace[A(x, t, p)X] +H(x, t, p)¡ jpjβµZ

K

θ(x, t, p, y)dy

¶where

A(x, t, p) = σ(x, t, p)σt(x, t, p)

with σ:O£ [0,1)£IRN !MN,k (the space of N£k matrices) is a bounded function,possibly discontinuous at p = 0, locally Lipschitz in x 2 O and p 2 IRN n f0g,continuous in t and satisfying for some C > 0, for all t 2 [0,1), for almost everyx 2 O and p 2 IRN n f0g

jDxσ(x, t, p)j · C and jDpσ(x, t, p)j · C

jpj .

The function b is assumed to be locally Lipschitz continuous in x and continuousin t, while H is assumed to be locally Lipschitz continuous in x 2 O and p 2 IRN ,continuous in t 2 [0,1), and satisfying for some C > 0, for all t 2 [0,1), for almostevery x 2 O, and p 2 IRN

jDxH(x, t, p)j · C(1+ jpj) and jDpH(x, t, p)j · C.

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The function θ 2 C(O£[0,1)£SN−1£O, [0,1)) (with SN−1 = fx 2 IRN : jxj = 1g)is assumed to be Lipschitz continuous in the x, p variables and β is nondecreasing andLipschitz continuous.If these operators are to be geometric (that is to satisfy condition (A4) that

follows) then we need also to require that A is homogeneous of degree 0 in p and His homogeneous of degree 1 in p (and also to add (I ¡ p − p) under the trace in thesecond example).In (ii) we have typically in mind the following two boundary conditions:

∂u

∂γ= 0 on ∂O £ (0,1) , (11)

where γ : ∂O£ [0,1)! IRN is a Lipschitz continuous vector Þeld such that γ(x, t) ¢n(x) > 0 on ∂O £ [0,1), n(x) being the unit exterior normal to ∂O at x, and thecapillarity type boundary condition

∂u

∂n= θ(x, t)jDuj on ∂O £ (0,1) , (12)

where θ, : ∂O £ [0,1)! IR is, say, a locally Lipschitz continuous function such thatjθ(x, t)j < 1 on ∂O £ [0,1).To deÞne, and prove existence and uniqueness of, viscosity solutions of (10), we

follow [Sl] where nonlocal equations with homogeneous Neumann boundary conditionswere studied. Using results on (local) parabolic equations with nonlinear Neumann-type boundary conditions by Ishii and Sato [IS] and Barles [B], we extend the resultsof [Sl] to equations of the form (10).

We now list the basic requirements on F and G. We point out that the main as-sumptions introduced because of the presence of the nonlocal term are the monotonic-ity with respect to set inclusion and the continuity of F with respect to the topologyon B.(A1) The function F is locally bounded on O£ [0,1)£ IRN £S(N)£B, continuous

on O £ [0,1) £ IRN n f0g £ S(N) £ B, and for all x 2 O, t 2 [0,1), andK,L 2 B,

¡1 < F ∗(x, t, 0, O,K) = F∗(x, t, 0, O, L) < +1. (13)

(A2) F satisÞes the degenerate ellipticity condition and is nonincreasing with respectto its set argument :

F (x, t, p,X,K) · F (x, t, p, Y, L) whenever X ¸ Y and L µ K (14)

for all x 2 O, t 2 [0,1), p 2 IRN n f0g, X, Y 2 S(N) and K,L 2 B, where ¸ stands for the usual partial ordering of symmetric matrices.

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(A3) For any T > 0, the function G is uniformly continuous on ∂O £ (0, T ] £ IRNand there exists a constant ν(T ) > 0 such that, for all λ > 0, x 2 ∂O, t 2 (0, T ]and p 2 IRN ,

G(x, t, p+ λn(x))¡G(x, t, p) ¸ ν(T )λ . (15)

The nonlocal equations we consider are in most instances level-set equations ofsome geometric evolutions. Such equations fall in the class of geometric equations,that is the ones that satisfy the following two conditions:

(A4) For any λ > 0, ν 2 IR and x 2 O, t 2 (0,1), p 2 IRN nf0g, X 2 S(N), K 2 B

F (x, t,λp,λX + νp− p,K) = λF (x, t, p,X,K). (16)

(A5) For all λ > 0, x 2 ∂O, t 2 (0,1) and p 2 IRN

G(x, t,λp) = λG(x, t, p). (17)

To be able to show the uniqueness and existence of solutions we need additionalassumptions. Since we are able to extend both the results of [B] and [IS], the condi-tions required in either of the papers (for local equations) are sufficient. We denotethese assumptions by (A6) and list them in the Appendix A.We will show that under the assumptions (A1)-(A3) and (A5)-(A6) we have

the existence and uniqueness of solutions of (10) with continuous initial data. In theremainder of the paper we will refer to any set of assumptions that includes (A4)and (A5) and implies existence and uniqueness of (10) for all u0 2 C(O) as theassumptions of the level-set approach.

We recall that if f : A ! IR, where A is a subset of some IRk, the upper- andlower-semicontinuous envelopes f ∗ and f∗ of f are given by

f∗(y) = lim supz→y

f(z) and f∗(y) = lim infz→y

f(z).

We deÞne the viscosity solutions to the problem (10), analogously to [Sl].

DeÞnition 1.1 An upper-semicontinuous function u : O £ [0, T ) ¡! IR [ f¡1g isa viscosity subsolution of (10) if for all x 2 O, u(x, 0) · u∗0(x) and for all (x, t) 2O £ (0, T ) and all functions ϕ 2 C∞(O £ (0, T )) such that u ¡ ϕ has maximum at(x, t), if x 2 O or x 2 ∂O and G(x, t,Dϕ(x, t)) > 0 then

ϕt(x, t) + F∗(x, t,Dϕ(x, t), D2ϕ(x, t), fy : u(y, t) ¸ u(x, t)g) · 0. (18)

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A lower-semicontinuous function v : O £ [0, T ) ¡! IR [ f1g is a viscosity superso-lution of (10) if for all x 2 O, v(x, 0) ¸ u0∗(x) and for all (x, t) 2 O£ (0, T ) and allfunctions ϕ 2 C∞(O £ (0, T )) such that v ¡ ϕ has minimum at (x, t), if x 2 O orx 2 ∂O and G(x, t,Dϕ(x, t)) < 0 then

ϕt(x, t) + F∗(x, t,Dϕ(x, t), D2ϕ(x, t), fy : u(y, t) > u(x, t)g) ¸ 0. (19)

A function u : O £ [0, T ) ¡! IR is a viscosity solution of (10) if u∗ is a subsolutionand u∗ is a supersolution.

We remark that using different test sets in the deÞnition of sub and supersolutionsis necessary for having the desired properties of viscosity solutions (in particularstability and existence).Viscosity sub and supersolutions deÞned as above, have the following properties:

(P1) Stability : If fungn=1,2,.. is a sequence of subsolutions (resp. supersolutions) of(10)(i)¡ (ii) bounded from above (resp. below) then u = lim sup∗ un is also asubsolution (resp. u = lim inf∗ un is a supersolution).

We recall the half-relaxed limits of a sequence of functions un :O £ [0, T ]! IRare deÞned by

lim sup∗ un(x, t) :=lim sup(y,s)→(x,t)n→∞

un(y, s) and lim inf∗ un(x, t) := lim inf(y,s)→(x,t)n→∞

un(y, s).

(P2) If u is a subsolution (resp. supersolution) of (10 i,ii) and ρ : IR ¡! IR is nonde-creasing then (ρ ± u)∗ is also a subsolution (resp. (ρ ± u)∗ is a supersolution).

Proofs of these properties can be found in [Sl]. Here we just remark that one of thekey assumptions to obtain both properties is the continuity of F on B with respectto the topology we mention at the beginning of this Section, moreover the secondproperty is a consequence also of the geometric nature of the equations (assumptions(A4) and (A5)), and just means that the equation is invariant under relabelings oflevel sets that preserve inclusion. The proofs of the comparison and existence resultsalso make use of the following fact about stability of level sets of semicontinuousfunctions.

Lemma 1.1 Let f : O £ (a, b) ¡! IRN be a lower (resp. upper) semicontinuousfunction. For every x 2 O, t 2 (a, b) and ε > 0 there exists δ > 0 such that

λ(ff( ¢ , t) > f(x, t))g n ff( ¢ , s) > f(x, t) + δg) < ε(resp. λ(ff( ¢ , s) ¸ f(x, t)¡ δg n ff( ¢ , t) ¸ f(x, t)g) < ε )

for s 2 (t¡ δ, t+ δ).

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Proof. As the two claims are proven analogously, let us show only the Þrst one.We can assume that f(x, t) = 0. Suppose that for some ε > 0 and t 2 (a, b) noappropriate δ can be found. Then there exist sequences δn converging to 0 and snconverging to t as n goes to inÞnity such that for all n 2 IN

λ(ff( ¢ , t) > 0)g nff( ¢ , sn) > δng) ¸ εSince f is lower semicontinuous,

ff( ¢ , t) > 0g µ lim infn→∞

ff( ¢ , sn) > δng =∞[n=1

∞\i=n

ff( ¢ , si) > δig.

Note that there exists n0 such that for all n > n0

λ

à ∞[k=1

∞\i=k

ff( ¢ , si) > δig n∞\i=n

ff( ¢ , si) > δig!< ε.

So for n > n0λ (ff( ¢ , t) > 0gnff( ¢ , sn) > δng) < ε.

Contradiction. ¤

The main result of this Section is the following Theorem in which we prove acomparison result between viscosity sub- and supersolutions of (10). Although theequations we have in mind are geometric the comparison holds for more generalequations, and we indicate that below.

Theorem 1.1 (Comparison). Assume (A1)-(A3), (A5) and (A6). Let u, v berespectively a viscosity sub- and supersolution of (10). If u(x, 0) · v(x, 0) for allx 2 O, then u(x, t) · v(x, t) for all x 2 O and all t 2 [0, T ).Proof. Under the set of assumptions that includes (A6a) the proof of comparison

given in [IS] extends, following [Sl], to nonlocal equations without any difficulties.However, there are some technical difficulties if (A6b) is satisÞed but nevertheless

enough control on the nonlocal term can be obtained so that the test function of [B],as well as techniques of [B], can be used. For simplicity we present the proof assumingthat the boundary condition G is independent of time. Handling of the nonlocal termis still the same if G depends on time.Let u be a subsolution and v a supersolution of (10). Assume that the comparison

principle does not hold, that is, that for some (x0, t0) 2 O£ (0, T ), u(x0, t0) > v(x0, t0).Let τ be a number in (t0, T ). Choose γ > 0 so that u(x0, t0)¡v(x0, t0)¡ γ

τ ¡ t0 > 0. Let(x0, t0) be a point where u(x, t)¡ v(x, t)¡ γ

τ ¡ t reaches its maximum on O £ [0, τ).

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For ε > 0, η > 0, and α > 0, consider the function

wη,ε,α(x, y, t) := u(x, t)¡ v(y, t)¡ ζη,ε,α(x, y)¡ γ

τ ¡ twhere ζε,η,α is the test function constructed by Barles in [B]. The test function israther complicated, so we only present its form, and list its properties:

ζη,ε,α(x, y) := [(ψη,ε(x, y))+]6 ¡ α( d(x) + d(y))

where d 2 W 3,∞(O, [0, 1]) coincides with the distance to ∂O in a neighborhood of ∂O,and ψη,ε is a C

2 function deÞned in the Appendix of [B]. It satisÞes the followinginequalities for ε and η small enough:

¡Cηε+ jx¡ yj2

Cε2· ψη,ε(x, y) · Cηε+ C jx¡ yj

2

ε2(20)

for some constant C independent of η and ε. Moreover, for any x, y 2 O such thatjx¡ yj < ηε,

¡Cηε+ 1

C

jx¡ yjε2

· jDxψη,ε(x, y)j,

jDyψη,ε(x, y)j · Cjx¡ yjε2

+ Cηε,

jDxψη,ε(x, y) +Dyψη,ε(x, y)j · Cjx¡ yj2ε2

+ Cηε,

¡Cε2I · D2ψη,ε(x, y) · C

ε2

∙I ¡I

¡I I

¸+ CηI, (21)

G(x,Dxψη,ε(x, y)) > 0 if x 2 ∂O,G(y,Dyψη,ε(x, y)) < 0 if y 2 ∂O.

Let (xη,ε,α, yη,ε,α, tη,ε,α) be the point where wη,ε,α reaches its maximum on O £O £ [0, τ). For the sake of simplicity of notations, from now on we drop the indexeswere possible. Standard argument shows that jx¡ yj2/ε2 ! 0 as ε! 0 and that forε, η,α small enough tη,ε,α > 0. By considering ε small enough (depending on η) wecan assume that jx ¡ yj · ηε so that the properties (21) hold for (x, y) = (x, y). Byparabolic Crandall-Ishii Lemma [CIL, Lemma 8.3] it follows that for any δ > 0 thereexist numbers a, b, p, q 2 IRN and symmetric matrices X, Y such that

(a, p, X) 2 P2,+u(x, t)

(b, q, Y ) 2 P2,−v(y, t)

p = Dxζη,ε,α(x, y) and q = ¡Dyζη,ε,α(x, y), (22)

12

a¡ b =γ

(τ ¡ t)2 (23)

¡µ1

δ+ kD2ζη,ε,α(x, y)k

¶I ·

µX 00 ¡Y

¶· ¡I + δD2ζη,ε,α(x, y)

¢D2ζη,ε,α(x, y)

The properties of ζη,ε,α insure that

G(x, p) > 0 if x 2 O and G(y, q) < 0 if y 2 OTherefore

a+ F∗(x, t, p, X, fu( ¢ , t) ¸ u(x, t)g) · 0and b+ F ∗(y, t, q, Y , fv( ¢ , t) > v(y, t)g) ¸ 0.

(24)

Since norms of Xα, Yα, pα and qα have bounds independent of α, they, as wellas xα, yα, and tα converge along a subsequence as α ! 0. Let us denote againthe limiting quantities by X, Y , p, q, x, y, and t. Note that u(xα, yα) ! u(x, t) andv(yα, tα) ! v(y, t) as α ! 0. along the same sequence. Otherwise wα(x, y, t) >wα(xα, yα, tα) for α small enough, which would contradict the fact that wα reachesits maximum at (xα, yα, tα).Lemma 1.1 now implies that

limα→0

λ(fu( ¢ , tα) ¸ u(xα, tα)gnfu( ¢ , t) ¸ u(x, t)g) = 0,

limα→0

λ(fv( ¢ , t) > v(y, t)gnfv( ¢ , tα) > v(yα, tα)g) = 0where the limit is taken along the sequence. Therefore since F∗ is lower and F ∗ isupper semicontinuous, and both are monotone in the set valued variable, (22) and(24) hold for new quantities with α = 0 and wη,ε,0 has a maximum at (x, y, t).Therefore for all x 2 O

u(x, t)¡ v(x, t) · u(x, t)¡ v(y, t)¡ [(ψη,ε(x, y))+]6 + [(ψη,ε(x, x))+]6 .Let us Þrst consider the case that for all η and ε small enough

[(ψη,ε(x, y))+]6 ¡ [(ψη,ε(x, x))+]6 > 0 for all x 2 O.

Then fu( ¢ , t) ¸ u(x, t)g µ fv( ¢ , t) > v(y, t)g which gives us the desired control onthe nonlocal term. Obtaining a contradiction from (24) and (A6b) follows a classicalarguments and we refer the reader to Barles[B] for the details. Now consider the casethat there is a sequence of η and ε converging to zero, such that there always existsx 2 O so that [(ψη,ε(x, y))+]6 · [(ψη,ε(x, x))+]6. Then from (20) it follows that

jx¡ yj2Cε2

¡ Cηε · ψη,ε(x, y) · ψη,ε(x, x) · Cηε.

13

Therefore jx¡ yj < 2Cεpηε. The estimates on derivatives of ψη,ε that we listed, thenyield that jDxψη,ε(x, y)j and jDyψη,ε(x, y)j are O(ε−1/2) and kD2ψη,εk is O(ε−2).Using that ζη,ε,0(x, y) = [(ψη,ε(x, y))

+]6 and that ψη,ε(x, y) = O(ε), along with(22) (with δ = kD2ζη,ε,α(x, y)k−1 for example) implies that X, Y , p, and q convergeto zero as ε and η converge to zero along the aforementioned sequence. Passing to asubsequence, we can assume that x, y, and t also converge as ε and η go to 0. Weuse the same notation for the limits. From (24) then follows, by the semicontinuityof F∗ and F ∗ and their monotonicity in the set-valued argument, that

a+ F∗(x, t, 0, 0, O) · 0 and b+ F ∗(y, t, 0, 0, ;) ¸ 0.The fact that a > b combined with (13) leads to contradiction. ¤The existence of a viscosity solution to (10) is obtained via the Perrons Method.

Although the application of Perrons Method is rather standard we provide a prooffor completeness and to outline the difficulties posed by the presence of nonlocal term.

Theorem 1.2 (Existence). Let F and G be functions satisfying the conditions(A1)-(A3), (A5) and (A6) and let u0 be a continuous function. Then the problem(10) has a unique continuous viscosity solution.

Proof. Letu(x, t) := supfw(x, t) : w a subsolution of (10)g (25)

We claim that u is the solution to problem (10). Showing that u∗(x, 0) = u0(x) =u∗(x, 0) on O requires construction of appropriate sub- and supersolutions (barriers),which because of nonlinear boundary conditions poses some difficulties. However theexisting constructions given by Ishii and Sato [IS] and mentioned by Barles [B], extendwith minimal modiÞcations to equations with nonlocal terms, so we omit them.Note that, u∗ is then, by stability, a viscosity subsolution of (10). The deÞnition

of u then implies that u = u∗. Let us show that u∗ is a supersolution. Assume itis not. Then there exists a smooth function ϕ on O £ (0, T ) such that u∗ ¡ ϕ has aminimum at (x0, t0) and

ϕt(x0, t0) + F∗(x0, t0, Dϕ(x0, t0), D2ϕ(x0, t0), fy : u∗(y, t0) > u∗(x0, t0)g) < ¡2ε

for some ε > 0 (and G(x0, t0, Dϕ(x0, t0) < ¡2ε if x0 2 ∂O). We can assume thatϕ(x0, t0) = u∗(x0, t0) = 0. Consider the function

u(x, t) = maxfu(x, t),ϕδ(x, t)gwhere ϕδ(x, t) := ϕ(x, t)+δ¡jx¡x0j4¡jt¡ t0j4 and δ > 0 is to be determined. Sinceu∗ and F∗ are lower semicontinuous, using Lemma 1.1, there exists a positive r < 1

14

such that if (x, t) 2 B((x0, t0), r) then t0/2 < t < T ¡ t0/2, jϕδt (x, t)¡ ϕt(x0, t0)j < εand

F∗(x, t,Dϕδ, D2ϕδ, fu∗( ¢ , t) > 2Mrg) < F∗(x0, t0, Dϕ,D2ϕ, fu∗( ¢ , t0) > 0g) + ε

where M := kϕkC1(U) + 1 with U := O £ [t0/2, (T + t0)/2]. By making r smaller ifnecessary we also require that if x0 2 O then r <dist(x0, ∂O) and if x0 2 ∂O thenG(x, t,Dϕδ) < G(x0, t0,Dϕ) + ε on B((x0, t0), r). Now take any positive δ < r4/2.Let us show now that u is a subsolution of (10). Let ψ be a smooth function suchthat u¡ ψ has a maximum at (x, t). If u(x, t) = u(x, t) then u¡ ψ has a maximumat (x, t) and since u is a subsolution

ψt + F∗(x, t,Dψ, D2ψ, fu( ¢ , t) ¸ u(x, t)g)· ψt + F∗(x, t,Dψ, D2ψ, fu( ¢ , t) ¸ u(x, t)g) · 0

(or G(x, t,Dψ) · 0 if x 2 ∂O).If u(x, t) > u(x, t) then u(x, t) = ϕδ(x, t) near (x, t). Since δ < r4/2, we have

δ ¡ jx¡ x0j4 ¡ jt¡ t0j4 < 0 when (x, t)62 B((x0, t0), r). Using the fact that u(x, t) >u(x, t) ¸ ϕ(x, t) we conclude that (x, t) 2 B((x0, t0), r). If x 2 O then using thatϕδ(x, t) · δ +M(j(x¡ x0, t¡ t0)j < r4 +Mr < 2Mr one obtains

ψt + F∗(x, t,Dψ,D2ψ, fu( ¢ , t) ¸ u(x, t)g)· ϕδt (x, t) + F∗(x, t,Dϕ

δ,D2ϕδ, fu( ¢ , t) ¸ ϕδ(x, t)g)· ϕt(x0, t0) + ε+ F∗(x, t,Dϕδ, D2ϕδ, fu∗( ¢ , t) > 2Mrg)· ϕt(x0, t0) + ε+ F∗(x0, t0, Dϕ, D2ϕ, fu∗( ¢ , t0) > 0g) + ε < 0.

If x 2 ∂O then Dψ(x, t) = Dϕδ(x, t)¡ λn for some λ ¸ 0. Here n is the unit outsidenormal to ∂O at x. Therefore using property (A3)

G(x, t,Dψ(x, t)) · G(x0, t0, Dϕ(0, t0)) + ε < 0.

Also note that u( ¢ , 0) · u0( ¢ ) on O since r < t0. Together with the inequalitiesabove, that implies that u is a subsolution of (10). Note that since u∗(x0, t0) = 0,there exists a sequence (xn, tn) converging to (x0, t0) such that u(xn, tn) convergesto 0 as n goes to inÞnity. But then for n large enough u(xn, tn) > u(xn, tn), whichcontradicts the deÞnition of u, since u is a subsolution.Therefore u∗ is a supersolution. The comparison result in Theorem 1.1 implies

that u · u∗, and hence u = u∗. Therefore u is a continuous solution of (10). Theuniqueness of the solution follows again from Theorem 1.1. ¤

15

2 A generalized deÞnition for nonlocal motion and

its connections with the level-set approach

In this section we consider the initial-boundary value problem (10) with F and Gsatisfying the assumptions of the level set approach. Our aim is to extend thegeometrical approach to the weak motion of hypersurfaces in bounded domains withan angle contact boundary condition introduced in [BDL] to the case of nonlocalnormal velocity depending not only on the normal direction and the curvature butalso on the volume of the set the front enclose. Moreover we show its connections withthe level-set approach. We Þrst brießy recall the basic ideas of the level-set approachconnected to the initial boundary value problem (10).The level-set approach for problems associated with Neumann type boundary

conditions (see e.g. [B, GS, IS]) can be described in a similar way to the IRN case(see e.g. [CGG, ES]). Let E be the collection of triplets (Γ, D+,D−) of mutuallydisjoint subsets of O such that Γ is closed and D± is open and O = Γ [ D+ [ D−.For any (Γ0, D

+0 , D

−0 ) 2 E , Þrst choose u0 2 C(O) so that

D+0 = fx 2 O : u0(x) > 0g , D−

0 = fx 2 O : u0(x) < 0g , Γ0 = fx 2 O : u0(x) = 0g.By the results of Section 1, for every T > 0 and u0 2 C(O), there exists a unique

viscosity solution u of (10) in C(O£[0, T ]). If, for all t > 0, we deÞne (Γt, D+t , D

−t ) 2 E

by

Γt=fx 2 O : u(x, t) = 0g , D+t = fx 2 O : u(x, t) > 0g , D−

t = fx 2 O : u(x, t) < 0g,then, because of (A4), (A5), and since a comparison result holds for (10), the col-lection f(Γt, D+

t ,D−t )gt≥0 is uniquely determined, independently of the choice of u0,

by the initial triplet (Γ0, D+0 ,D

−0 ).

The properties of the generalized level set evolution have been the object of ex-tended study, at least in IRN . One of the most intriguing issues rather important inthe study of the asymptotics of reaction-diffusion equations is whether the so-calledfattening phenomena occurs or not, i.e. whether the set Γt develops an interior or not.Following the IRNcase, we say that the no-interior condition for f(x, t) : u(x, t) = 0gholds if and only if

f(x, t) : u(x, t) = 0g = ∂f(x, t) : u(x, t) > 0g = ∂f(x, t) : u(x, t) < 0g. (26)

The question when no-interior condition holds is a difficult one. For some condi-tions on when it holds and examples when it fails (for fronts in IRN ) see [BSS], [K] andreferences therein. The importance of the no-interior condition and its connectionwith more geometrical approaches than the level-set one is explained in the following

16

result, proved in IRN in [BSS] and which can be easily extended to the case of nonlocalequations and with nonlinear Neumann boundary conditions. In this result, if A is asubset of some IRk, 11A denotes the indicator function of A, i.e., 11A(x) = 1 if x 2 Aand 11A(x) = 0 if x 2 Ac.

Theorem 2.1 Under the assumptions of the level-set approach, the functions 11D+t ∪Γt¡

11D−tand 11D+

t¡ 11D−

t ∪Γt are respectively the maximal subsolution (and solution) andthe minimal supersolution (and solution) of (10) associated respectively with the ini-tial data u0 = 11D+

0 ∪Γ0¡ 11D−0 and u0 = 11D+

0¡ 11D−0 ∪Γ0

. Moreover, if Γ0 has an empty

interior, 11D+t¡ 11D−

tis the unique discontinuous solution of (10) associated with the

initial data u0 = 11D+0¡ 11D−

0if and only if the property (26) holds .

In fact the main consequence of Theorem 2.1 is that, if (26) holds, the problemis well-posed geometrically since the evolution of the indicator function is uniquelydetermined.Now we turn to the geometrical deÞnition. To do so and to simplify the presen-

tation, we have to introduce some notations.If A is a subset of some IRk, we denote by Int(A) the interior of A and if x 2 A

and r > 0, we set BA(x, r) := B(x, r) \ A (the open ball in the topology of A),BcA(x, r) := B

c(x, r) \ A, BA(x, r) := B(x, r) \ A (the closed ball in the topology ofA) and ∂BA(x, r) := ∂B(x, r) \A.In the sequel we denote by (Ωt)t∈(0,T ) a family of open subsets of O and we set

Γt = ∂Ωt. The signed-distance function d(x, t) from x to Γt deÞned by

d(x, t) =

½d(x,Γt) if x 2 Ωt,¡d(x,Γt) otherwise,

where d(x,Γt) denotes the usual nonnegative distance from x 2 IRN to Γt. If Γt is asmooth hypersurface, then d is a smooth function in a neighborhood of Γt, and forx 2 Γt, n(x, t) = ¡Dd(x, t) is the unit normal to Γt pointing away from Ωt.Hereafter we consider geometric operators F of the form

F (x, t, p,X,K) := F (x, t, p,X,λ(K)) (27)

where F is a real-valued locally bounded function on O £ [0, T ] £ IRN £ S(N) £ IRsatisfying the assumptions of the level set approach and being decreasing with respectto the last variable.One of the main examples we have in mind is

F (x, t, p,X,K) := ¡Trace[(I ¡ p− p)X] + b(x, t) ¢ p¡ c0jpj(λ(K)¡ λ(Kc))

17

with c0 > 0.We remark that all the results of this Section hold also in the case of nonlocal

operators depending on any measure which is absolutely continuous with respect tothe Lebesgue one.We premise the following result which motivates the deÞnition of generalized

super- and sub-ßow with nonlocal normal velocity ¡F and angle boundary condi-tion G we will give later.

Theorem 2.2 Suppose that the assumptions of the level set approach hold.(i) Let (Ωt)t∈(0,T ) be a family of open subsets of O such that the set Ω := [t>0Ωt£ftgis open in O£ (0, T ). Then the function χ = 11Ω¡ 11Ωc is a viscosity supersolution of(10)(i)¡ (ii) iff for all smooth functions θ: [0, T ]! IR such that θ(t) · λ(Ωt), for allt 2 (0, T ) and for all α > 0, χ is a supersolution of½

(i) ut + F (x, t,Du,D2u, θ(t)¡ α) = 0 in O £ (0, T ),

(ii) G(x, t,Du) = 0 in ∂O £ (0, T ), (28)

(ii) Let (Ft)t∈(0,T ) be a family of closed subsets of O such that the set F := [t>0Ft£ftgis closed in O£ (0, T ). Then χ = 11F ¡ 11Fc is a viscosity subsolution of (10)(i)¡ (ii)iff for all smooth functions θ: [0, T ]! IR such that λ(Ft) · θ(t) for all t 2 (0, T ) andfor all α > 0, χ is a subsolution of½

(i) ut + F (x, t,Du,D2u, θ(t) + α) = 0 in O £ (0, T ),

(ii) G(x, t,Du) = 0 in ∂O £ (0, T ). (29)

Remark 2.1 One can show that for any family (Ωt)t∈(0,T ) (resp. (Ft)t∈(0,T )) of open(resp. closed) subsets of O, the function χ = 11∪tΩt×t ¡ 11(∪tΩt×t)c (resp. χ =11∪tFt×t ¡ 11(∪tFt×t)c) is lower (resp. upper) semicontinuous if and only if

Ωt µ[ε>0

lim infs→t

(Ωs ¡ ε) (resp.\ε>0

lim sups→t

(Fs + ε) µ Ft)

where

lim infs→t

Ωs :=[δ>0

\0<|s−t|≤δ

Ωs (resp. lim sups→t

Fs :=\δ>0

[0<|s−t|≤δ

Fs).

Thus if (Ωt)t∈(0,T ) (resp. (Ft)t∈(0,T )) satisÞes the hypotheses of Theorem 2.2 then themap t! λ(Ωt) (resp. t! λ(Ft)) is lower semicontinuous (resp. upper semicontin-uous).

18

Proof of Theorem 2.2. We only prove (i), the case (ii) being analogous. We Þrstassume that χ is a viscosity supersolution of (10)(i)¡ (ii). Let (x0, t0) 2 O £ (0, T )be a strict global minimum point of χ ¡ φ where φ 2 C∞(O £ [0, T ]). We have toshow that for all smooth functions θ: [0, T ] ! IR, such that θ(t) · λ(Ωt) for everyt 2 (0, T ) and for all α > 0 we have

∂φ

∂t(x0, t0) + F

∗(x0, t0, Dφ(x0, t0),D2φ(x0, t0), θ(t0)¡ α) ¸ 0.This inequality is certainly true if (x0, t0) is in the interior of either the set fχ = 1g orthe set fχ = ¡1g since in these two cases χ is constant in a neighborhood of (x0, t0).Hence

∂φ

∂t(x0, t0) = 0, Dφ(x0, t0) = 0, D

2φ(x0, t0) · 0 and F ∗(x0, t0, 0, 0, θ(t0)¡α) =0. Assume that (x0, t0) 2 ∂fχ = 1g \ ∂fχ = ¡1g). The lower semicontinuity of χyields χ(x0, t0) = ¡1. In this case the inequality follows directly from (A2).Conversely suppose that for all smooth functions θ: [0, T ]! IR, such that θ(t) ·

λ(Ωt) for every t 2 (0, T ) and for all α > 0, χ is a supersolution of (28). Let (x0, t0) 2O £ (0, T ) be a strict global minimum point of χ¡ φ where φ 2 C∞(O £ [0, T ]). Weconsider again only the case (x0, t0) 2 O £ (0, T ). We have to show the inequality

∂φ

∂t(x0, t0) + F

∗(x0, t0, Dφ(x0, t0), D2φ(x0, t0), fy : χ(y, t0) > χ(x, t0)g) ¸ 0.This inequality is obvious if (x0, t0) is in the interior of either the set fχ = 1g or theset fχ = ¡1g for the reasons above. Assume that (x0, t0) 2 ∂fχ = 1g \ ∂fχ = ¡1g)and suppose by contradiction that, for some γ > 0, we have

∂φ

∂t(x0, t0) + F

∗(x0, t0, Dφ(x0, t0),D2φ(x0, t0), fz : χ(z, t0) > ¡1g) < ¡γand hence

∂φ

∂t(x0, t0) + F

∗(x0, t0,Dφ(x0, t0), D

2φ(x0, t0),λ(Ωt0)) < ¡γ.Since the function t 7! λ(Ωt) is lower semicontinuous (see Remark 2.1) it is thesupremum of a family of smooth functions. Therefore there exists a smooth functionθ(¢) satisfying θ(t) · λ(Ωt) and a small positive constant α such that λ(Ωt0)¡θ(t0)¡αis so small that (using the upper semicontinuity of F

∗)

∂φ

∂t(x0, t0) + F

∗(x0, t0, Dφ(x0, t0), D

2φ(x0, t0), θ(t)¡ α) < ¡3γ4.

But this contradicts the assumption that χ is a supersolution of (28), which concludesthe proof. ¤Now we give the deÞnition of generalized super- and sub-ßow in bounded domains

with the nonlocal normal velocity ¡F and angle boundary condition G.

19

DeÞnition 2.1 A family (Ωt)t∈(0,T ) (resp. (Ft)t∈(0,T )) of open (resp. closed) sub-sets of O is called a generalized super-ßow (resp. sub-ßow) with normal velocity¡F (x, t,Dd,D2d, fd(y, t) > d(x, t)g) and angle condition G(x, t,Dd) iff for all smoothfunctions θ: [0, T ]! IR such that for all t 2 (0, T )

θ(t) · λ(Ωt) (resp. λ(Ft) · θ(t))

and for all α > 0, (Ωt)t is a generalized super-ßow (resp. sub-ßow) in the senseof DeÞnition 1.1 in [BDL] with normal velocity ¡F (x, t,Dd,D2d, θ(t) ¡ α) (resp.¡F (x, t,Dd,D2d, θ(t) + α)) and angle condition G(x, t,Dd).

We denoteF−(x, t,Dd,D2d) := F (x, t,Dd,D2d, θ(t)¡ α)

andF+(x, t,Dd,D

2d) := F (x, t,Dd,D2d, θ(t) + α).

We recall the deÞnition in [BDL] of a generalized super and sub-ßow with normalvelocity given respectively by ¡F− and ¡F+ and angle boundary condition G.

DeÞnition 2.2 A family (Ωt)t∈(0,T ) (resp. (Ft)t∈(0,T )) of open (resp. closed) sub-sets of O is called a generalized super-ßow (resp. sub-ßow) with normal velocity¡F−(x, t,Dd,D2d) (resp. ¡F+(x, t,Dd,D2d)) and angle condition G(x, t,Dd) ifand only if, for any x0 2 O, t 2 (0, T ) , r > 0, h > 0 and for any smooth func-tion φ : O £ [0, T ]! IR such that

(i)∂φ

∂t+ (F−)∗(y, s,Dφ, D2φ) < 0, (resp.

∂φ

∂t+ (F+)∗(y, s,Dφ, D2φ) > 0 ) in

BO(x0, r)£ [t, t+ h],(ii) G(y, s,Dφ) < 0 (resp. G(y, s,Dφ) > 0 ) in ∂O \B(x0, r)£ [t, t+ h],(iii) For any s 2 [t, t+ h], fy 2 BO(x0, r) : φ(y, s) = 0g 6= ; and

jDφ(y, s)j 6= 0 on f(y, s) 2 BO(x0, r)£ [t, t+ h] : φ(y, s) = 0g,

(iv) fy 2 BO(x0, r) : φ(y, t) ¸ 0g ½ Ωt (resp. fy 2 BO(x0, r) : φ(y, t) · 0g ½ F ct ),

(v) for all s 2 [t, t+ h]

fy 2 ∂BO(x0, r) : φ(y, s) ¸ 0g ½ Ωs(resp.

fy 2 ∂BO(x0, r) : φ(y, s) · 0g ½ F cs),

20

then we havefy 2 BO(x0, r) : φ(y, t+ h) > 0g ½ Ωt+h

(resp.fy 2 BO(x0, r) : φ(y, t+ h) < 0g ½ Fct+h).

The next result gives the relationship between the notion of generalized super-and sub-ßow and the level-set evolutions related to (10). Since it is a straightforwardconsequence of Theorem 2.2 and Theorem 1.2 in [BDL] about the equivalence ofDeÞnition 2.2 with the evolutions related to (29) or (28), we omit its proof.

Theorem 2.3 Suppose that the assumptions of the level set approach hold.(i) Let (Ωt)t∈(0,T ) be a family of open subsets of O such that the set Ω := [t>0Ωt£ftgis open in O£ [0, T ]. Then (Ωt)t∈(0,T ) is a generalized super-ßow with normal velocity¡F and angle boundary condition G if and only if the function χ = 11Ω ¡ 11Ωc is aviscosity supersolution of (10)(i)¡ (ii).(ii) Let (Ft)t∈(0,T ) be a family of closed subsets of O such that the set F := [t>0Ft£ftgis closed in O£ [0, T ]. Then (Ft)t∈(0,T ) is a generalized sub-ßow with normal velocity¡F and angle boundary condition G if and only if the function χ = 11F ¡ 11Fc is aviscosity subsolution of (10)(i)¡ (ii).

3 Applications to the asymptotics of nonlocal

reaction-diffusion equations

3.1 The abstract method

In this section, we describe the abstract method to study the asymptotics of solu-tions to nonlocal semilinear reaction-diffusion equations set in bounded domains withNeumann-type boundary conditions. In the asymptotic problems we have in mind, weare given a family (uε)ε of bounded functions on O£ [0, T ], typically the solutions ofreaction-diffusion equations with Neumann type boundary condition and with a smallparameter ε > 0. The aim is to show that there exists a generalized ßow (Ωt)t∈[0,T ] onO with a certain nonlocal normal velocity ¡F (x, t,Dd,D2d,Ωt) and angle boundaryG(x, t,Dd) on ∂O such that, as ε! 0,

uε(x, t)! b(x, t) if (x, t) 2 Ω :=[

t∈(0,T )Ωt £ ftg,

anduε(x, t)! a(x, t) if (x, t) 2 Ωc

21

where, for all (x, t), a(x, t), b(x, t) 2 IR can be interpreted as local equilibria of thissystem. In order to be more speciÞc and to present the main steps of the method, weintroduce the sets

Ω1 = Intf(x, s) 2 O £ [0, T ] : lim inf∗ [uε ¡ b] (x, s) ¸ 0g , (30)

andΩ2 = Intf(x, s) 2 O £ [0, T ] : lim sup∗ [uε ¡ a] (x, s) · 0g . (31)

Then we are going to consider the families (Ω1t )t and (Ω2t )t deÞned by

Ω1t = Ω1 \ (O £ ftg) Ω2t = Ω

2 \ (O £ ftg) . (32)

For simplicity of notations, for i = 1, 2, we identify Ωit and (Ωit)c with their projections

in O.It is worth noticing that Ω1, Ω2 are deÞned as subsets of O£ (0, T ], they are open

by deÞnition and disjoint. In particular we remark that, by construction the functionsχ = 11Ω1¡11(Ω1)c and χ = 11(Ω2)c¡11Ω2 are respectively lower and upper semicontinuous

functions in O £ (0, T ] where, in fact, Ω1 has to be read here as[

t∈(0,T ]Ω1t £ ftg and

Ω2 as[

t∈(0,T ]Ω2t £ftg. We Þnally point out that χ, χ can be extended either by lower

semicontinuity or by upper semicontinuity to O£ [0, T ] and we keep below the samenotations for these extensions.As in [BDL], our method can be described in three steps.

1. Initialization : we have to determine the traces Ω10 and Ω20 of Ω

1 and Ω2 for t = 0.A convenient way to deÞne these traces are through the function χ and χ

Ω10 = fx 2 O : χ(x, 0) = 1g and Ω20 = fx 2 O : χ(x, 0) = ¡1g . (33)

2. Propagation : we have to show that (Ω1t )t and ((Ω2t )c)t are respectively super- and

sub-ßow with normal velocity ¡F and angle condition G.3.Conclusion : we use the following corollary whose proof is a straightforward conse-quence of Theorem 2.3 and therefore we omit it.

Corollary 3.1 Assume that the assumptions of the level-set approach hold and thatthe above (Ω1t )t and ((Ω

2t )c)t are respectively super and sub-ßow with normal velocity

¡F and angle boundary condition G and suppose there exists (∂Ω+0 ,Ω+0 ,Ω

−0 ) 2 E

such that Ω+0 µ Ω10 and Ω−0 µ Ω20. Then if (Γt,Ω

+t ,Ω

−t ) is the level-set evolution of

(∂Ω+0 ,Ω+0 ,Ω

−0 ) we have

(i) for all t > 0,Ω+t ½ Ω1t ½ Ω+t [ Γt , Ω−t ½ Ω2t ½ Ω−t [ Γt .

22

(ii) If [tΓt £ ftg satisÞes the no interior condition, then for all t > 0, we haveΩ+t = Ω

1t and Ω−t = Ω

2t .

We turn to comment the Þrst two steps of our method. We Þrst point out that themain difficulty to prove these steps comes from the nonlocal feature of the velocityand our strategy consists in replacing in a suitable way the volume of Ω1t and Ω

2t

with smooth functions . More precisely for every smooth function θ: [0, T ]! IR suchthat θ(t) · λ(Ω1t ) (resp. λ((Ω2t )c) · θ(t)) and all constants α > 0 we introduce thefollwing sets

eΩ1(α, θ) = Intf(x, s) 2 O £ [0, T ] : lim inf∗ ∙uε ¡ bε,αε

¸(x, s) ¸ 0g , (34)

eΩ2(α, θ) = Intf(x, s) 2 O £ [0, T ] : lim sup∗ ∙uε ¡ aε,αε

¸(x, s) · 0g (35)

where (aε,α)ε and (bε,α)ε are suitable sequences of real-valued functions deÞned in

O£ [0, T ] such that aε,α ! a, bε,α ! b, as ε! 0, uniformly in O£ [0, T ], and for allα > 0 small enough.In the Subsection 2.3 of [BDL] the set (34) (resp. (35)) is shown to be a gen-

eralized super-ßow (resp. sub-ßow) in the sense of DeÞnition 2.2 with respect to¡F (x, t,Dd,D2d, θ(t) ¡ α) (resp. ¡F (x, t,Dd,D2d, θ(t) + α)) and angle bound-

ary condition G(x, t,Dd). Thus our main aim is to show that Ω1 = eΩ1(α, θ) andΩ2 = eΩ2(α, θ). Indeed if the previous equalities hold we automatically obtain boththe initialization and the propagation of the front with the nonlocal normal velocity¡F (x, t,Dd,D2d, fd(y, t) > d(x, t)g) and angle boundary condition G(x, t,Dd) as aconsequence of the results in [BDL].

3.2 The asymptotics of nonlocal Allen-Cahn equation withmonotonicity

This section is devoted to the study of the model case of nonlocal Allen-Cahn Equationset in a bounded domain with Neumann type boundary conditions, which will be alsothe occasion of giving the reader a more precise idea of how the abstract methodworks. More precisely we will focus our attention to the following initial boundaryvalue problem

(i) uε,t ¡∆uε + b(x, t) ¢Duε + ε−2f(uε, εROuε) = 0 in O £ (0, T ),

(ii) G(x, t,Duε) = 0 on ∂O £ (0, T ),

(iii) uε = g on O £ f0g,

(36)

23

where g is a real-valued continuous function in O, G satisÞes the conditions for thelevel-set approach to hold, in particular (A4), and (A5), and b:O ! IRN is a Lip-schitz continuous vector Þeld. As far as the reaction term f : IR2 ! IR goes, through-out the paper, we assume that f 2 C2(IR2, IR) and f0(u) := f(u, 0) satisÞes

f0 has exactly three zeroes m− < m0 < m+,

f0(s) > 0 in (m−,m0) and f0(s) < 0 in (m0,m+),

f 00(m±) > 0, f 000 (m−) < 0 and f 000 (m+) > 0.

(37)

We observe that, for sufficiently small v there exist h−(v) < h0(v) < h+(v) suchthat

f(h−(v), v) = f(h0(v), v) = f(h+(v), v) = 0.

f(r, v) > 0 on (h−(v), h0(v)) and on (h+(v),+1) ,f(r, v) < 0 on (h0(v), h+(v))and on(¡1, h−(v))

fu(r, v) ¸ γ > 0; on (¡1, h−(v) + γ] [ [h+(v)¡ γ,+1) , (38)

for some γ > 0 independent of v.We assume that for all (r, v) 2 IR£ IR

fv(r, v) · 0 (39)

We note that as a consequence of the above assumptions on f we have:

h±(v)! m±, h0(v)! m0 as v ! 0 (40)

Since, for Þxed v, the function u7! f(u, v) satisÞes the hypotheses of Aronson andWeinberger [AW] and Fife and McLeod [FM], there exists a unique pair (q(r, v), c(v))such that

qrr(r, v) + c(v)qr(r, v) = f(q(r, v), v) (41)

andlimr±∞

q(r, v) = h±(v) and q(0, v) = h0(v). (42)

We continue listing some technical assumptions that we will be making on (q(¢, v), c(v)) :

q(¢, v) and c(v) depend smoothly on v (43)

24

and(i) limv→0 supr[jvj(jqvj+ jqvvj) + jqrvj] = 0,

(ii) jrj−2jqrrj+ jrj−1jqrj · Ke−Kδ, for all jrj ¸ δ

(iii) q(r, v)! h±(v) exponentially fast as r ! §1

(iv) qr(r, v) > 0

(v) qv(r, v) = O(1), as v ! 0 locally uniformly wrt r

(44)

Finally we impose

c(0) = 0 and ¡c(v)v! c0 > 0. (45)

One example we have in mind is the one considered in [CHL] where

f(u, v) = 2u(u2 ¡ 1) + vh(u) (46)

with h < 0. In particular if h(u) ´ ¡C < 0 then by analogous computations in [BSS]one can show that c0 =

3

2C.

We recall that the notion of viscosity solution can be extended to integro-differentialequations of the form (36)(i) with f satisfying the monotonicity condition (39) (see,i.e. [AT]).Below for the readers convenience we give a sketch of proof of the comparison

result between viscosity bounded sub and supersolutions of the problem (36) for Þxedε > 0. Then the comparison result and the Perrons Method yield the existence of aunique continuous viscosity solution of (36). The existence and the uniqueness of asmooth solution of (36) without the assumption (39) is obtained in [CHL] in the caseof homogeneous Neumann boundary condition.

Lemma 3.1 Assume (37) and (39). Let u, v be respectively bounded lower and uppersemicontinuous viscosity sub- and supersolution of (36) (i)-(ii) with Þxed ε = ε0 > 0and u(x, 0) · v(x, 0). Then

u(x, t) · v(x, t) for all (x, t) 2 O £ [0, T ].

Proof.1. Without loss of generality, we can Þx ε0 = 1. For simplicity, we assume that

u, v are smooth functions and ∂O is C2. (For general case, one can combine thearguments in [CIL] and in [AT]).

25

2. Suppose that u¡v has a maximum δ0 > 0 at (x0, t0) in O£ [0, T ]. Fix a positiveconstant L and let

u(x, t) = e−Ltu(x, t), v(x, t) = e−Ltv(x, t).

Then u(x, t)¡v(x, t) has a maximum δ(L) > 0 in O£[0, T ]. Note that if the maximumis attained at a point (xL, tL), then e

LtLδ(L) · δ0.3. Consider the signed distance function d(x, ∂O) with d > 0 in O and extend

this function from a neighborhood of ∂O to O such that the extended function d(x)is C2 in O. Then for α > 0 the function

w(x, t) = u(x, t)¡ v(x, t) + εd(x)

has a maximum at Pα = (xα, tα) in O£ [0, T ], which converges to (xL, tL), tL > 0 withu ¡ v = δ(L) at (xL, tL). Suppose that xα 2 ∂O. Since at Pα we have Du(xα, tα) =Dv(xα, tα) + εn(xα) + λn(xα) with λ ¸ 0, by (A3) we have

G(xα, tα, Du) ¸ G(xα, tα, Dv + αn(xα)) > G(xα, tα,Dv),

which contradicts the deÞnition of u, v.4. Therefore xα 2 O and we have the following inequality at Pα:L(u¡ v) · ∆(u¡ v) + b(xα, tα) ¢ (Du¡Dv) + e−Ltα [f(v,

ROv)¡ f(u, R

Ou)]

· O(α) + e−Ltα[f(v, ROv)¡ f(u, R

Ou)].

Since u · v + eLtαδ(L), eLtαδ(L) is bounded and f(u, ¢) is nonincreasing, by sendingα! 0 we get

Lδ(L) · e−LtLK(1+ jOj)eLtLδ(L) =M(1+ jOj)δ(L),

where M is the global Lipschitz constant of the function f (since u, eLtLδ(L) arebounded and f is C2, we may assume that f is globally Lipschitz without loss ofgenerality). Thus we get a contradiction if we choose L > M(1+ jOj). ¤In the following Lemma we show some estimates satisÞed by the (uε)ε by using

the comparison result in Lemma 3.1.

Lemma 3.2 Under the assumptions of Lemma 3.1 the family (uε)ε satisÞes

m− + o(1) · uε(x, t) · m+ + o(1) as ε! 0 (47)

uniformly in O £ [0, T ].

26

Proof. We just sketch the proof, being it quite standard. We Þrst show that family(uε)ε is uniformly bounded in (x, t) 2 O £ [0, T ]. To this purpose, one observes thatthe functions u := Ct + M and u := ¡Ct ¡M with M,C > 0 large enough arerespectively super and subsolution of (36). Thus by comparing uε with u and u weobtain jjuεjj∞ · K for some K > 0.Now we prove that uε(x, t) · m++o(1) as ε! 0 (the other inequality being proved

in a similar way). Since jjuεjj∞ · K and fv · 0 we have that uε is a supersolution of(i) uε,t ¡∆uε + b(x, t) ¢Duε + ε−2f(uε, εKλ(O)) = 0 in O £ (0, T )

(ii) G(x, t,Duε) = 0 on ∂O £ (0, T ),

(iii) uε = u0 on O £ f0g,

(48)

We consider the solution of the odeúζ(s, ξ) + ε−2f(ζ, εKλ(O)) = 0 s 2 (0,1)

ζ(0, ξ) = ξ(49)

where ξ > max(jjgjj∞, jm+j). By using the properties of f one can show thatζ(s, ξ) · h+(εKλ(O)) + [ξ ¡ h+(εKλ(O))] exp(¡ε−2Ct)

for some C > 0. Since h+(εKλ(O))! m+ as ε! 0, we get ζ(s, ξ) · (m+ + o(1)) asε! 0. Finally having by construction uε(x, 0) · ξ we obtain uε(x, t) · m+ + o(1) asε! 0 and we conclude. ¤Under the current assumptions we expect that the front evolution associated with

the asymptotics of (36) is motion by mean curvature and an additional term depend-ing on the volume enclosed by the front. The corresponding geometric pde isut ¡ tr[(I ¡ cDu− cDu)D2u] + b(x, t) ¢Du¡ c0jDujµ(x, t, u(¢, t)) = 0 in O £ (0, T )

G(x, t,Du) = 0 on ∂O £ (0, T ),

u = u0 on O £ f0g,(50)

where µ(x, t, u(¢, t)) := m+λ(Ω+x,t(u))+m−λ((Ω+x,t(u))

c), Ω+x,t(u) = fu(y, t) > u(x, t)g,and c0 > 0 is deÞned in (45).

We premise the following Lemma where we show a key property of G which isused in the sequel to check the Neumann boundary condition. To formulate it, weuse the following notation : for p 2 IRN and x 2 ∂O, T (p) := p¡ p ¢ n(x)n(x). T (p)represents the projection of p on the tangent hyperplane to ∂O at x.

27

Lemma 3.3 Assume that (A2) and (A4) hold and that, for some x 2 ∂O, t 2 (0, T )and p 2 IRN , we have G(x, t, p) · 0 (resp. G(x, t, p) ¸ 0), then there exists a constantK(T ) such that, if p ¢ n(x) · ¡K(T )jT (p)j, then

G(x, t, p+ p) · 0 .(resp. if p ¢ n(x) ¸ K(T )jT (p)j, then

G(x, t, p+ p) ¸ 0 .)

We refer the reader to [BDL] for the proof of Lemma 3.3, we only remark that, by(A4), G(x, t, 0) ´ 0 and therefore the above result holds with p = 0.The main result of this Section is

Theorem 3.1 Assume (37), (44) and let uε be the solution of (36) where g : O! IRis a continuous function such that the set Γ0 = fx 2 O : g(x) = m0g is a nonemptysubset of O. Then, as ε! 0,

uε(x, t)! m+ fu > 0g,

locally uniformly inm− fu < 0g,

where u is the unique viscosity solution of (50) with u0 = d0, the signed distance toΓ0, which is positive in the set fg > m0g and negative in the set fg < m0g. If, inaddition, the no interior condition (26) holds, then, as ε! 0,

uε(x, t)!m+ fu > 0g,

locally uniformly in

m− fu > 0gc.Proof of Theorem 3.1. We consider the open sets Ω1 and Ω2 of sets deÞned inSection 3.1 by (30), (31) with b(x, t) ´ m+ and a(x, t) ´ m−.By following the abstract method in Section 3.1 we have to show that Ω10 and Ω

20

are not empty and the families (Ω1t )t>0, and ((Ω2t )c)t>0 are respectively a generalized

super-ßow and sub-ßow with normal velocity ¡F and angle condition G. We willconsider the Ω1-case, the Ω2-case being treated in a similar way. To this purpose weproceed as follows.1. Let us consider a smooth function θ: [0, T ] ! IR satisfying for all t 2 (0, T ),θ(t) · λ(Ωt) and a constant α > 0. By deÞnition of Ω1t and Lemma 3.2 for smallε > 0 we haveZ

O

uε dx ¸ZO

m+11Ωt +m−11Ωct dx¡ (m+ ¡m−)α

¸ (m+ ¡m−)θ(t) +m−λ(O)¡ (m+ ¡m−)α (51)

28

We denote byµ(t) := (m+ ¡m−)(θ(t)¡ α) +m−λ(O) (52)

and we set f ε(u, t) := f(u, εµ(t)) (for clarity of notations we drop the the dependenceon α and θ in µ and f ε). From (39) it follows that uε is a supersolution also of

(i) uε,t ¡∆uε + b(x, t) ¢Duε + ε−2f ε(uε, t) = 0 in O £ (0, T )

(ii) G(x, t,Duε) = 0 on ∂O £ (0, T ),

(iii) uε = u0 on O £ f0g,

(53)

Let mε,α+ and mε,α

− be the greatest and the smallest stable equilibria for f ε(u, t). Weintroduce the following set (depending on α and θ):

eΩ1(α, θ) := Intf(x, s) : lim inf∗ ∙uε(x, s)¡mε,α+ (s)

ε

¸¸ 0g (54)

and for all t 2 (0, T ] we seteΩ1t (α, θ) := eΩ1(α, θ) \ (O £ ftg). (55)

2. Now we use the following Lemma whose proof is postponed.

Lemma 3.4 For all α > 0 and for every smooth function θ: [0, T ] ! IR such that

θ(t) · λ(Ω1t ) for all t 2 (0, T ) we have Ω1 = eΩ1(α, θ).3. In [BDL] it is proved that eΩ1t (α, θ) 6= ; for t > 0 small, and (eΩ1t (α, θ))t isa generalized super-ßow in the sense of DeÞnition 2.2 with respect to the velocity¡F−(x, t,Dd,D2d) and angle boundary condition ¡G(x, t,Dd). Thus by combiningthis fact and Lemma 3.4 we get that Ω10 6= ; (namely the initialization of the front) and(Ω1t )t is a generalized super-ßow (namely the propagation of the front) with respectto the nonlocal normal velocity ¡F (x, t,Dd,D2d,Ω1t ) and angle boundary condition¡G(x, t,Dd) according the DeÞnition 2.1. The conclusion then follows from Corollary3.1. ¤

Remark 3.1 We remark that the initialization procedure can be proved also directlywithout introducing the set (54). More precisely one can reproduce the same argu-ments of the Þrst step of Theorem 2.3 in [BDL] by constructing globally in O sub-and supersolutions of (53) with f ε(u) := f(u, 0) § εK 0λ(O)), for a suitable choiceof K 0, associated with radially symmetric moving fronts. This can be done with-out the monotonicity condition (39), only the assumption that (uε)ε is uniformlybounded being necessary in order to replace in the equation (36) f(uε, ε

ROuε) with

f(u, 0)§ εK 0λ(O).

29

Now we turn to the proof of Lemma 3.4. We remark that this Lemma is aconsequence of the estimates on uε that the initialization step in [BDL] gives andwhich have been properly adapted to this case.

Proof of Lemma 3.4.It is enough to show that Ω1 µ eΩ1(α, θ), the other inclusion being trivially

satisÞed by the deÞnition of these sets. More precisely we are going to show thefollowing inequality

f(x, s) : lim inf∗ [uε(x, s)¡m+] ¸ 0g µ f(x, s) : lim inf∗∙uε(x, s)¡mε,α

+ (s)

ε

¸¸ 0g(56)

from which the inclusion Ω1 µ eΩ1(α, θ) follows (by considering the interior of theabove sets). To this end we follow the strategy of proof of Proposition 2.1 in [BDL].Let (x0, t0) 2 O £ (0, T ] be such that

lim inf∗ [uε(x0, t0)¡m+] ¸ 0.Then, by deÞnition of lim inf∗ , for any γ > 0 there exist h > 0, r > 0, ε > 0 such thatfor all ε < ε, x 2 BO(x0, r) and jt¡ t0j < h we have uε(x, t) ¸ m+ ¡ γ.We suppose that x0 2 ∂O (the case x 2 O being similar and even simpler). By

the smoothness of O, if η is small enough and if x := x0 ¡ ηn(x0) then B(x, η) µ Oand B(x, η) \ ∂O = fx0g. Consider the function ψη(x) = η2 ¡ jx ¡ xj2. We observethat Dψη(x0) ¢ n(x0) = ¡η < 0. Thus we can Þnd R > η and δ > 0 such thatB(x, R) µ B(x0, r) and the function

ψ(x) = R2 ¡ jx¡ xj2 (57)

satisÞes Dψ(x) ¢ n(x) < 0 on fx 2 O : jd(x)j < δg, d(¢) being the signed distancefunction to the set fx : ψ(x) = 0g. By Lemma 3.3, choosing R close enough to η, wemay have also G(x, t,Dψ(x)) < 0 on fx 2 ∂O : jd(x)j < δg for, say, any t · 1.We introduce the function Ψ:O £ [0, T ]! IR given by

Ψ(x, s) = ψ(x)¡ C(s¡ t0 + h), (58)

with C > 0 which will be chosen later and denote by d(¢, s) the signed distance to theset fΨ(¢, s) = 0g which is normalized to have the same signs of Ψ in O£ [0, T ]. Here

d(x, s) = [(R2 ¡ C(s¡ t0 + h))+]1/2 ¡ jx¡ xj .

By the choice of R, there is 0 < δ0 <(m+ ¡m0)

2^ δ such that for all 0 < δ < δ0 and

t 2 (t0 ¡ h, t0 + h) we haveuε(x, t) ¸ (m+ ¡ γ)11BO(x0,r)

¡K11(BO(x0,r))c¸ (m0 + 2δ)11Ψ(·,t)>0 ¡K11Ψ(·,t)≤0

30

where K is an upper bound of jjuεjj∞.Now we need the following two Lemmae whose proof is postponed in the Appendix

B.

Lemma 3.5 Under the assumptions of Theorem 3.1, for any β > 0, there are con-stants τ > 0 and ε (depending on β) such that for all 0 < ε · ε and for allt 2 (t0 ¡ h, t0 + h) we haveuε(x, t+tε) ¸ (mε,α

+ (t+tε)¡βε)11d(x,t+tε)≥β+(mε,α− (t+tε)¡βε)11d(x,t+tε)<β on O .

where tε = τε2j log εj.

Lemma 3.6 There exist h < h, β > 0, depending only on ψ deÞned in (57) suchthat, if β · β(ψ) and ε · ε(β,ψ), then there exits a subsolution wε,β of (53) inO £ (t0 ¡ tε, t0 + h) such that,wε,β(x, t0¡tε) · [mε

+(t0¡tε)¡βε]11d(·,t0−tε)≥β+[mε−(t0¡tε)¡βε]11d(·,t0−tε)<β in O.

Moreover, if (x, t) 2 O £ (t0 ¡ tε, t0 + h) satisÞes d(x, t) > 2β, then

lim inf∗

∙wε,β(x, t)¡mε,α

+ (t)

ε

¸¸ ¡2β .

We Þrst observe that if in Lemma 3.5 we choose t = t0 ¡ 2tε then for ε smallenough we have

uε(x, t0¡tε) ¸ (mε,α+ (t0¡tε)¡βε)11d(x,t−tε)≥β+(mε,α

− (t¡tε)¡βε)11d(·,t−tε)<β in O .

Then Lemma 3.6 yields a subsolution wε,β of (53)(i)-(ii) such that

wε,β(x, t0 ¡ tε) · uε(x, t0 ¡ tε) in O

thus by the maximum principle we have

wε,β(x, t) · uε(x, t) in O £ (t0 ¡ tε, t0 + h).Morever from (76) it follows that if t 2 (t0 ¡ tε, t0 + h), x 2 O \ BO(x0, r) andd(x, t) > 2β we have

lim inf∗

∙uε(x, t)¡mε,α

+ (t)

ε

¸¸ ¡2β .

Since β is arbitrary and does not depend on h and d(x0, t0) > 0 (by construction),we have

lim inf∗

∙uε(x0, t0)¡mε,α

+ (t0)

ε

¸¸ 0.

Thus we have proved the inequality (56) and we conclude the proof of Lemma 3.4.¤

31

Remark 3.2 The asymptotic result of this subsection continues to hold also in thecase of nonlinearities f depending on (x, t, u, v) provided the monotonicity condition(39) is satisÞed for all (x, t, u, v).

3.3 Some remarks on the asymptotics of nonlocal Allen-Cahnequation without monotonicity

In this subsection we extend the asymptotic result of the subsection 3.2 under asuitable relaxation of the monotonicity assumption (39).As we have already pointed out in the subsection 3.2 the monotonicity condition

on f allows us to have a rather simple proof of the asymptotics by replacing thenonlocal term with some suitable smooth functions of the time. On the other hand,as we observed in Remark 3.1, the initialization step can be proved without themonotonicity condition (39) and the proof of Lemma 3.6, which is related to thepropagation step, can be readily extended also to the case when uε is a supersolutionof

uε,t ¡∆uε + b(x, t) ¢Duε + ε−2f(uε, εµ(t)) = ε−1o(1) in O £ (0, T ), (59)

where µ(t) is the function deÞned in (52) and o(1) ! 0 as ε ! 0 locally uniformlyin (x, t) (that is trivially satisÞed in the monotone case). Indeed if we examine theproof of Lemma 3.6, we build a function wε,β satisfying for some positive constant ν(independent on ε)

wε,βt ¡∆wε,β + b(x, t) ¢Dwε,β + ε−2f(wε,β, εµ(t)) · ¡ε−1ν +O(1), as ε! 0 , (60)

locally in the space and in suitable small intervals (see (75)). Thus for ε > 0 smallenough we can compare uε and w

ε,β.In other words the asymptotic behaviour of (36) does not change if the error

between f(uε, εROuε) and f(uε, εµ(t)) is of the type εo(1) as ε! 0.

Now we give some sufficient conditions on f implying (59). Typically we have inmind the case when the monotonicity condition (39) holds between the two stableequilibria of W such as

f(u, v) = 2(u+ θ(x, t)v)(u¡m−)(u¡m+)

f(u, v) = 2(u+ h(u)v)(u¡m−)(u¡m+)

with θ(x, t), h(u) > 0.In this subsection we assume the hypotheses of subsection 3.2 except (39) and we

add the following assumptions on f :

(H1) f(m±, v) = 0 for all v 2 IR.

32

(H2) There exists C > 0 such that for every δ 2 (0,m+¡m−) we have fv(u, v) · Cδfor all u 2 [m− ¡ δ,m+ + δ] and for all v 2 IR.

(H3) fv(u, v) · 0 for all u 2 [m−,m+] and for all v 2 IR.(H4) For all ε > 0 there exists a smooth solution uε of (36) and the family u

ε isuniformly bounded in O £ [0, T ].

Proposition 3.1 Under the current assumptions the family (uε)ε satisÞes

m− ¡ L exp(¡ε−2Kt) · uε(x, t) · m+ + L exp(¡ε−2Kt)

for suitable L,K > 0 and for (x, t) 2 O £ (0, T ).

Proof. We prove the inequality

uε(x, t) · m+ + L exp(¡ε−2Kt)

the other one being proved in an analogous way. Let us introduce the functionM ε+(t) := supO(u

ε(x, t)¡m+)+ and we claim that it is a viscosity subsolution of the

following variational inequality:

min(ζ(t), úζ(t) + ε−2γ(ζ(t))) = 0 for all t 2 (0, T ), (61)

where γ > 0 is the constant appearing in (38). To this purpose, let ψ be a smoothfunction of the time t such that Mε

+(t) ¡ ψ(t) has a global maximum at t 2 (0, T ),and M ε

+(t) = ψ(t). If Mε+(t) = 0 then (61) is trivially satisÞed. Otherwise M

ε+(t) =

(uε(x, t)¡m+)+ with x 2 O.We Þrst assume thatM ε

+(t) = uε(x, t)¡m+ with x 2 O.

We note that uε ¡ ψ has a maximum at (x, t) as well. Thus ψt(t) = uεt(x, t) andDuε(x, t) = Dψ(t) = 0. Since uε is a smooth solution of (36) we have

ψt(t) + ε−2f(uε(x, t), ε

ZO

uε(x, t) dx) = 0.

On the other hand, since uε(x, t) > m+ and (38), (H1) hold, we have

f(uε(x, t), ε

ZO

uε(x, t) dx) ¸ fu(ξ, ε

ZO

uε(x, t) dx)(uε(x, t)¡m+)

¸ γ(uε(x, t)¡m+)

where m+ · ξ · uε(x, t). Therefore we have

min(ψt(t),ψt(t) + ε−2γ(M ε

+(t))) · 0. (62)

33

Now we assume that M ε+(t) = u

ε(x, t)¡m+ with x 2 ∂O. We introduce the function

Ψ(x, t) = uε(x, t)¡ ψ(t) + αd(x),

where α > 0 and d(¢) is the signed distance function from ∂O. Let (xα, tα) be asequence of maximum points of Ψ. By the same arguments of Lemma 3.1 one canshow that xα 2 O and uε(xα, tα) ¡m+ converges to M

ε+(t) as α ! 0. We may also

assume that uε(xα, tα)¡m+ ¸ 0. Thus we have

0 = ψt(tα) + ε−2f(uε(xα, tα), ε

ZO

uε(x, tα) dx)

¸ ψt(tα) + ε−2γ(uε(xα, tα)¡m+)

By letting α! 0 we get (62). Hence we prove the claim.Now let us consider the solution of the ode:½

úζ(t, ξ) + ε−2γζ(t, ξ) = 0 in (0, T )ζ(0, ξ) = ξ

with ξ ¸ jjuεjj∞. Easy computations show that

ζ(t, ξ) · L exp(¡ε−2Kt) for all t 2 (0, T )

for some positive constants L,K independent on ε. We note that ζ is a supersolutionof min(ζ, úζ(t, ξ)+ε−2γζ(t, ξ)) = 0. SinceM ε(0) · ζ(0) then by maximum principle wehave M ε(t) · ζ(t) for all t > 0. In particular we get uε(x, t) · m+ + L exp(¡ε−2Kt)for all (x, t) 2 O £ (0, T ) and we conclude. ¤To conclude this subsection, we observe that Proposition 3.1, the hypothesis (H2)

and the fact that jjuεjj∞ ·M imply

f(uε, ε

ZO

uε(x, t) dx)¡ f(uε, εµ(t)) · εC exp(¡ε−2Kt) (63)

for some C > 0 (depending on jjuεjj∞, m±) and for all t 2 (0, T ). Hence for allt0 2 (0, T ) and for all h > 0, uε is a supersolution also of

uε,t ¡∆uε + b(x, t) ¢Duε + ε−2f ε(uε, t) = o(ε) in O £ (t0 ¡ h, t0 + h). (64)

Thus the proofs of Lemma 3.5 and 3.6 can be adapted under the current assumptionsand consequently Lemma 3.4 continues to hold.

34

Appendix A

In this appendix we list two sets of additional assumptions on the open set O, andthe functions F and G under which the Comparison Theorem 1.1 holds: the Þrst setof conditions is imposed in Ishii and Sato [IS] and the second one is imposed in Barles[B].

(A6a) (Ishii and Sato [IS])

² G(x, p) 2 C(IRN £ IRN ) \ C1,1(IRN £ IRNnf0g).² There exists a function ω : [0,1)! [0,1) continuous at 0 satisfying ω(0) = 0such that if X,Y 2 S(N) and µ1, µ2 2 [0,1) satisfyµ

X 00 ¡Y

¶· µ1

µI ¡I¡I I

¶+ µ2

µI 00 I

¶,

then

F (y , t, q, Y,K)¡ F (x, t, p,X,K) ·ω³µ1(jx¡ yj2 + ( jp¡ qjjpj ^ jqj ^ 1)

2) + µ2 + jp¡ qj+ jx¡ yj(jpj _ jqj+ 1)´

for all t 2 (0, T ] , x, y 2 O, p, q 2 IRNnf0g and K 2 B, where jpj ^ jqj =min(jpj, jqj) and jpj _ jqj = max(jpj, jqj).

(A6b) (Barles [B])

² ∂O is W 3,∞.

² For every T > 0, there exists ϑ(T ) > 0 such that for all t, s 2 (0, T ], x, y 2 ∂Oand p, q 2 IRN

jG(x, t, p)¡G(y, s, q)j · ϑ(T ) [jp¡ qj+ (jpj+ jqj)(jx¡ yj+ jt¡ sj)]

² For every C > 0, there exists a function ωC : [0,1) ! [0,1), continuousat 0 with ωC(0) = 0, such that for all η > 0, for all t 2 [0,1), x, y 2 O,p, q 2 IRNnf0g, and X, Y 2 S(N) satisfying

¡Cηε2

µI 00 I

¶·

µX 00 ¡Y

¶· Cη

ε2

µI ¡I¡I I

¶+ Cη

µI 00 I

¶jp¡ qj · Cε(jpj ^ jqj)jx¡ yj · Cηε,

35

the following holds for all K 2 B:F (y, t, q, Y,K) ¡ F (x, t, p,X,K)

· ωC

µη +

jx¡ yj2ε2

++jx¡ yj(jpj _ jqj+ 1)¶

The main difference between these two sets of assumptions is that the conditionsof Ishii and Sato [IS] require less regularity on the domain (that is that ∂O is C1 whichis also our basic assumption on O), whereas the conditions in Barles [B] require lessregularity on the boundary condition (that is G is uniformly continuous, which is apart of (A3)), moreover G may depend also on t. For example in the case of theoblique derivative boundary condition G(x, t, p) = γ(x, t) ¢ p, in [IS] it is requiredthat γ does not depend on t and it is C1,1 whereas in [B] it is assumed that γ 2W 1,∞(∂O £ (0,1)).

Appendix B

In this appendix we give the proof of Lemmae 3.5 and 3.6.Proof of Lemma 3.5. We essentially follow the lines of the proof of Step 1 ofTheorem 2.3 in [BDL] and we reproduce here some key points for sake of completenessand refer the reader to Subsection 2.3 in [BDL] for all the details.1. We modify the function f ε taking in account the t-dependence and, to do so, wedo it here in the following way : because of the assumptions on f ε, there exists afunction r 7! fδ(u) (0 < δ < δ0) such that, for every T > 0, if ε is small enough,fδ(u) ¸ f ε(u, t) + 2ε for any u 2 IR and t 2 [0, T ]. Moreover fδ is a cubic typenonlinearity satisfying (37) with three zeros which are m−¡ δ, m0+ δ/2 and m+¡ δ.We modify the function f ε in two steps; we Þrst introduce a smooth cut-off func-

tion ζ1 2 C∞0 (IR) such that 0 · ζ1 · 1 in IR, ζ1(u) = 1 in (m0 ¡ δ,m0 + δ) andζ1(u) = 0 for u · m0 ¡ 2δ and u ¸ m0 + 2δ. We set

f εδ (u, t) = ζ1(u)fδ(u) + (1¡ ζ1(u)) [f ε(u, t) + εβ)] , (65)

where 0 < β · 1. Using the assumptions on f ε, it is easy to see that, for δ smallenough, f εδ has the same regularity properties as the f

ε and has exactly three zeros,mε,α− (t) + O(βε), m0 + δ/2, m

ε,α+ + O(βε); moreover f εδ ¸ f ε on IR with f εδ (u) =

f ε(u, t) + εβ if ju¡m0j ¸ 2δ and f εδ is independent of t for ju¡m0j · δ. 2. Then weconsider another cut-off function ζ2 2 C∞0 (IR) such that 0 · ζ2 · 1 in IR ζ2(u) = 0in (¡1,m0+ δ/4][ [m0+ δ,+1) and ζ2(s) = 1 in [m0+ δ/3,m0+2δ/3]. Finally weconsider

f εδ (u, t) = (1¡ ζ2(u)) f εδ (u, t) + ζ2(u)δ2+m0 ¡ uj log εj .

36

We note that, because again of the properties of fε, fεδ has exactly three zeros :mε,α− (t)+O(βε), m0+δ/2, m

ε,α+ (t)+O(βε); moreover, for ε small enough, f

εδ ¸ f ε+βε

in IR and f εδ = fε + εβ for ju¡m0j ¸ 2δ.

2. We consider the solution χ(ξ, ¢, t) of the ode½úχ+ fεδ (χ, t) = 0χ(ξ, 0, t) = ξ 2 IR, (66)

3. Tedious computations show that χ satisÞes

χξ(ξ, s, t) > 0 in IR£ [0,+1)£ [0,+1),for all β > 0, T > 0, there exists a(β, δ, T ) > 0 such that

χ(ξ, s, t) ¸ mε+(t)¡ βε for s ¸ aj log εj and ξ ¸ δ +m0

for all t 2 [0, T ],

(67)

andfor every a, T > 0, there exists M(a, T ) 2 IR such that, for ε small enough,

(χξ(ξ, s, t))−1jχξξ(ξ, s, t)j · ε−1M(a, T ) for 0 < s · aj log εj,

for all t 2 [0, T ].(68)

4. If δ is small enough, for every a > 0 and T > 0, there exists fM(a, T ) > 0 suchthat, for ε small enough and for 0 < s · aj log εj, we have

jχt(ξ, s, t)j · fM(a, T )ε .5. Let ϕ be a smooth function such that

¡K · ϕ · m0+2δ in IR, ϕ(z) = K in fz < 0g and ϕ(z) = m0+2δ on fz ¸ δg.(69)

Fix t 2 (t0 ¡ h, t0 + h) and deÞne w:O £ [0, T ]! IR by

w(x, s) = χ(ϕ(d(x, s))¡ M(s¡ t)ε

,s¡ tε2

, s).

Then w is a viscosity subsolution of 53(i)¡(ii) in O£(t, t+ tε), where tε = aε2j log εj.6. By construction we have

uε(x, t) ¸ (m0 + 2δ)11d(x,t)≥δ ¡K11d(x,t)<δ on O .

37

and on the other hand,

w(x, t) = χ(ϕ(d(x, t)), 0, x, t) = ϕ(d(x, t))

· (m0 + 2δ)11d(x,t)≥δ ¡K11d(x,t)<δ.Thus, the maximum principle yields that

w(x, s) · uε(x, s) on O £ [t, t+ aε2j log εj] . (70)

Evaluating (70) for s = t+ aε2j log εj and for x 2 O such that d(x, s) ¸ δ we getχ(m0 + 2δ ¡Kaεj`nεj, aj log εj, x, aε2j log εj) · uε(x, aε2j log εj+ t)

But, since for ε small enough

m0 + 2δ ¡Kaεj log εj ¸ m0 + δ,

it follows from (67) that

mε,α+ (t+ tε) +O(βε) · uε(x, t+ tε) if d(x, t+ tε) ¸ δ .

7. Finally, because of the properties of fεδ , if a is large we have also χ(ξ, aj log εj, x, aε2j log εj+t) ¸ mε,α

− (t+ tε) +O(βε) for all bounded ξ. Therefore for all x 2 O we havemε,α− (t+ tε) + O(βε) · uε(x, t+ tε) for any x 2 O

and

[mε,α+ (t+tε)+O(βε)]11d(x,t+tε)≥δ+[m

ε,α− (x, t+tε)+O(βε)]11d(x,t+tε)<δ · uε(x, t+tε).

Then the result holds for τ = a by taking β < δ small enough in order to replace ifnecessary O(βε) by βε. ¤Proof of Lemma 3.6.Also in this case, since the proof is very similar to the ones of Lemma 2.3 and

Theorem 2.4 in [BDL], we just outline the kea ideas by refering the reader to [BDL]for the details.We consider the smooth function Ψ given by (58) and we observe that for C > 0

large enough one has for some % > 0

∂Ψ

∂t(x, s) + F ∗(x, s,DΨ, D2Ψ, µ(s)) < ¡% in O £ (0, T ) .

On an other hand, we have also

G(x, s,DΨ(x, s)) < ¡% ,

38

on a ∂Oneighborhood of fΨ = 0g and for small h. Using the smoothness of Ψ andthe fact that, for small h, DΨ(x, s)6= 0 if Ψ(x, s) = 0, there exist γ > 0 and h · hsuch that d is smooth in the set Qγ,h = f(x, s) : jd(x, s)j · γ , t0 ¡ h · s · t0 + hgand jDΨj 6= 0 in Qγ,h. We note that on the set [t0−h≤s≤t0+hfΨ(x, s) = 0g, d satisÞes

dt + F∗(x, s,Dd,D2d, θ(s)¡ α) = dt ¡∆d+ b(x, s) ¢Dd

¡c0jDdj[(m+ ¡m−)(θ(s)¡ α) +m−λ(O)] · ¡ %

2jDΨj .

Moreover recalling the properties of Ψ on ∂O, we have also

G(x, t,Dd) · ¡ %

2jDΨj on ∂O \Qγ,h . (71)

We note that we can choose C, h such that d(x0, t0) > 0 and we may assume that

jDdj = 1 and D2dDd = 0 in Qγ,h .

Let (qε(r, s), cε(s)) be the unique pair satisfying

qεrr(r, s) + cε(s)qεr = f

ε(qε, s). (72)

We consider in Qγ,h a function of the form

vε(x, s) = qε(ε−1(d(x, s)¡ 2β), s)¡ 2βε (73)

We verify that vε is a viscosity solution of (53)(i)-(ii) in Qγ,h. As far as the boundarycondition (53)(ii) is concerned, we Þrst observe that Dv(x, s) = ε−1qrDd and thusG(x, s,Dv(x, s)) = ε−1qεrG(x, s,Dd(x, s)) < 0 because of (A4) and (71).Moreover we have

vεt ¡∆vε + b(x, s) ¢Dvε + ε−2f ε(vε, s) = ε−2Iε + ε−1IIε + IIIε , (74)

where

Iε = ¡qεrr ¡ cε(s)qεr + f ε(qε, s),IIε = qεr(dt ¡∆d+ b(x, s) ¢Dd+ ε−1cε(s))¡ 2βf εu(qε, s)IIIε = qεt +O(1).

We note that ε−1cε(s)! ¡c0µ(s) as ε! 0 locally uniformly in s and for all α.By analogous computations to the one in [BS], one can see that if β is small

enough then vε satisÞes for some constant ν(%,β) < 0

vεt ¡∆vε + b(x, s) ¢Dvε + ε−2fε(v, s) · ε−1ν(%,β) +O(1) as ε! 0 (75)

for all (x, s) 2 Qγ,h. Next we extend the subsolution vε to O£ (t0¡ tε, t0+ h) and wedo it in two steps.First we have

39

Lemma 3.7 For ε small enough, the functions gε± deÞned on [0, T ] by gε,α± (s) =mε,α± (s)¡ εβ are viscosity subsolutions of (53)(i)-(ii).We leave the proof of Lemma 3.7 to the reader since it follows rather easily from

the properties of f ε, mε,α+ and mε,α

− .The next step is to deÞne the function vε : f(x, s) 2 O£ [t0¡ tε, t0+ h] : d(x, s) ·

γg ! IR by

vε(x, s) =

sup(vε(x, s), gε,α− (s)) if d(x, s) > ¡γ

gε,α− (s) otherwise

By similar computations of Lemma 4.2 in [BS] and using Lemma 3.7, it is easy toprove that vε is a viscosity subsolution of (53)(i)-(ii).Then we choose a smooth function ϕ : IR ! IR such that ϕ0 · 0 in IR, ϕ = 1 in

(¡1, γ/2), 0 < ϕ < 1 in (γ/2, 3γ/4), ϕ = 0 in (3γ/4,+1), and, Þnally, ϕ00 · 0 in aneighborhood of γ/2.The function wε,β : O £ [t0 ¡ tε, t0 + h]! IR deÞned by

wε,β(x, s) =

ϕ(d(x, s))vε(x, s) + (1¡ ϕ(d(x, s)))gε,α+ (s) if d(x, s) < γ,

gε,α+ (s) otherwise,

is a viscosity subsolution of (53)-(10)(ii) on O £ (t0 ¡ tε, t0 + h), if ε and h aresufficiently small. Moreover

wε,β(¢, t0¡tε) · (mε,α+ (t0¡tε)¡βε)11d(x,t0−tε)≥β+(mε,α

− (t0¡tε)¡βε)11d(x,t0−tε)<β in O

and if s 2 (t0 ¡ tε, t0 + h), x 2 O \ B(x0, r) and d(x, s) > 2β, then

lim inf∗

∙wε,β(x, s)¡mε,α

+ (s)

ε

¸¸ ¡2β . (76)

Thus we can conclude.

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