Homogenization of a semilinear heat equation

We consider the homogenization of a semilinear heat equation with vanishing viscosity and with oscillating positive potential depending on $u/\varepsilon$. 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 dimension $n=1$, whereas in dimension $n>1$ we 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.


Introduction
We consider the following problem: where α ≥ 0 and the potential g is a periodic, Lipschitz continuous and positive function.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 appendix 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 where c : [0, +∞) → [0, +∞) is a continuous, nondecreasing and nonnegative function, which satisfies c(0) = and the second inequality is strict if g is nonconstant.Since the solution u to (2) is Lipschitz in x, with the same Lipschitz constant of the initial datum, necessarily the average speed is less than c( ∇u 0 ∞ ).
In the case α > 1 the limit problem is very simple and reads In particular the solutions to (1) converge locally uniformly to u 0 (x) + t 1 0 g(s) −1 ds −1 .In the case 0 < α < 1, the limit problem is In the limiting case α = 0, the limit problem is given by The functions c− and F are both discontinuous functions, such phenomenon is unusual in homogenization problems, and makes the analysis of this limit more challenging.Due to the lack of uniqueness of solutions to Hamilton-Jacobi equations with discontinuous Hamiltonian, in this case we prove that along subsequences the solution u ε α to (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 to x) of the limit function.This result allows us to obtain a complete description of the limit function for some classes of initial data.In particular, if the initial data is either monotone in one direction or convex, we prove that the solutions to (1) converge locally uniformly to u 0 (x) + t 1 0 g(s)ds when α ∈ (0, 1), and to the solution to u t − ∆u = 1 0 g(s)ds with initial datum u 0 when α = 0. If, on the other hand, the initial data is a radially simmetric function, which has a unique maximum point, then the limit function is given, for α ∈ (0, 1), by min u 0 (x) + t 1 0 g(s)ds, max u 0 + t . As a consequence we show further properties of the limit function u, 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 (19), 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 function g which possibly changes sign and satisfies 1 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. when 1 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) converges locally uniformly to the initial datum u 0 (x).More precisely, in [15] the following long time rescaling of (1) has been considered: showing that u ε converge locally uniformly to a solution of a quasilinear parabolic equation, see (50), which for α > 1 is the level set mean curvature equation.In [1], the 1-dimensional case has been considered for α = 1, in a more general setting.Homogenization of periodic structures has been studied by viscosity solution methods 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 (periodically) 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 homogenization of ordinary differential equations such as u ′ 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), at the microscopic scale.Pulsating wave solutions with (average) slope p ∈ R n are solutions to (1) with ε = 1 of the form φ(x, t) − c(p)t, where φ(x, t) − p • x is 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), with ε = 1, since typically they are the long time attractors of such solutions, see for instance [6,7,10,11,9,19].In particular in [19], it is proved the existence of horizontal (e.g. with slope p = 0) pulsating wave solutions to u t = ∆u + g(x, u, ∇u), where g is a positive function, which is periodic in x, u.The same argument also applies to get existence of pulsating wave solutions for rational slopes p ∈ Q n .In [9] a similar problem has been studied in the plane, that is existence for any slope p of pulsating wave solutions (which are traveling horizontally) to with g strictly positive.The authors also provide a complete description of the asymptotic speed of propagation c(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 (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 solution.In Section 3, we introduce the problem and the assumptions and provide a priori estimates on solutions to (1) and on their uniform limits.Section 4 is devoted to the solution to the cell problem, in the case α = 1 and then in the case α = 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.

Notation and preliminary definitions
Given z ∈ R, we will denote with [z] the smallest integer bigger than z: Given a smooth function u(x, t) : R n × (0, +∞) → R, we will denote with u t the partial derivative with respect to t, with ∇u, ∇ 2 u, ∆u resp.the gradient, the Hessian and the Laplacian of u with respect to x.

Assumptions and basic estimates
We assume the following conditions on the forcing term g : R → R: g is Lipschitz continuous , Z periodic, and g(y) > 0 for every y.
We consider the following Cauchy problem where α ≥ 0 and u 0 is a Lipschitz continuous function, with Lipschitz constant L.
We can assume without loss of generality that L ∈ N. In the case α = 0, we make the additional assumption that u 0 ∈ C 1,1 .
Proof.Due to the Lipschitz regularity of g, a standard comparison principle among sub and supersolutions to ( 10) holds (see [21]).So, existence and uniqueness of solutions to (10) follow easily, and the regularity comes from standard elliptic regularity theory (see [21]).Assume now that the initial datum u 0 has bounded Hessian.Indeed it is not restrictive, 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 (10) Let C = ∇ 2 u 0 ∞ .Then for every ε, the functions u 0 (x) ± ( g ∞ + ε α C)t are respectively super and subsolution to (10), which implies by the comparison principle that Hence, again applying the comparison principle, we get that for every t, s ≥ 0 This implies that u ε are equi-lipschitz in t.
Finally, condition ( 14) is equivalent to require that u 0 (x + ηr) − δr ≥ u 0 (x) for all x ∈ R n and every r > 0. Let us fix r > 0 and define, for all ε > 0, the function Moreover, by the periodicity of g, v ε it is also a solution to equation in (10).So, by comparison principle we get Passing to the limit as ε → 0, we obtain that which gives the thesis.
We now recall a well-known result of the theory of viscosity solutions (see [2]).

Cell problem and asymptotic speed of propagation
To study the homogenized limit of solutions to (10), 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 ∈ R n we look for a function v p (x, t) moving with average speed c in the vertical direction, which has average slope p, which means that v p (x, t) − ct − p • x is space-time periodic.Since the equation is homogeneous (it does not depend on x) we look for functions v p of the following form: where the function χ p : R → R is such that Finding a pulsating wave of the form (17) for equation (10) reduces to showing that, for every p ∈ R n , there exists a unique constant cε α (p) = cε α (|p|) such that the following problem has a solution χ = χ ε p : Observe that, if g is constant, that is g ≡ ḡ, then cε (|p|) = ḡ for every p and every ε and χ ε p (z) = z.Note that the cell problem ( 19) can be reformulated in a more standard way as follows.Given p ∈ R n , α ≥ 0, ε > 0, find the constant cε α (|p|) for which the equation admits a periodic solution w ε p .Therefore, once we have a solution χ to (19), we can extend it to the whole R in 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 We say that the solution to (19) is unique up to horizontal translations if every solution to (19) is the restriction in the interval (0, 1) to a solution to (21).

Case α = 1, effective Hamiltonian
In this section we consider the case α = 1.Under this assumption, the cell problem reads as follows: for every p ∈ R n , show that there exists a unique constant c(|p|) such that there exists a solution χ = χ p (•) We can also state the cell problem using the equivalent formulation: given p ∈ R n , find the constant c(|p|) for which there exists a periodic solution w p to In the following theorem we show that the cell problem has a (unique) solution.
Theorem 4.1.For every p there exists a unique c(|p|) such that there exists a monotone increasing solution χ p to (22), which is also unique up to horizontal translations.Moreover, the map |p| → c(|p|) is continuous, increasing and positive, In particular The proof is divided in several steps.
Step 1: construction of a solution for p = 0.
For p = 0, we rewrite (22) as follows We integrate the equation between 0 and 1 and we get which gives the representation formula (24), and the uniqueness of c(0).The solution χ 0 is defined implicitly by the formula Step 2: construction of a solution for p = 0.For |p| = 0, we perform the change of variable where c(|p|) = c(|p|)/|p|, which is equivalent to Given c > 0 and a > 0, let h a,c be the unique solution to the ODE: We multiply ( 29) by e cz and we estimate g from above and below with max g and min g respectively.Integrating in (0, z) the two inequalities we get the estimate Let z := sup{z : h a,c (z) < 1} ∈ (0, +∞).Notice that from (30) it follows that for c small enough there holds h ′ a,c (z) > a for all z > 0, whereas for c big enough we have h ′ a,c (z) < a for all z > 0. As a consequence, for all a > 0 there exists c(a) > 0 such that From ( 30 Hence for all |p| > 0 there exists at least one a(|p|) such that z(a(|p|)) = 1/|p|, and the solution of (29) with a = a(|p|) and c = c(a(|p|)) is also a solution of (28).
The case p = 0 has already been considered in Step 1. Assume by contradiction that there exists p ∈ R n , p = 0, such that the problem (23) admits two periodic solutions w 1 , w 2 , with constants c 1 < c 2 .Let z a minimum point of w 1 − w 2 .Note that if w is a periodic solution to (23), then w(z) = w(z + k) + k is still a periodic solution of the same equation for all k ∈ R.So we can assume that w 1 (z) = w 2 (z) and w ′ 1 (z) = w ′ 2 (z).At this minimum point, recalling that χ ′ (z) = w ′ (z) + 1 > 0, we have ) − g(w 2 (z) + z) = 0 which gives a contradiction and proves the uniqueness of c(|p|).
Let now w 1 , w 2 be two solutions to (23), as above, with w 1 (z) = w 2 (z) and w ′ 1 (z) = w ′ 2 (z) for some z.By uniqueness of solutions to the Cauchy Problem associated to (23), it follows that w 1 = w 2 , which yields the uniqueness of χ p up to horizontal translations.
In the limiting case that 1 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 (22) with c(|p|) ≡ 0 for every p.

Case α = 1, the weak and strong diffusion regimes
In this section we analyze the solution of the cell problem (22) in the case α = 1.

Convergence of solutions
In this section we study the asymptotic limit as ε → 0 of the solutions to (10) in the different regimes, α = 1, α > 1, 0 < α < 1 and α = 0.According to Proposition 3.1, the solutions u ε α to (10) converge locally uniformly, up to subsequences, to a Lipschitz function u.Our aim is to show that the limit u is a viscosity solution of an effective equation, given by u t − c(|∇u|) = 0.The effective operator has been defined in Theorem 4.1 for α = 1, and it coincides with the continuous function c(|p|).In the case α = 1, the effective operator has been defined in Proposition for p = 0. We consider also the limiting case α = 0, where the effective equation is given by u t − F (∇u, ∇ 2 u) = 0.
We start with a preliminary estimate which follows from the comparison principle for (10) .
Proposition 5.1.Let u ε α be the solution to (10) with α ≥ 0. Then every uniform limit u of u ε α satisfies where inf R n u 0 = −∞ if u 0 is not bounded from below and sup R n u 0 = +∞ if u 0 is not bounded from above.
Proof.It is enough to prove the result when u 0 ≡ k, for some constant k ∈ R. The thesis then follows by comparison principle for (10).
Recall that c(0) = and observe that, if χ 0 is the solutions to (27), the functions are respectively a sub and a supersolution to (10), for every α ≥ 0. So, by comparison Letting ε → 0 and recalling that χ 0 (z)/z → 1 as z → +∞, we get the conclusion.

Case α = 1
Theorem 5.2.Let u ε be the solution to (10) for ε > 0 and α = 1.Then u ε converges as ε → 0 locally uniformly to the unique Lipschitz continuous viscosity solution to Proof.By Proposition 3.1, up to passing to subsequences u ε → u locally uniformly, where u is a Lipschitz continuous function which satisfies (12).So, if we prove that u is a solution to (34), we conclude using uniqueness of solutions to (34) as stated in Proposition 3.2 the convergence of the whole sequence u ε to u.We show that u is a subsolution to the effective equation in (34), the proof of the supersolution property being completely analogous.
Let (x 0 , t 0 ) and φ a smooth function such that u−φ has a strict maximum at (x 0 , t 0 ) and u(x 0 , t 0 ) = φ(x 0 , t 0 ).Let R > 0 and let B the closed ball centered at (x 0 , t 0 ) and with radius R. Define a family of perturbed test functions, parametrized by a parameter s ∈ R, as follows: where χ p is a solution to ( 22) with p = ∇φ(x 0 , t 0 ).By the properties of χ p , φ ε s+1 (x, t) = φ ε s (x, t) + ε.Note that φ ε s → φ as ε → 0, locally uniformly in x, t, s.So for every s there exists a sequence (x ε s , t ε s ) → (x 0 , t 0 ) as ε → 0 such that (x ε s , t ε s ) is a maximum point for Therefore by continuity there exists s ε such that m(s ε ) = 0.
From now on we fix the test function 1) be the fractional part of s ε , then by the properties of χ p we get that (x ε sε , t ε sε ) = (x ε s ε , t ε s ε ).So the conclusion follows by the locally uniform convergence of and Plugging these quantities into Equation (10) computed at (x ε , t ε ), we obtain Using the fact that χ p solves (22), we get Computing ( 22) at minima and maxima of χ ′ p we deduce that Moreover, from equation ( 22), we deduce that also Therefore, as ε → 0, we get that the terms in (35), (36) go to zero by the smoothness of φ, and we are left with φ t (x 0 , t 0 ) − c(|∇φ(x 0 , t 0 )|) ≤ 0.
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 ε α is converging locally uniformly to some function u (eventually depending on the subsequence).
We recall that due to the discontinuity of the operator, differently to the case α ≥ 1, viscosity solutions to (4) are in general not unique.
Theorem 5.4.Let u ε α be the solution to (10) with 0 < α < 1.Every locally uniformly limit u of u ε α is a Lipschitz continuous function, which satisfies (12), and solves in the viscosity sense the problem Moreover, i) u satisfies in the viscosity sense in every open set Ω ⊂ R n × [0, +∞), such that ∇u = 0 a.e. in Ω.
ii) u is a viscosity subsolution to ds at every point (x 0 , t 0 ) such that (0, X, λ) ∈ J + u(x 0 , t 0 ) with X < 0 in the sense of matrices.
Proof.The fact that, up to a subsequence, u ε α 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.
We now prove assertions i), ii), iii).Proof of i).First of all observe that repeating the proof of Case 1, we get that that u t = 1 0 g(s)ds almost everywhere in Ω.If this is true, then u t = 1 0 g(s)ds in the viscosity sense in Ω.Indeed, let ρ δ be a sequence of standard mollifiers.So u δ = u * ρ δ → u uniformly and (u δ ) t = (u t * ρ δ ) = 1 0 g(s)ds everywhere in Ω.The conclusion then follows from the stability of viscosity solutions.
We now show an analogous result for general Lipschitz continuous initial data.
Theorem 5.12.Let n = 1, let u 0 be a Lipschitz function, and let u ε α be the solution to (10).Then u ε α converges locally uniformly to a unique function u.Proof.By Corollary 5.11 the result is true if u 0 ∈ L δ for some δ > 0.