Large Time Behavior of Periodic Viscosity Solutions for Uniformly Elliptic Integro-Differential Equations
In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem (or cell problem), i.e. we construct solutions of the form . We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of “mixed operators” for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (ii) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis.
Key words and phrases:Parabolic nonlinear integro-differential equations, Ergodic problem, Long time behavior, Strong maximum principle, Lipschitz estimates
2010 Mathematics Subject Classification:35B40, 35R09, 35D40, 35D10
In this paper, we provide new results on the large time behavior of viscosity solutions for parabolic integro-differential equations (PIDE in short).
1.1. A model equation
In order to describe our approach and our results, we consider the following model example
where is the unknown function depending on with , , and . denotes the derivative of with respect to , while stands for its gradient with respect to the space variable . The operator is the usual Laplacian with respect to the -variable, while denotes the fractional Laplacian of exponent with respect to the -variable
where is the gradient of with respect to the -variable and is the unit ball in . Finally, we assume that and , () are real-valued, Lipschitz continuous functions which are respectively - and - periodic. Because of this last assumption, the solution is expected to be -periodic if the initial datum is.
For such a PIDE, our aim is to show that, for large times (), the solution asymptotically behaves like where is a solution of the associated ergodic (or additive eigenvalue) problem which, for (1), reads
A key result in this direction is that there exists a unique such that this ergodic problem has a periodic (Lipschitz) continuous solution . With such a result in hand, one has to prove the convergence, namely that for every solution of (1), there exists a solution of the ergodic problem such that as , uniformly in .
To do so, we follow a by-now rather classical method which was systematically developed in . To carry out this method, the two key ingredients are estimates on the modulus of continuity (Lipschitz estimates in our case), and a Strong Maximum Principle, both for equations (1) and (2).
These needed results were obtained in previous papers. Lipschitz and Hölder estimates were obtained in  where an emphasis was made on “mixed operators”, i.e. on equations like (1) where the “uniform ellipticity” comes from both integral and differential terms, namely the and terms. This particular form of the equation creates also difficulties for the Strong Maximum (or Comparison) Principle, see .
In addition to this specific difficulty coming from “mixed operators”, we also want to handle the “superlinear case”, namely the case when in (1). This requires additional work and ideas both for solving (2) and for the convergence proof since one needs Lipschitz estimates to “linearize” this term. For this reason, we distinguish two cases: a sublinear and a superlinear one, with respect to the gradient growth.
In the sublinear case the estimates on the modulus of continuity come from the ellipticity of the equation. Indeed, as shown in , even though the equation is completely degenerate in the local term and in the nonlocal term, their combination render the diffusion uniformly elliptic. In the superlinear case, although most of these estimates are derived through the same type of arguments under suitable structure conditions on the nonlinearities, there are some situations where they come from the gradient term (cf. the proof of Lemma 1).
It is worth pointing out that some of our results (in particular in the sublinear case) could be proved in an easier way since they do not require such Lipschitz estimates but we have chosen to systematically use them in order to unify the paper and to keep it with a reasonable length.
1.3. The general framework
Let us now present the general framework of our analysis. Although not the most general one, we have chosen this framework since it carries the key difficulties. Extensions to a larger framework are given in Section 5. We consider parabolic PIDE of the form
(where denotes the Hessian matrix with respect to , ) subject to the initial condition
where and are -periodic functions, is a continuous function in and are nonlinear “elliptic” terms (precise assumptions are given in Section 2). We point out that each nonlinear term involves second-order derivatives and a nonlocal operator of “order” . These operators are of Lévy-Itô type: if is a smooth bounded function, then
where are the jump functions and the Lévy measures (here we also refer the reader to Section 2 for precise assumptions). Then we set
We recall that these operators are “natural” generalizations of diffusion operators of the form where is the diffusion matrix and denotes its transpose; in particular, they can be characterized as the infinitesimal generators of some solutions of stochastic differential equations driven by a general Lévy process, instead of a Brownian motion; see  for more details.
In the case of (1), we have
By choosing such a framework, we want to shed some light on the fact that each nonlinear term can have different forms of (local or nonlocal) diffusions in the different sets of variables while the -term carries the possible super-linearity (see Assumptions (H-a) and (H-b) below). Of course this simplified framework can be generalized and we refer the reader to Section 5 devoted to the extensions to see how this can be done for both the -term but also for and .
1.4. Known results
The study of large time behavior of solutions of Hamilton-Jacobi and fully nonlinear parabolic equations has attracted a lot of attention, in different contexts. It follows closely the development of the viscosity solution theory. Indeed, the first result one could cite about the large time behavior of solutions of first-order independent equations is already contained in  (see also ).
When studying the large time behavior of solutions of such equations, one has first to construct solutions of the form ; equivalently, one has to solve a stationary equation depending on a parameter which is unknown. This is the so-called cell (or ergodic) problem. Then one has to prove that the solution of the Cauchy problem indeed converges towards for some solution of the cell problem. The first step was completed in the seminal (yet unpublished) work of Lions, Papanicolaou and Varadhan  about homogenization of first-order Hamilton-Jacobi equations (in the case of coercive Hamiltonians). Indeed, solving the cell problem is the way the “effective” (or averaged) Hamilton-Jacobi Equation is determined.
The second step was first completed, for Hamilton-Jacobi Equations, by Namah and Roquejoffre  for equations with a particular structures and then by Fathi  in the general case of convex and coercive Hamilton-Jacobi Equations in the periodic setting. It is worth pointing out that the proof of  is based on pde methods while the results of  relies on dynamical systems arguments (“Weak KAM method”). Afterwards J.-M. Roquejoffre  and A. Davini and A. Siconolfi in  refined the approach of A. Fathi and they studied the asymptotic problem for Hamilton-Jacobi Equations on a compact manifold.
Barles and Souganidis extended in  the previous results and showed the asymptotic behavior under weaker convexity assumptions, using viscosity solutions techniques. They also gave counterexamples in  on the asymptotic behavior of solutions of Hamilton-Jacobi Equations, when the initial datum is not periodic anymore. Motivated by the latest works, Ishii established in  a general convergence result for Hamilton-Jacobi equations on the whole space. For asymptotic behavior of solutions of various boundary value problems for Hamilton-Jacobi Equations we refer to the works [28, 32, 35, 10, 26, 9] based either on weak KAM theory or on PDEs techniques. Lastly, we mention that Dirr and Souganidis showed in  that the asymptotic behavior remains true if one perturbes with additive noise viscous or non-viscous Hamilton-Jacobi equations, periodic in space.
As we already mentioned, our approach follows the ideas systematically developed by Barles and Souganidis in , where they described the long time behavior of space-time periodic solutions of quasilinear PDEs. This behavior has also been established for semilinear equations, with methods of degree theory, by Namah and Roquejoffre in . Long time behavior and ergodic problems for second order PDEs with Neumann boundary conditions have been studied in the series of papers [6, 18, 7]. Recently, viscous Hamilton-Jacobi equations have been treated by Tchamba in the superquadratic case , and Barles, Porretta and Tchamba in the subquadratic case .
We already mentioned that, when studying homogenization of Hamilton-Jacobi or fully nonlinear elliptic equation, it is necessary to solve a cell problem. As far as nonlinear integro-differential equations with periodic data are concerned, we can first mention a series of papers devoted to the homogenization of dislocation dynamics, see e.g. [30, 22]. Some results were also obtained for linear integro-differential equations with periodic data in [16, 2] by analytical methods and in [42, 25, 23, 24] by probabilistic ones. Homogenization of Markov processes governed by certain Levy operators was discussed by Horie Inuzuka and Tanaka in  and general results on their convergence in law were established in  (see also ). In , the author studies the convergence in law of a rescaled solution of a stochastic differential equation driven by an -stable Lévy process, . The author points out that such an homogenization question was raised in [15, p. 531]. It is also mentioned in the introduction of  that they are very few such results.
For nonlinear equations, Schwab  established homogenization results for a large class of equations. In the previously mentioned papers, both in the linear and the nonlinear cases, an ergodic problem has to be solved. In , since the equation is linear, linear equation techniques are used such as the study of the resolvent. In papers such as [30, 22, 2, 40], viscosity solutions / maximum principle techniques are used.
1.5. Organization of the article
The paper is organized as follows. In Section 2, assumptions on the nonlocal operator and the nonlinearities are given. We also point out which known results from [8, 17, 4] can be used with such a set of assumptions. In Section 3, we solve the stationary ergodic problem. In Section 4, we state and prove the convergence result for solutions of the Cauchy problem. In Section 5, we explain how to extend the previous results to even more equations. In Section 6, we give examples of applications of our results on specific equations.
Before stating our assumptions, we want to point out that, in order to avoid too many technicalities, we are going to use simplified assumptions for the main results, in particular some strong assumptions on the homogeneity of and . Then in Section 5 we explain how to extend our results to more general situations, under weaker homogeneity and growth assumptions.
Thus, the set of hypotheses below may not always seem consistent when considered as a whole. Typically, assuming the homogeneity assumption (F0) would simplify the general growth hypothesis (F2) a lot. But we keep these assumptions as such since in the extension section, we shall use some of them in their general versions.
We begin with assumptions on the singular measures and jump functions.
(Integrability) For , is a Lévy measure, i.e. there exists such that,
(Regularity of the measures) There exists and, for , there exists such that, for small enough
where is the ball centered at and of radius .
(Jump size) There exist two constants such that, for any and
(Regularity of the jumps) There exist a constant such that, for any
(Nondegeneracy) There exists a constant and for , there exist such that, for every , there exist such that the following holds for any and
Now we turn to the assumptions on the nonlinearities , , and the source term . Since these assumptions are the same for , we write them for a general and in , having in mind that they hold for , with . We denote by the space of symmetric matrices.
(Homogeneity) For any , , we have
(Periodicity-Continuity) and are continuous and -periodic in .
(Ellipticity-Growth conditions) There exist two bounded functions and a constant such that and some constants , , such that for any , , and any
if satisfy the inequality
with , for some unit vector , and .
As we mention it at the beginning of the section, (F2) does not seem to be consistent with (F0), nor will be the next assumption (F3). However, we will comment the more general framework in Section 5 dedicated to extensions. In the sequel, we use the notations for the quantities appearing in (F2) when they are related to .
(Lipschitz Continuity) is Lipschitz continuous, uniformly in .
(Regularity) There exists a modulus of continuity such that for any
for all , satisfying the matrix inequality (5) with and .
Finally, on the Hamiltonian , we assume one of the two following hypotheses.
(Sublinearity) is locally Lipschitz continuous and there exists a Hamiltonian , -positively homogeneous such that
(Superlinearity) is locally Lipschitz continuous and there exists , , and such that, for all and
We also use below a consequence of (H-b), namely the fact that there exists , such that, for all and all ,
We leave the proof of (6) to the reader: for , it comes from (H-b) while, for , it can be deduced from the first case, taking small enough.
Comments on the list of assumptions
We need a long list of assumptions in order to apply known results about uniqueness, regularity, Strong Maximum/Comparison Principle etc. for different equations. In order to convince the reader that we can indeed apply all these theorems, we next make a precise list of the ones we will use and we justify that our assumptions imply theirs.
To fit the framework of viscosity solutions (to ensure the existence of continuous solutions when combined with Perron’s method, but not only), we will be using the Comparison Principle for both the evolution equation (3) and for the perturbed stationary equation (8) introduced in the next section. Comparison principle has been shown in  to hold if a series of assumptions (A1)-(A4) were satisfied. In our case (A1) in  comes from (M1), (M3), (M4) in the present paper; (A2) is trivially satisfied; (A3-2) in  comes from (F4) in the present paper; (A4) in  comes from (F3) in the present paper.
Both for uniqueness of the solution for the ergodic problem corresponding to equation (7) and for establishing the long time behavior of solutions of equation (3) we will be using Strong Comparison Principle of Lipschitz sub- and supersolutions from [17, Theorem 32]; (H) in  comes from (F2), (F3) and (H-a)/(H-b) in the present paper. We would like to point out that, from a rigorous point of view, (F3) does not yield (H) in . However, in view of the proof of this result, see the very end of it, it is clear that (F3) is enough to conclude.
As mentioned in the introduction, we would like to deal with Lipschitz solutions for the some of the equations that will appear in the proof. Regularity of solutions for Eq. (10), and in particular Lipschitz estimates, follows from [4, Corollary 7]. Indeed, take which satisfies (H0) and (H2) from ; (H1) for and in  follows from (F2) in the present paper; (H2) in  follows from (F3) in the present paper; (H3) in  follows from (F4) in the present paper; (J1)-(J5) in  follows from (M1)-(M5) in the present paper. We also need Lipschitz estimates given by [4, Corollary 7] for solutions of equation (11)in the sublinear case. Choose , and replace with . Then trivially satisfies (H0) and (H2) from ; (H1) for in  follows from (F3) and (H-a) in the present paper; (M3)-(J4) in  follows from (M3)-(M4) in the present paper.
Finally, to solve the ergodic problem in the sublinear case, we will make usage of the Strong Maximum/Comparison Principle from [17, Theorem 20] for Eq. (12). This equation is degenerate elliptic and nonlinearities are continuous, i.e. it satisfies (E) from ; moreover, it is -homogeneous thanks to (F0) and (H-a); in particular, it satisfies the scaling assumption (S) of . Thanks to (M5) and (F3), it also satisfies .
3. The stationary ergodic problem
In this section we discuss the solvability of the stationary ergodic problem. For the sake of simplicity we write below
Our result is the following.
Assume that (M1)-(M5), (F0)-(F4) with , and either (H-a) or (H-b) holds. There exists a unique constant for which the stationary ergodic problem
has a Lipschitz continuous periodic viscosity solution . Moreover, is the unique Lipschitz continuous solution of (7), up to an additive constant.
For any , we consider as in  the following approximate equation
If , we notice that and are respectively sub- and supersolutions of the above approximated equation. Then it follows from Perron’s method for integro-differential equations as described for instance in , and from the comparison principle (Theorem 3 of ) that there exists a unique bounded viscosity solution which satisfies
Assumptions (M1),(M3),(M4) and (F4) on , and for imply that there is a comparison result for (8).
By the periodicity of and , and are both solutions of the above problem, for any . Then the uniqueness of the solution implies that they are equal; hence, is -periodic.
We next consider . The following proposition states the uniform boundedness of this sequence of normalized functions. It is the crucial technical part of the analysis of the ergodic problem.
The sequence is uniformly bounded.
The proof of the proposition is postponed. We remark that satisfies
We derive from Proposition 1 and results from  that is also equi-Lipschitz continuous (at this point, the whole set of assumptions is required). Therefore, we can use Ascoli’s Theorem to extract a subsequence which converges locally uniformly (and therefore uniformly, because of the periodicity) to a Lipschitz continuous -periodic function . On the other hand, (9) implies that is bounded; hence, up to extracting again a subsequence, we can further assume that . By the (continuous) stability of viscosity solutions of integro-differential equations, see e.g. , we conclude that is a solution of (7).
From the comparison principle for (3), we conclude that, for all ,
Dividing by and letting , this implies . Since and are arbitrary, we conclude that .
Finally, thanks to (F3), (H-a)/(H-b) and the Lipschitz continuity of and , we can apply the Strong Comparison Principle from [17, Theorem 32] to the Lipschitz solutions and of (3) and conclude that is constant. It is worth pointing out that this step uses in a crucial way the Lipschitz continuity of and because of the linearization procedure (but which has to be used only in the superlinear case). This completes the proof of Theorem 1. ∎
Solving the stationary ergodic problem is still possible for some cases when , as we will point out in some examples, see Section 6. As a matter of fact, the solutions would be Hölder continuous and the stationary ergodic problem would have a Hölder continuous solution. In this case, if the uniqueness of the ergodic constant remains true, it is not clear anymore that the solutions of the ergodic problem are unique up to an additive constant. However, it is worth pointing out that the Lipschitz continuity of the solutions of the ergodic problem is needed (in general) in order to prove the asymptotic result for the PIDE.
We now turn to the proof of the proposition. We distinguish the sublinear and the superlinear case.
Proof of Proposition 1 in the sublinear case.
We argue by contradiction: we assume that we can find a subsequence, that we still denote by , for which the associated sequence of norms blows up, i.e.
We next consider
which, by (F0), satisfies
Since for all , is equi-Lipschitz continuous by the results of . Hence, we can extract a locally uniformly converging subsequence (hence globally by periodicity); we denote by the limit which is a -periodic, Lipschitz continuous function with .
By the (continuous) stability result for viscosity solutions, see e.g. , Assumption (H-a) implies that is a solution of
The limiting equation (12) is -homogeneous; in particular, it satisfies the scaling assumption (S) of . If we can check that it satisfies from , then this will imply that the equation enjoys the Strong Maximum Principle (cf. [17, Theorem 22]).
This gives the contradiction: indeed, on one hand, we know that and, by the continuity and periodicity of , its maximum/minimum value is attained and equal to . Henceforth, by the Strong Maximum Principle, the function must be constant equal to . On the other hand since for all , which is the desired contradiction.
We now show that indeed, assumption of  is satisfied, and that it results from from (F2) and (M5). Fix ; we must check that for all and for all we have that
as , uniformly for , . Here the constant is given by (M1) and the cone is slightly different from the one in (M5). Namely, it has the form
Going back to the original form of the nonlinearity and using the ellipticity - growth assumption (F2) we get the following lower bound for ,
Using now the structure of the cone, and noting that , we get
Employing now the nondegeneracy assumption (M5) and setting by
we further have that
where is bounded when . Therefore as uniformly in , and and the proof is now complete in the sublinear case. ∎
Proof of Proposition 1 in the superlinear case..
The beginning of the proof in the superlinear case goes along the same lines as in the sublinear case. We assume that along a subsequence and reach a contradiction. For simplicity we still keep the notation for the subsequence and assume (with no restriction) that for all . We consider as before the rescaled functions
so that for any . Rewriting the equation with and using again the 1-positive homogeneity of nonlinearities, cf. (F0), we see that still satisfies (11). In particular, we get from (H-b) (and more precisely from (6)) that
We next claim that the following holds true.
The family is equicontinuous in .
The proof of the lemma is postponed and we complete the proof of the proposition. Since , using Ascoli’s Theorem, we can extract a subsequence which converges locally uniformly (hence globally) towards a continuous -periodic function . Using standard stability results for viscosity solutions together with Estimate (9), and (F4), we get
From this we deduce that in the viscosity sense. But there we get a contradiction from the continuity of since and . The proof of the proposition is now complete. ∎
Proof of Lemma 1.
We first fix a parameter and claim that
For any , there exists a (large) constant such that for any , we have
It is classical that such a result implies that is uniformly continuous and that the modulus of continuity only depends on and . To prove the claim, we consider the function
Since is -periodic, this function reaches its maximum at some point that we may consider in the same cell. If this maximum is nonpositive, then we are done since this means that for any .
Otherwise there are two options: either ; or .
In case we have for any
which implies, together with , that
and the claim holds.
In case , we can use the viscosity inequalities since the functions and are smooth near and respectively. Notice that which gives the estimate
The equation for can be derived from (11)
where with and and . Now we subtract both inequalities and we obtain
Using computations from , we get the following estimate of the higher order terms.
There exists depending on constants appearing in (M1)-(M5) such that
Combining (F3)-(F4) with Lemma 2, we thus get
where is the Lipschitz constant coming from (F3) and . We therefore have