Homogenization of a semilinear heat equation

Homogenization of a semilinear heat equation Tome 4 (2017), p. 633-660.

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HOMOGENIZATION OF A SEMILINEAR HEAT EQUATION

by Annalisa Cesaroni, Nicolas Dirr & Matteo Novaga

Abstract. — We consider the homogenization of a semilinear heat equation with vanishing viscosity and with oscillating positive potential depending onu/ε. According to the rate between the frequency of oscillations in the potential and the vanishing factor in the viscosity, we obtain different regimes in the limit evolution and we discuss the locally uniform convergence of the solutions to the effective problem. The interesting feature of the model is that in the strong diffusion regime the effective operator is discontinuous in the gradient entry. We get a complete characterization of the limit solution in dimensionn= 1, whereas in dimensionn >1we discuss the main properties of the solutions to the effective problem selected at the limit and we prove uniqueness for some classes of initial data.

Résumé(Homogénéisation d’une équation de la chaleur semi-linéaire). — Nous considérons l’homogénéisation d’une équation de la chaleur semi-linéaire avec viscosité tendant vers0et un potentiel positif oscillant dépendant deu/ε. Suivant le rapport entre la fréquence des oscillations dans le potentiel et le facteur tendant vers0dans la viscosité, nous obtenons différents régimes de l’évolution limite et nous discutons la convergence uniforme locale des solutions du problème effectif. L’aspect intéressant du modèle est que, dans un régime à forte diffusion, l’opérateur effectif est discontinu comme fonction du gradient. Nous obtenons une caractérisation complète de la solution limite en dimensionn = 1, alors qu’en dimensionn > 1 nous analysons les propriétés principales des solution du problème effectif sélectionné à la limite, et nous montrons l’unicité pour certaines classes de données initiales.

Contents

1. Introduction. . . 634

2. Notation and preliminary definitions. . . 637

3. Assumptions and basic estimates. . . 638

4. Cell problem and asymptotic speed of propagation . . . 640

5. Convergence of solutions. . . 645

6. Open problems. . . 658

References. . . 659

Mathematical subject classification (2010). — 35B27, 74Q10, 35D40.

Keywords. — Homogenization, parabolic equations, vanishing viscosity.

A. C. was partially supported by the GNAMPA Project 2016 “Fenomeni asintotici e omogeneiz- zazione” and by the research project of the University of Padova “Mean-Field Games and Nonlinear PDEs”. N. D. was partially supported by the University of Padova, the Leverhulme Trust RPG-2013- 261 and EPSRC EP/M028607/1. M. N. was partially supported by the University of Pisa via grant PRA 2017 “Problemi di ottimizzazione e di evoluzione in ambito variazionale”.

1. Introduction We consider the following problem:

(1.1)

(u^εt−ε^α∆u^ε−g(u^ε/ε) = 0 in Rⁿ×(0,+∞) u^ε(x,0) =u0(x) in Rⁿ

whereα>0and the potentialgis a periodic, Lipschitz continuous and positive func- tion. This is a simple model for the motion of an interface in a heterogeneous medium, modeled by g. These kind of equations arise e.g. in the study of the propagation of flame fronts in a solid medium having horizontal periodic striations, see the appen- dix of [19] for a survey of the physical background motivating this equation (see also [9, 1]).

In this paper we show that, depending on the value of α, different regimes arise in the limit evolution. If α = 1, then u^ε converges locally uniformly to the unique Lipschitz continuous viscosity solution to

(1.2)

(ut−c(|∇u|) = 0 in Rⁿ×(0,+∞) u(x,0) =u0(x),

wherec: [0,+∞)→[0,+∞)is a continuous, nondecreasing and nonnegative function, which satisfies

c(0) = Z 1

0

1 g(s)ds

⁻¹

and lim

|p|→+∞c(|p|) = Z 1

0

g(s)ds.

In particular,

Z 1 0

g(s)⁻¹ds ⁻¹

6c(|p|)6 Z 1

0

g(s)ds,

and the second inequality is strict ifg is nonconstant. Since the solutionuto (1.2) is Lipschitz inx, with the same Lipschitz constant of the initial datum, necessarily the average speed is less thanc(k∇u0k^∞).

In the caseα >1the limit problem is very simple and reads

(1.3)





 ut=

Z 1 0

g(s)⁻¹ds ⁻¹

inRⁿ×(0,+∞) u(x,0) =u0(x).

In particular, the solutions to (1.1) converge locally uniformly to u0(x) +t

Z 1 0

g(s)⁻¹ds ⁻¹

.

In the case0< α <1, the limit problem is (1.4)

(ut−c−(|∇u|) = 0 in Rⁿ×(0,+∞) u(x,0) =u0(x),

where

c−(|p|) =









 Z 1

0

g(s)ds p6= 0 Z 1

0

g(s)⁻¹ds ⁻¹

p= 0.

In the limiting caseα= 0, the limit problem is given by (1.5)

(ut−F(∇u,∇²u) = 0 in Rⁿ×(0,+∞) u(x,0) =u0(x)

where

F(p, X)) =







trX+R1

0 g(s)ds p6= 0

min

trX+ Z 1

0

g(s)ds, Z 1

0

g(s)⁻¹ds −1

p= 0.

The functions c− and F are both discontinuous functions. Such a phenomenon is unusual in homogenization problems, and makes the analysis of this limit more chal- lenging.

Due to the lack of uniqueness of solutions to Hamilton-Jacobi equations with dis- continuous Hamiltonian, in this case we prove that along subsequences the solutionu^εα

to (1.1) converges locally uniformly to a viscosity solution of the limit problem. We also provide a quite detailed description of which are the solutions of the discontinuous problem selected in the limit, and we identify the asymptotic speed of propagation at strict maxima, at strict minima and at saddle points (with respect tox) of the limit function. This result enables us to obtain a complete description of the limit function for some classes of initial data. In particular, if the initial data are either monotone in one direction or convex, we prove that the solutions to (1.1) converge locally uniformly to u0(x) +tR1

0 g(s)dswhen α∈(0,1), and to the solution of ut−∆u=R1 0 g(s)ds with initial datumu0whenα= 0. If, on the other hand, the initial datum is a radially symmetric function, which has a unique maximum point, then the limit function is given, forα∈(0,1), by

min

u0(x) +t Z 1

0

g(s)ds, maxu0+t Z 1

0

g(s)⁻¹ds ⁻¹

.

As a consequence we show further properties of the limit functionu, when the initial datum is bounded from above.

In particular, in the one-dimensional case, we are able to prove the full convergence of the solutions u^ε_α and the uniqueness of the limit function u for α ∈ (0,1), see Theorem 5.12.

Our homogenization results are based on maximum principle type arguments. In particular, we provide the effective limit problem through the solution of the so-called cell problem, and then we prove the convergence of solutions by a suitable adaptation of the perturbed test function method proposed by Evans. The cell problem in our case reduces to an ordinary differential equation, see (4.3), which permits to define

the limit differential operator (in terms of the asymptotic speed of propagation of pulsating traveling waves) and to introduce the so-called correctors, which play the role of local barriers for the evolution.

Throughout the paper we shall assume that the potential g is strictly positive.

Nevertheless we expect that similar results hold also in the case of a functiongwhich possibly changes sign and satisfiesR1

0 g(s)ds >0. In this case, though, the analysis of the cell problem is much more involved. In the limiting case, i.e., whenR1

0 g(s)ds= 0, the cell problem has been studied in [15] and [1], for α>1, and it is possible to prove that the solution u^ε(x, t) of (1.1) converges locally uniformly to the initial datum u0(x). More precisely, in [15] the following long time rescaling of (1.1) has been considered:

(1.6) u^ε_t−∆u^ε+ 1

ε^αg(u^ε/ε) = 0,

showing that u^ε converge locally uniformly to a solution of a quasilinear parabolic equation, see (6.1), which forα >1 is the level set mean curvature equation. In [1], the1-dimensional case has been considered forα= 1, in a more general setting.

Homogenization of periodic structures has been studied by viscosity solution meth- ods in a long series of papers, we just recall [4, 5, 8, 16] and references therein.

However, only few papers deal with homogenization of equations depending (period- ically) on u/ε, as in our case, besides the already cited works [15, 1]. For first order Hamilton-Jacobi equations we recall [14, 3]. Eventually, in [13, 20] the homogeniza- tion of ordinary differential equations such asu⁰_ε(t) =g(t/ε, uε/ε)have been studied, using respectively viscosity solutions and G-convergence methods.

One of the main step to solve the homogenization problem is the identification of the limit operator, as we already noted. This is done by solving a suitable defined cell problem, or equivalently, by looking to pulsating wave solutions to the equation (1.1), at the microscopic scale. Pulsating wave solutions with (average) slope p ∈ Rⁿ are solutions to (1.1) withε= 1of the formφ(x, t)−c(p)t, whereφ(x, t)−p·xis a space- time periodic function and c(p) is the (average) speed of the solution. Notice that, since g depends only on u, these pulsating waves are in fact traveling waves which moves horizontally in the p-direction. Such solutions are related to the correctors used in homogenization problems and are very important in the analysis of long time behavior of the solutions to (1.1), with ε= 1, since typically they are the long time attractors of such solutions, see for instance [7, 6, 10, 11, 9, 19]. In particular, in [19], the existence of horizontal (e.g. with slope p= 0) pulsating wave solutions to ut =

∆u+g(x, u,∇u)is proved, wheregis a positive function, which is periodic inx, u. The same argument also applies to get existence of pulsating wave solutions for rational slopesp∈Qⁿ. In [9] a similar problem has been studied in the plane, that is existence for any slopepof pulsating wave solutions (which are traveling horizontally) to

(1.7) ut=δuxx+g(u)p

1 +u²_x,

withgstrictly positive. The authors also provide a complete description of the asymp- totic speed of propagationc(p), showing that it is increasing with respect to|p|(as in

our case) and looking also at the limit behavior as the viscosity is vanishing, that is, δ → 0. Eventually, in [17, 18] a geometric variant of (1.7) has been considered, for which the author is able to construct planar and V-shaped pulsating waves.

The paper is organized as follows. In Section 2 we recall some notation used in the paper, including the definition of viscosity solutions. In Section 3, we introduce the problem and the assumptions and provide a priori estimates on solutions to (1.1) and on their uniform limits. Section 4 is devoted to the solution to the cell problem, in the caseα= 1and then in the caseα6= 1, and on the analysis of qualitative properties of the limit operators. In Section 5 we prove the main results, that is the homogenization limits. Eventually in Section 6 we discuss some open problems, which in our opinion could be interesting to investigate.

Acknowledgements. — The second and third authors wish to thank the University of Padova for the kind hospitality during the preparation of this work.

2. Notation and preliminary definitions

Givenz∈R, we will denote with[z]the smallest integer bigger than z:

[z]∈Z, [z]−1< z6[z].

Given a smooth function u(x, t) : Rⁿ ×(0,+∞) → R, we will denote with ut the partial derivative with respect tot, with∇u,∇²u,∆uresp. the gradient, the Hessian and the Laplacian ofuwith respect tox.

Given a continuous functionu:Rⁿ×(0,+∞)→R, we recall the definition of the sub and superjets ofuat a point(x0, t0)∈Rⁿ×(0,+∞)(see [2], [12]):

J⁺u(x0, t0) :=

(∇φ(x0, t0),∇²φ(x0, t0),φt(x0, t0)) :

φ∈C², φ>u, φ(x0, t0) =u(x0, t0) , J⁻u(x0, t0) :=

(∇φ(x0, t0),∇²xφ(x0, t0),φt(x0, t0)) :

φ∈C², φ6u, φ(x0, t0) =u(x0, t0) . We recall the definition of viscosity solution for a parabolic system

(2.1)

(ut−F(∇u,∇²u) = 0 inRⁿ×(0,+∞) u(x,0) =u0(x) inRⁿ,

where the differential operatorF is possibly discontinuous (see [12]). Given a contin- uous functionu:Rⁿ×[0,+∞)→R, then

– u is a subsolution to (2.1) if u(x,0) 6u0(x) and λ−F^?(p, X) 6 0, for every (x0, t0)∈Rⁿ×(0,+∞)and(p, X, λ)∈J⁺u(x0, t0),

– uis a supersolution to (2.1) if u(x,0) >u0(x)and λ−F?(p, X)>0, for every (x0, t0)∈Rⁿ×(0,+∞)and(p, X, λ)∈J⁻u(x0, t0),

where F^? andF? denote respectively the upper and lower semicontinuous envelopes ofF.

3. Assumptions and basic estimates

We assume the following conditions on the forcing termg:R→R: (3.1) gis Lipschitz continuous ,Zperiodic, andg(y)>0for everyy.

We consider the following Cauchy problem (3.2)

(u^εt−ε^α∆u^ε−g(u^ε/ε) = 0 in Rⁿ×(0,+∞) u^ε(x,0) =u0(x) in Rⁿ

whereα>0and

(3.3) u0is a Lipschitz continuous function, with Lipschitz constantL.

We can assume without loss of generality thatL∈N. In the caseα= 0, we make the additional assumption thatu0∈C^1,1.

Proposition 3.1. — Assume (3.1) and (3.3) and let α > 0. Then (3.2) admits a unique solution u^ε_α∈C^2+γ,1+γ/2 for allγ∈(0,1). Moreover, up to a subsequence,

u^ε_α−→u locally uniformly inRⁿ×[0,+∞).

Forα >0, every limit function uis a Lipschitz continuous function, which satisfies (3.4) |u(x, t)−u(y, s)|6L|x−y|+kgk^∞|t−s| ∀x, y∈Rⁿ, t, s>0.

Forα= 0, under the additional assumption thatu0∈C^1,1(Rⁿ), every limit functionu is a Lipschitz continuous function, which satisfies

(3.5) |u(x, t)−u(y, s)|6L|x−y|+(kgk^∞+k∇²u0k^∞)|t−s| ∀x, y∈Rⁿ, t, s>0.

Finally, if there existη∈Rⁿ, with|η|= 1, andδ >0 such that (3.6) ∇u0(x)·η>δ for a.e.x∈Rⁿ, then∇u(x, t)·η>δ for a.e.(x, t).

Proof. — Due to the Lipschitz regularity ofg, a standard comparison principle among sub and supersolutions to (3.2) holds (see [21]). So, existence and uniqueness of solu- tions to (3.2) follow easily, and the regularity comes from standard elliptic regularity theory (see [21]).

Assume now that the initial datum u0 has bounded Hessian. Indeed it is not re- strictive, since we can uniformly approximate the initial datum with a sequence of smooth functions with bounded Hessian. The comparison principle implies that the associated sequence of solutions converges locally uniformly to the solution to (3.2) in Rⁿ×[0,+∞).

Let C = k∇²u0k^∞. Then for every ε, the functions u0(x)±(kgk^∞+ε^αC)t are respectively super and subsolution to (3.2), which implies by the comparison principle that

|u^εα(x, t)−u^εα(x,0)|6(kgk^∞+Cε^α)t.

Hence, again applying the comparison principle, we get that for every t, s>0 (3.7) |u^ε_α(x, t+s)−u^ε_α(x, t)|6sup

x |u^ε_α(x, s)−u^ε_α(x,0)|6(kgk^∞+Cε^α)s.

This implies thatu^εare equi-lipschitz int.

We prove now uniform equi-continuity inx. Let us consider the functions w^ε_±(x, t) :=u^ε_α(x+z, t)±

L|z|/ε ε.

Notice thatw_±^ε are both solutions to the equation in (3.2), due to the periodicity ofg.

Moreover, we have

w^ε₊(x,0) =u0(x+z) + L|z|/ε

ε>u0(x+z) +L|z|>u0(x) w₋^ε(x,0) =u0(x+z)−

L|z|/ε

ε6u0(x+z)−L|z|6u0(x).

and

By the comparison principle this implies that (3.8) |u^εα(x+z, t)−u^εα(x, t)|6

L|z|/ε

ε6L|z|+ε ∀x, z∈Rⁿ, t>0.

In particular, if we takez=εzwherez∈Zⁿ, then (3.8) gives

|u^ε_α(x+εz, t)−u^ε_α(x, t)|

ε|z| 6L ∀x∈Rⁿ, z∈Zⁿ, t>0.

For everyε >0we consider a Lipschitz continuous functionue^ε_α, which satisfies (3.7),

|eu^ε_α(x, t)−ue^ε_α(y, t)|6L|x−y| for everyx, y∈Rⁿ and t>0,and such that u^ε_α≡eu^ε_α on the latticeεZⁿ×(0,+∞). This implies thatku^εα−eu^εαk^∞6Kε. Indeed, letx∈Rⁿ andt >0. Fix yε∈εZⁿ such that |x−yε|6ε. Then, using (3.8) and the definition ofue^ε_α, we get

|u^ε_α(x, t)−ue^ε_α(x, t)|6|u^ε_α(x, t)−u^ε_α(yε, t)|+|eu^ε_α(x, t)−eu^ε_α(yε, t)| 6L|x−yε|+ε+L|x−yε|6(2L+ 1)ε.

By Ascoli-Arzelá Theorem, up to subsequencesue^εα→uuniformly, then alsou^εαcon- verges uniformly to the same function, which, by (3.8) and (3.7), satisfies (3.4) if α >0and (3.5) if α= 0.

Finally, Condition (3.6) is equivalent to require that u0(x+ηr)−δr>u0(x)for allx∈Rⁿ and everyr >0. Let us fixr >0and define, for allε >0, the function

v^ε(x, t) :=u^ε_α(x+ηr, t)− δr/ε

ε.

Then

v^ε(x,0) =u0(x+ηr)− δr/ε

ε>u0(x+ηr)−δr>u0(x)

for everyx∈Rⁿ. Moreover, by the periodicity ofg,v^εit is also a solution to equation in (3.2). So, by the comparison principle we get

v^ε(x, t) =u^ε_α(x+ηr, t)− δr/ε

ε>u^ε_α(x, t) ∀(x, t).

Passing to the limit asε→0, we obtain that

u(x+ηr, t)−δr>u(x, t) ∀(x, t),

which gives the thesis.

We now recall a well-known result of the theory of viscosity solutions (see [2]).

Proposition3.2. — Let c: [0,+∞)→[0,+∞)be a continuous function, and let u0

as in (3.3). Then there exists a unique Lipschitz continuous viscosity solution to (ut−c(|∇u|) = 0 inRⁿ×(0,+∞)

u(x,0) =u0(x) inRⁿ.

4. Cell problem and asymptotic speed of propagation

To study the homogenized limit of solutions to (3.2), first of all we look for special solutions governing the behavior of the equation on a rescaled framework, that is almost linear pulsating wave solutions (see [10, 19]). In particular, for every vector p ∈ Rⁿ we look for a function vp(x, t) moving with average speed c in the vertical direction, which has average slopep, which means thatvp(x, t)−ct−p·xis space-time periodic. Since the equation is homogeneous (it does not depend on x) we look for functions vp of the following form:

(4.1) vp(x, t) =ε χ^ε_pp·x+ct ε

where the function χp:R→Ris such that

(4.2) lim

z→±∞

χp(z) z = 1.

Finding a pulsating wave of the form (4.1) for Equation (3.2) reduces to showing that, for every p∈Rⁿ, there exists a unique constant c^ε_α(p) =c^ε_α(|p|) such that the following problem has a solutionχ=χ^ε_p:

(4.3)











ε^α−1χ⁰⁰(z)|p|²−c^ε_α(|p|)χ⁰(z) +g(χ(z)) = 0 z∈(0,1) χ(1) =χ(0) + 1

χ⁰(0) =χ⁰(1).

Observe that, ifgis constant, that is,g≡g, thencε(|p|) =gfor everypand everyεand χ^ε_p(z) =z. Note that the cell problem (4.3) can be reformulated in a more standard way as follows. Given p∈ Rⁿ,α>0, ε >0, find the constantc^εα(|p|)for which the equation

(4.4) − |p|²ε^α−1w⁰⁰(z) +c^ε_α(|p|)(w⁰(z) + 1)−g(w(z) +z) = 0

admits a periodic solutionw^εp. Therefore, once we have a solutionχ to (4.3), we can extend it to the whole Rin the following way: w(z) = χ(z)−z can be extended by periodicity to the whole space R, and then χ(z) = w(z) +z : R →R is a function such that

(4.5)

(ε^α−1χ⁰⁰(z)|p|²−c^εα(|p|)χ⁰(z) +g(χ(z)) = 0 z∈R limz→±∞χp(z)/z= 1.

We say that the solution to (4.3) is unique up to horizontal translations if every solution to (4.3) is the restriction in the interval(0,1)to a solution to (4.5).

4.1. Case α = 1, effective Hamiltonian. — In this section we consider the case α= 1. Under this assumption, the cell problem reads as follows: for everyp∈Rⁿ, show that there exists a unique constantc(|p|)such that there exists a solutionχ=χp(·)

(4.6)











χ⁰⁰(z)|p|²−c(|p|)χ⁰(z) +g(χ(z)) = 0 z∈(0,1) χ(1) =χ(0) + 1

χ⁰(0) =χ⁰(1).

We can also state the cell problem using the equivalent formulation: given p∈ Rⁿ, find the constantc(|p|)for which there exists a periodic solutionwp to

(4.7) − |p|²w⁰⁰(z) +c(|p|)(w⁰(z) + 1)−g(w(z) +z) = 0.

In the following theorem we show that the cell problem has a (unique) solution.

Theorem4.1. — For everypthere exists a uniquec(|p|)such that there exists a mono- tone increasing solutionχp to(4.6), which is also unique up to horizontal translations.

Moreover, the map|p| 7→c(|p|)is continuous, increasing and positive,

(4.8) c(|p|) =









 Z 1

0

g(χp(z))dz= R1

0 g(s)ds R1

0(χ⁰_p(z))²dz p6= 0 Z 1

0

1 g(s)ds

⁻¹

p= 0.

In particular,

|p|→0lim c(|p|) = Z 1

0

1 g(s)ds

⁻¹

and lim

|p|→0χp(z) =χ0(z)inC(R) (4.9)

|p|→+∞lim c(|p|) = Z 1

0

g(s)ds and lim

|p|→+∞χp(z) =z inC¹(R), (4.10)

with c(|p|)<R1

0 g(s)dsif g is nonconstant.

Proof. — The proof is divided in several steps.

Step 1: construction of a solution forp= 0. — Forp= 0, we rewrite (4.6) as follows

(4.11)











c(0)χ⁰(z)−g(χ(z)) = 0 z∈(0,1) χ(1) = 1, χ(0) = 0

χ⁰(0) =χ⁰(1).

We integrate the equation between0and 1and we get Z 1

0

dχ g(χ) = 1

c(0)

which gives the representation formula (4.8), and the uniqueness of c(0). The solu- tionχ0 is defined implicitly by the formula

Z χ₀(z) 0

ds g(s) =z

Z 1 0

ds g(s).

Step 2: construction of a solution forp6= 0. — For|p| 6= 0, we perform the change of variableχp(z) =−h(−z/|p|), so the cell problem (4.6) reads











h⁰⁰(z) +c(|p|)h⁰(z)−g(h(z)) = 0 z∈(−1/|p|,0) h(0) = 0, h(−1/|p|) =−1

h⁰(−1/|p|) =h⁰(0),

wherec(|p|) =c(|p|)/|p|, which is equivalent to

(4.12)











h⁰⁰(z) +c(|p|)h⁰(z)−g(h(z)) = 0 z∈(0,1/|p|) h(0) = 0, h(1/|p|) = 1

h⁰(1/|p|) =h⁰(0).

Givenc >0anda >0, letha,c be the unique solution to the ODE:

(4.13)











h⁰⁰_a,c(z) +ch⁰_a,c(z)−g(ha,c(z)) = 0 z >0 ha,c(0) = 0

h⁰_a,c(0) =a.

We multiply (4.13) bye^cz and we estimateg from above and below withmaxg and ming respectively. Integrating in(0, z)the two inequalities

h⁰⁰_a,c(z)e^cz+ch⁰_a,c(z)e^cz>e^czming, h⁰⁰_a,c(z)e^cz+ch⁰_a,c(z)e^cz6e^czmaxg we get the estimate

(4.14) 0< ae^−cz+ming

c 1−e^−cz

6h⁰_a,c(z)6ae^−cz+maxg

c 1−e^−cz .

Letz:= sup{z: ha,c(z)<1} ∈(0,+∞). Notice that from (4.14) it follows that forc small enough there holdsh⁰_a,c(z)> afor allz >0, whereas forc big enough we have h⁰a,c(z)< afor all z >0. As a consequence, for alla >0 there existsc(a)>0 such that

(4.15) ming

a 6 c(a)6maxg

a h⁰_a,c(a)(z(a)) =a.

From (4.14) and (4.15) it also follows that

(4.16) ming

c(a) 6h⁰_a,c(a)(z)6maxg

c(a) ∀z.

SinceRz(a)

0 h⁰_a,c(a)(z)dz= 1, (4.16) yields c(a)

maxg 6z(a)6 c(a) ming, which gives

(4.17) ming

maxg 1

a 6z(a)6 maxg ming

1 a. In particular, there holds

a→0limz(a) = +∞ and lim

a→+∞z(a) = 0.

Hence for all |p|>0 there exists at least one a(|p|) such thatz(a(|p|)) = 1/|p|, and the solution of (4.13) witha=a(|p|)andc=c(a(|p|))is also a solution of (4.12).

Step 3: uniqueness ofc(|p|)andχp. — The case p = 0 has already been considered in Step 1. Assume by contradiction that there exists p∈ Rⁿ, p 6= 0, such that the problem (4.7) admits two periodic solutions w1, w2, with constants c1 < c2. Let z a minimum point of w1−w2. Note that if w is a periodic solution to (4.7), then w(z) =e w(z+k) +kis still a periodic solution of the same equation for allk∈R. So we can assume thatw1(z) =w2(z)andw₁⁰(z) =w⁰₂(z).

At this minimum point, recalling thatχ⁰(z) =w⁰(z) + 1>0, we have 0 =−|p|²w⁰⁰₁(z) +c1(w⁰₁(z) + 1)−g(w1(z) +z)

6−|p|²w⁰⁰₂(z) +c1(w⁰₂(z) + 1)−g(w2(z) +z)

<−|p|²w⁰⁰₂(z) +c2(w⁰₂(z) + 1)−g(w2(z) +z) = 0 which gives a contradiction and proves the uniqueness ofc(|p|).

Let noww1, w2be two solutions to (4.7), as above, withw1(z) =w2(z)andw₁⁰(z) = w⁰₂(z) for some z. By uniqueness of solutions to the Cauchy problem associated to (4.7), it follows that w1 = w2, which yields the uniqueness of χp up to horizontal translations.

Step 4: properties of c(|p|). — Note that integrating the equation (4.6) in (0,1) we get c(|p|) = R1

0 g(χp(z))dz and from integrating (4.6) multiplied by χ⁰_p we get c(|p|)R1

0(χ⁰p(z))²dz = R1

0 g(s)ds, and then the representation formulas (4.8). In particular, from c(p) = R1

0 g(χp(z))dz we deduce that c(p) > ming > 0. Moreover, note that, if gis nonconstant, thenχ⁰p(z)cannot be constant and

Z 1 0

(χ⁰p(z))²dz >

Z 1 0

χ⁰p(z)dz ²

= 1.

So, by (4.8), we deduce thatc(p)<R1

0 g(s)dsfor everyp.

We prove continuity in 0, since continuity in p6= 0 is much simpler and follows the same argument. Let |pn| →0, with|pn| 6= 0 for everyn. So c(pn) is a bounded sequence and, by (4.8),

Z 1

0 |χ⁰_pn(z)|²dz6 R1

0 g(s)ds ming .

We recall thatχp_n(z)∈[0,1]forz ∈[0,1], so these estimates give an apriori bound in H¹(0,1) for χpn. So, up to passing to a subsequence, we get thatc(pn)→ ec and χp_n→χlocally uniformly. Then, by stability of viscosity solutions,χis a solution to (4.11), and by uniquenessec=c(0)andχ=χ0. Moreover, since bothχp_n(z)−z and χ0(z)−zare periodic functions such that their difference converges locally uniformly to 0, then we can conclude using periodicity that the convergence is uniform on R. This gives (4.9).

Now we prove (4.10). Reasoning as above, we get thatc(|p|)andχ⁰_pare equibounded respectively inRand inL²(0,1), uniformly with respect to|p|. By Equation (4.6) we

getχ⁰⁰_p = (c(|p|)χ⁰−g(χ))/|p|², so the uniformL² bound onχ⁰_p implies a uniformL² bound on χ⁰⁰_p, uniform in |p| >1. Eventually passing to a subsequence, c(|p|) → c, χp→χandχ⁰p→χ⁰locally uniformly as|p| →+∞. By stability of viscosity solutions, we get thatχsolvesχ⁰⁰(z) = 0, withχ(0) = 0,χ(1) = 1. Soχ(z) =z. Moreover, since χp(z)−z is a periodic function converging locally uniformly inC¹ to0, we get that actually it converges uniformly inC¹to0in the wholeR. Thereforeg(χp(z))→g(z) uniformly and then we conclude, using (4.8), that

|p|→+∞lim c(|p|) = lim

|p|→+∞

Z 1 0

g(χp(z))dz= Z 1

0

g(s)ds.

Finally we prove monotonicity ofc(|p|). Assume by contradiction that there exist p1, p2 ∈Rⁿ,|p1|>|p2|, such thatc(|p1|)< c(|p2|). Let w1, w2 two solutions to (4.7) associated to p1, p2. Letz a minimum point ofw1−w2, reasoning as in Step 3, we can assumew1(z) =w2(z). Then, at this minimum point,

0 =−|p1|²w⁰⁰₁(z) +c(|p1|)(w⁰₁(z) + 1)−g(w1(z) +z)

<−|p2|²w⁰⁰₂(z) +c(|p2|)(w⁰₂(z) + 1)−g(w2(z) +z) = 0,

which gives a contradiction. Thereforec(|p1|)>c(|p2|).

Remark4.2. — We expect that the same result holds also for R1

0 g(s)ds >0. In this case though, the ODE arguments are much more involved. Observe that, ifg changes sign, then necessarily we havec(0) = 0andχ0(z)≡s0forz∈(0,1), wheres0∈[0,1]

is such thatg(s0) = 0.

In the limiting case that R1

0 g(s)ds = 0, the same cell problem has been solved in [15], see also [1, Prop. 1.3], showing that there exists a solution to (4.6) with c(|p|)≡0for every p.

4.2. Case α 6= 1, the weak and strong diffusion regimes. — In this section we analyze the solution of the cell problem (4.6) in the caseα6= 1.

The solution to the cell problem is an easy corollary to Theorem 4.1. Moreover, we can also compute the asymptotic behavior asε → 0 to the solutions to the cell problem.

Proposition4.3. — Letα6= 1andε >0. Then there exists a unique constantc^ε_α(|p|) such that (4.3)admits a solutionχ^εp, which is monotone increasing, and unique up to horizontal translations. Moreover,

(i) if α >1 then we have that

ε→0lim⁺c^ε_α(|p|) =c+(|p|) :=

Z 1 0

g⁻¹(s)ds ⁻¹

∀p∈Rⁿ

andχ^εp →χ0 uniformly in C(R), for everyp, whereχ0 is the solution to (4.11);

(ii) if α <1 then we have that

ε→0lim⁺c^ε_α(|p|) =c−(|p|) :=









 Z 1

0

g⁻¹(s)ds ⁻¹

p= 0 Z 1

0

g(s)ds p6= 0

and, for p6= 0,χ^ε_p(z)→z uniformly in C¹(R), whereas χ^ε₀=χ0.

Proof. — Note that (4.3) coincides with the cell problem (4.6) associated to pε = pε^(α−1)/2. Therefore by uniqueness ofc proved in Theorem 4.1, for every ε >0 and every α 6= 1, there exists a unique c^ε_α(|p|) = c(|p|ε^(α−1)/2), such that there exists a solutionχ^ε_p to (4.3). Note thatχ^ε_p=χpε^(α−1)/2.

Moreover, ifα >1, since|p|ε^(α−1)/2→0 for everyp, then by (4.9),c^εα(|p|)→c(0) andχ^ε_p →χ0 uniformly.

Ifα <1, then forp6= 0,|p|ε^(1−α)/2→+∞and then, by (4.10),c^εα(|p|)→R1 0 g(s)ds andχ^ε_p(z)→zuniformly inC¹forp6= 0as ε→0.

5. Convergence of solutions

In this section we study the asymptotic limit asε→0of the solutions to (3.2) in the different regimes,α= 1,α >1,0< α <1 andα= 0.

According to Proposition 3.1, the solutionsu^ε_αto (3.2) converge locally uniformly, up to subsequences, to a Lipschitz function u. Our aim is to show that the limituis a viscosity solution of an effective equation, given byut−c(|∇u|) = 0. The effective operator has been defined in Theorem 4.1 forα= 1, and it coincides with the con- tinuous functionc(|p|). In the case α6= 1, the effective operator has been defined in Proposition 4.3. It coincides in the caseα >1with the constant value

c+(|p|)≡ Z 1

0

(g(s))⁻¹ds ⁻¹

,

whereas in the case0< α <1, it isc−(|p|), which coincides with R1

0 gforp6= 0, and with R1

0(g(s))⁻¹ds−1

forp= 0. We consider also the limiting caseα= 0, where the effective equation is given byut−F(∇u,∇²u) = 0.

We start with a preliminary estimate which follows from the comparison principle for (3.2) .

Proposition5.1. — Let u^ε_α be the solution to (3.2)with α>0. Then every uniform limit uof u^εαsatisfies

inf

Rⁿ

u0+t Z 1

0

1 g(s)ds

⁻¹

6u(x, t)6sup

Rⁿ

u0+t Z 1

0

1 g(s)ds

⁻¹

where inf_Rⁿu0 =−∞ if u0 is not bounded from below andsup_Rnu0 = +∞ if u0 is not bounded from above.

Proof. — It is enough to prove the result whenu0≡k, for some constantk∈R. The thesis then follows by the comparison principle for (3.2).

Recall thatc(0) = R1 0

1 g(s)ds−1

and observe that, if χ0is the solutions to (4.11), the functions

v^ε(x, t) =εχ0(tc(0)/ε) +ε k/ε

−ε and V^ε(x, t) =εχ0(tc(0)/ε) +ε k/ε are respectively a sub and a supersolution to (3.2), for everyα>0. So, by comparison

v^ε(x, t)6u^ε_α(x, t)6V^ε(x, t).

Lettingε→0and recalling thatχ0(z)/z →1asz→+∞, we get the conclusion.

5.1. Caseα= 1

Theorem 5.2. — Let u^ε be the solution to (3.2) for ε > 0 and α = 1. Then u^ε converges as ε → 0 locally uniformly to the unique Lipschitz continuous viscosity solution to

(5.1)

(ut−c(|∇u|) = 0, u(x,0) =u0(x).

Proof. — By Proposition 3.1, up to passing to subsequencesu^ε→ulocally uniformly, whereuis a Lipschitz continuous function which satisfies (3.4). So, if we prove thatu is a solution to (5.1), we conclude using uniqueness of solutions to (5.1) as stated in Proposition 3.2 the convergence of the whole sequenceu^εtou.

We show thatuis a subsolution to the effective equation in (5.1), the proof of the supersolution property being completely analogous.

Let (x0, t0) and φ a smooth function such that u−φ has a strict maximum at (x0, t0) and u(x0, t0) = φ(x0, t0). Let R > 0 and let B the closed ball centered at (x0, t0)and with radiusR. Define a family of perturbed test functions, parametrized by a parameters∈R, as follows:

φ^ε_s(x, t) =εχp

φ(x, t) ε +s

where χp is a solution to (4.6) with p = ∇φ(x0, t0). By the properties of χp, φ^ε_s+1(x, t) =φ^ε_s(x, t) +ε. Note thatφ^ε_s→φas ε→0, locally uniformly in x, t, s. So for everysthere exists a sequence(x^ε_s, t^ε_s)→(x0, t0)as ε→0 such that (x^ε_s, t^ε_s)is a maximum point for u^ε−φ^ε_s in B and (u^ε−φ^ε_s)(x^ε_s, t^ε_s) → u(x0, t0)−φ(x0, t0) = 0.

We claim that for everyε >0we can chooses^εsuch that(u^ε−φ^εs^ε)(x^εs^ε, t^εs^ε) = 0. In- deed, letm(s) = max_B(u^ε−φ^ε_s). Note thatm(s)is continuous andm(s+k) =m(s)−εk for everyk∈Z. Therefore by continuity there existss^εsuch thatm(s^ε) = 0.

From now on we fix the test functionφ^ε=φ^εs^ε and the maximum point(x^εs^ε, t^εs^ε) = (x^ε, t^ε). So, u^ε(x^ε, t^ε) =φ^ε(x^ε, t^ε), u^ε 6 φ^ε in B and (x^ε, t^ε) → (x0, t0) as ε → 0.

Indeed, let esε ∈ [0,1) be the fractional part of s^ε, then by the properties of χp

we get that (x^ε

es^ε, t^ε

es^ε) = (x^ε_sε, t^ε_sε). So the conclusion follows by the locally uniform convergence ofφ^εs^ε toφ.

Let us denotez^ε= (φ(x^ε, t^ε)/ε) +s^ε, so thatu^ε(x^ε, t^ε) =εχp(z^ε). We compute u^ε_t(x^ε, t^ε) =φ^ε_t(x^ε, t^ε) =χ⁰_p(zε)φt(x^ε, t^ε)

and

−ε∆u^ε(x^ε, t^ε)>−ε∆φ^ε(x^ε, t^ε) =−εχ⁰_p(zε)∆φ(x^ε, t^ε)−χ⁰⁰_p(z^ε)|∇φ(x^ε, t^ε)|². Plugging these quantities into Equation (3.2) computed at(x^ε, t^ε), we obtain

0 =u^εt−ε∆u^ε−g(u^ε/ε)>χ⁰p(z^ε)φt(x^ε, t^ε)−εχ⁰p(z^ε)∆φ(x^ε, t^ε)

−χ⁰⁰_p(z^ε)|∇φ(x^ε, t^ε)|²−g(χp(z^ε)).

Using the fact thatχp solves (4.6), we get 0>χ⁰p(zε)(φt(x0, t0)−c(|∇φ(x0, t0)|))

−χ⁰_p(zε) (φt(x0, t0)−φt(x^ε, t^ε) +ε∆φ(x^ε, t^ε)) (5.2)

−χ⁰⁰p(z^ε) |∇φ(x^ε, t^ε)|²− |∇φ(x0, t0)|² . (5.3)

Computing (4.6) at minima and maxima ofχ⁰_p we deduce that

(5.4) χ⁰p(z)∈

ming R1

0 g(s)ds,maxg ming

∀p, ∀z.

Moreover, from Equation (4.6), we deduce that also (5.5) kχ⁰⁰pk^∞6maxg−ming

|p|² ifp6= 0 and kχ⁰⁰₀k^∞6 kgk^∞kg⁰k^∞ c(0)² . Therefore, asε→0, we get that the terms in (5.2), (5.3) go to zero by the smoothness ofφ, and we are left withφt(x0, t0)−c(|∇φ(x0, t0)|)60.

5.2. Caseα >1

Theorem5.3. — Let u^εα be the solution to (3.2)withα >1. Then

ε→0limu^εα(x, t) =u0(x) +t Z 1

0

1 g(s)ds

⁻¹

locally uniformly.

Proof. — The argument is similar (in fact easier) of that in the proof of Theorem 5.2.

We sketch it briefly. Up to subsequences, we know that u^εα is converging locally uni- formly to some functionu(eventually depending on the subsequence).

We show that ut 6 R1 0

1 g(s)ds−1

in the viscosity sense. A completely analo- gous argument shows that ut > R1

0 1 g(s)ds−1

in the viscosity sense. Recalling that u(x,0) =u0(x), we conclude that thereforeu(x, t) =u0(x) +t R1

0 1 g(s)ds−1

.

Let (x0, t0) and φ a smooth function such that u−φ has a strict maximum at (x0, t0) and u(x0, t0) = φ(x0, t0). Let R > 0 and let B the closed ball centered at (x0, t0)and with radiusR. We define a perturbed test function as follows:

φ^ε(x, t) =εχ0

φ(x, t) ε +s

where χ0 is the solution to (4.11) and the parameter s is chosen as in the proof of Theorem 5.2. So, (x^ε, t^ε) is a maximum point for u^ε−φ^ε in B and u^ε(x^ε, t^ε) = φ^ε(x^ε, t^ε),u^ε6φ^ε inB and(x^ε, t^ε)→(x0, t0)asε→0.

Let us denotez^ε= (φ(x^ε, t^ε)/ε) +s, so thatu^ε(x^ε, t^ε) =εχ0(z^ε). So, using the fact that(x^ε, t^ε)is a maximum point for u^ε−φ^ε, we plugφ^ε into (3.2) and we obtain

0 =u^ε_t−ε^α∆u^ε−g(u^ε/ε)

>χ⁰₀φt(x^ε, t^ε)−ε^αχ⁰₀∆φ(x^ε, t^ε)−ε^α−1χ⁰⁰₀|∇φ(x^ε, t^ε)|²−g(χ0).

By regularity of φ and using the estimates (5.4), (5.5), we get that, as ε → 0, ε^αχ⁰₀∆φ(x^ε, t^ε) → 0 and ε^α−1χ⁰⁰₀|∇φ(x^ε, t^ε)|² → 0. So, we conclude recalling that χ⁰₀>0 and thatχ0 solves (4.11) that

0>φt(x0, t0)−c(0) +O(ε).

5.3. Case 0 < α < 1. — In this case, the limit differential operator c−(|p|) is not continuous, but just lower semicontinuous. In particular, the lower semicontinuous envelope of c− coincides with the function itself, whereas the upper semicontinuous envelope is the constant functionc−(|p|)^∗≡R1

0 g(s)ds.

We now show that every limit of u^ε_α is a viscosity solution of the limit problem (1.4). According to the definition recalled in Section 2, this means the following. Ifφ is a smooth test function such thatu(x0, t0) =φ(x0, t0)andu6φ, thenφt(x0, t0)6 R1

0 g(s)ds. If, on the other hand, u > φ, then φt(x0, t0) > c−(|∇φ(x0, t0)|), so in particular φt(x0, t0) > R1

0 g(s)ds at points where ∇φ(x0, t0) 6= 0 and φt(x0, t0) >

c−(0) = (R1

0 g⁻¹(s)ds)⁻¹ at points where∇φ(x0, t0) = 0.

We recall that, due to the discontinuity of the operator, differently to the case α>1, viscosity solutions to (1.4) are in general not unique.

Theorem5.4. — Letu^ε_αbe the solution to(3.2)with0< α <1. Every locally uniform limit u of u^ε_α is a Lipschitz continuous function, which satisfies (3.4), and solves in the viscosity sense the problem

(ut−c−(∇u) = 0 in Rⁿ×(0 +∞) u(x,0) =u0(x) in Rⁿ.

Moreover,

(i) usatisfies in the viscosity sense ut=

Z 1 0

g(s)ds

in every open setΩ⊂Rⁿ×[0,+∞), such that∇u6= 0a.e. inΩ.

(ii) uis a viscosity subsolution to ut=

Z 1 0

1 g(s)ds

⁻¹

at every point (x0, t0) such that (0, X, λ) ∈J⁺u(x0, t0) with X <0 in the sense of matrices.

(iii) uis a viscosity supersolution to ut=

Z 1 0

g(s)ds

at every point(x0, t0)such that(0, X, λ)∈J⁻u(x0, t0), and there existη∈Rⁿr{0}, δ >0, such thatη^tXη>δ|η|².

Proof. — The fact that, up to a subsequence, u^εα converges locally uniformly to a Lipschitz function u is proved in Proposition 3.1. Since u is Lipschitz continuous, then it is differentiable almost everywhere.

We prove now that u is a viscosity solution to the limit problem. We show the statement for supersolutions, since for subsolutions is completely analogous.

Fixφa smooth test function such that u(x0, t0) =φ(x0, t0)and u < φelsewhere.

We consider two cases, depending on the value of∇φ(x0, t0).

Case 1:∇φ(x0, t0) =p6= 0. — In this case we shall prove thatφt(x0, t0)>R1

0 g(s)ds.

Definep^ε=ε^(α−1)/2pandχp^ε the solution to (4.6), withc(ε^(α−1)/2|p|). We define the perturbed test function as in the proof of Theorem 5.2:

φ^ε(x, t) =εχp^ε

φ(x, t) ε +s

.

Sinceχp^ε(z)converges uniformly tozasε→0by Proposition 4.3, we get thatφ^ε→φ locally uniformly for everys. Reasoning as in the proof of Theorem 5.2, we get that there exists^ε,x^εt^ε such thatu^ε(x^ε, t^ε) =φ^ε(x^ε, t^ε), u^ε>φ^ε and(x^ε, t^ε)→(x0, t0) asε→0. So, pluggingφ^εinto Equation (3.2) computed at(x^ε, t^ε)we obtain

0 =u^εt−ε^α∆u^ε−g(u^ε/ε)6χ⁰p^εφt−ε^αχ⁰p^ε∆φ−χ⁰⁰p^εε^α−1|∇φ|²−g(χp^ε).

Using the fact thatχp^ε solves (4.6), we get 06χ⁰p^ε

φt(x0, t0)− Z 1

0

g(s)ds

−χ⁰p^ε

c(ε^(α−1)/2|p|)− Z 1

0

g(s)ds

−χ⁰_pε(φt(x0, t0)−φt(x^ε, t^ε) +ε^α∆φ(x^ε, t^ε))

−χ⁰⁰_pεε^α−1 |∇φ(x^ε, t^ε)|²− |∇φ(x0, t0)|² . Using (5.4), (5.5), (4.10) and the regularity of φ, letting ε → 0 we conclude that φt(x0, t0)>R1

0 g(s)ds.

Case 2:∇φ(x0, t0) = 0. — In this case we shall prove thatφt(x0, t0)> R1

0g⁻¹(s)ds−1

. As in Case 1, we let

φ^ε(x, t) =εχp

φ(x, t) ε +s^ε

,

where p ∈ Rⁿ will be determined later. As above, there exist s^ε, x^ε t^ε, depending continuously onp, such thatu^ε(x^ε, t^ε) =φ^ε(x^ε, t^ε),u^ε>φ^εand(x^ε, t^ε)→(x0, t0)as