A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations

We introduce a new PDE approach to establishing the large time asymptotic behavior of solutions of Hamilton-Jacobi equations, which modifies and simplifies the previous ones (Barles and Souganidis, 2000; Barles, Ishii and Mitake, 2012), under a refined"strict convexity"assumption on the Hamiltonians. Not only such"strict convexity"conditions generalize the corresponding requirements on the Hamiltonians in Barles and Souganidis (2000), but also one of the most refined our conditions covers the situation studied in Namah and Roquejoffre (1999).


Introduction
In this article we introduce a new PDE approach to establishing the large time asymptotic behavior of solutions of Hamilton-Jacobi equations.
In the last two decades there have been major developments in the study of the large time asymptotics of solutions of Hamilton-Jacobi equations, initiated by the work by Namah and Roquejoffre [18] and by Fathi [8].
The approach by Fathi is based on the weak KAM theory and the representation of solutions of the Hopf-Lax-Oleinik type or, in other words, as the value functions of optimal control, and has a wide scope which is different from the one in Namah-Roquejoffre [18]. The optimal control/dynamical approach of Fathi has been subsequently developed for further applications and technical improvements by many authors (see, for instance, [7,9,11,13,14,16,17]).
At the beginning of the developments mentioned above, another approach has been introduced by the first author and Souganidis [5], which does not depend on the representation formulas of solutions and thus applies to a more general class of Hamilton-Jacobi equations including those with non-convex Hamiltonians. We refer for recent developments in this direction to [4,3].
We also refer [3] for further comments and references related to the large time asymptotics of solutions of Hamilton-Jacobi equations.
Our aim here is to modify and slightly simplify the main ingredient in the PDE approach by the first author and Souganidis [5] as well as to refine the requirements on the Hamiltonians.
To clarify and simplify the presentation, we consider the asymptotic problem in the periodic setting. We are thus concerned with the Cauchy problem (CP) u t (x, t) + H(x, D x u(x, t)) = 0 in Q, where Q := R n ×(0, ∞), u represents the unknown function on Q, u t = u t (x, t) = (∂u/∂t)(x, t), D x u(x, t) = ((∂u/∂x 1 )(x, t), ..., (∂u/∂x n )(x, t)) and u 0 represents the initial data. The functions u(x, t) and u 0 (x) are supposed to be periodic in x.
We make the following assumptions throughout this article: (A1) The function u 0 is continuous in R n and periodic with period Z n . (A2) H ∈ C(R n × R n ).
Our notational conventions are as follows. We may regard functions f (x) on R n (resp., g(x, y) on R n × V , where V is a subset of R m ) periodic in x ∈ R n with period Z n as functions on the torus T n (resp., T n × V ). In this viewpoint, we write C(T n ), C(T n × V ), etc, for the subspaces of all functions f (x) in C(R n ), of all functions g(x, y) in C(R n × V ), etc, periodic in x with period Z n . We denote the sup-norm (or the L ∞ -norm) of a function f by f ∞ and f L ∞ interchangeably. Regarding the notion of solution of Hamilton-Jacobi equations, in this article we will be only concerned with viscosity solutions, viscosity subsolutions and viscosity supersolutions, which we refer simply as solutions, subsolutions and supersolutions. For any R > 0, B R denotes the open ball of R n with center at the origin and radius R. For any X ⊂ R n , UC (X) and Lip (X) denote the spaces of all uniformly continuous functions and all Lipschitz continuous functions on X, respectively.
We now recall the following basic results.
Theorem 3. Under the hypotheses (A2)-(A4), there exists a unique constant c ∈ R such that the problem These theorems are classical results in viscosity solutions theory. For instance, the existence part of Theorem 1 is a consequence of Corollaire II.1 in [1]. Under assumptions (A2) and (A3), as is well known, the comparison principle holds between bounded semicontinuous sub and supersolutions of (CP) if one of them is Lipschitz continuous. This comparison result and the existence part of Theorem 1 assure that for each continuous solution u of (CP) there is a sequence {u k } k∈N of Lipschitz continuous solutions of (CP), with u 0 replaced by u k (·, 0), which converges to u uniformly in Q. The existence of such a sequence of Lipschitz continuous solutions of (CP) and the comparison principle for Lipschitz continuous solutions of (CP) guarantees the Theorem 2 holds. Theorem 3 and its proof can be found in [15].
The problem of finding a pair (c, v) ∈ R × C(T n ), where v satisfies (EP) in the viscosity sense, is called an additive eigenvalue problem or ergodic problem. Thus, for such a pair (c, v), the function v (resp., the constant c) is called an additive eigenfunction (resp., eigenvalue).
The assumptions above are some kind of strict convexity requirements and they are satisfied if H is strictly convex in p. Indeed in this case, since q = θ −1 (p + θ(q − p)) + (1 − θ −1 )p, and ψ measures how strict is this inequality. We point out that, for (A6) − , this argument is valid if p = q and the inequality is obvious if p = q, while in the case of (A6) + clearly we have always p = q.
One may have another interpretation of these assumptions, namely that the function H(x, r), as a function of r, grows more than linearly on the line segment connecting from q to p + θ 0 (q − p) for some θ 0 > 1. (Notice that this growth rate is negative in the case of (A6) − . ) We conclude these remarks on (A6) ± by pointing out that (A6) + is an assumption on the behavior of H on the set {H ≥ 0} while (A6) − is an assumption on the behavior of H on the set {H ≤ 0}. We refer to Section 3 for more precise comments in this direction.
We establish the following theorem by a PDE approach which modifies and simplifies the previous ones in [5,3].
A generalization of the theorem above is given in Section 4 (see Theorem 11), which covers the main result in [18] in the periodic setting.
In Section 2, we give an explanation of the new ingredient in our new PDE method, a (hopefully transparent) formal proof of Theorem 4 by the new PDE method and its exact version. In Section 3, we make comparisons between (A6) ± and its classical versions, and discuss convexity-like properties of the Hamiltonians H implied by (A6) ± as well as a couple of conditions equivalent to (A6) ± . In Section 4, we present a theorem, with (A6) ± replaced by refined conditions, which includes the situation in [18] as a special case.

Proof of Theorem 4
Throughout this section, we assume that (A1)-(A5) hold. The first step consists in reducing to the case when u 0 ∈ Lip (T n ) and therefore u is Lipschitz continuous on T n × [0, ∞).

Lemma 5.
If the result of Theorem 4 holds for any u 0 ∈ Lip (T n ) then it holds for any u 0 ∈ C(T n ).
Proof. For a general u 0 ∈ C(T n ) we select a sequence {u 0,j } j∈N ⊂ Lip (T n ) which converges to u 0 uniformly in R n . For each j ∈ N let u j ∈ Lip (T n × [0, ∞)) be the unique solution of (CP), with u 0,j in place of u 0 . By Theorem 2, we have Since Theorem 4 holds for any initial data in Lip (T n ), we know that for each j ∈ N there exists a function u ∞,j ∈ C(T n ) such that lim t→∞ u j (x, t) = u ∞,j (x) uniformly in R n . This implies Hence there is a function u ∞ ∈ C(T n ) such that lim j→∞ u ∞,j (x) = u ∞ (x) uniformly in R n . Observe by using Theorem 2 that for any j ∈ N, from which we conclude that lim t→∞ u(·, t) − u ∞ ∞ = 0. By the stability property of viscosity solutions, we see that u ∞ is a solution of (1) and, consequently, u ∞ ∈ Lip (R n ) by Theorem 3.
Now we turn to the proof of Theorem 4 when u 0 ∈ Lip (T n ). By Theorem 1, there exists a unique solution u ∈ Lip (T n × [0, ∞)) of (CP) and we have to prove that u(x, t) converges uniformly in R n to a function u ∞ (x) as t → ∞.
Henceforth in this section we assume that u 0 ∈ Lip (T n ) and hence the solution u of (CP) is in Lip (T n × [0, ∞)). Also, we fix a solution v 0 ∈ Lip (T n ) of (1). Such a function v 0 exists thanks to Theorem 3. We set L : If we set z(x, t) = v 0 (x) and invoke Theorem 2, then we get which shows that u is bounded in Q. We may assume by adding a constant to v 0 if needed that for some constant C 0 > 0,
Our proof of Theorem 4 follows the outline of previous works like [5,3] where a key result is an asymptotic monotonicity property for u. This asymptotic monotonicity is a consequence of Proposition 6 which, roughly speaking, implies that min{u t , 0} → 0 as t → ∞. This is rigourously stated in Lemma 8 and its consequence in (27). With assumption (A6) − , this is also the case but with a different monotonicity (i.e., max{u t , 0} → 0 as t → ∞).
For this reason, the function w defined by (3) is a kind of Lyapunov function in our asymptotic analysis in a broad sense. The main new aspect in this article, compared to [5,3], is indeed the simpler form of our w, which is defined by taking supremum in s of the function whose functional dependence on u and v 0 is linear. In the previous works, the function , played the same role as our function w, and the value depends nonlinearly in u and v 0 . One might see that the passage from the function given by (5) to w given by (3) bears a resemblance that from the Kruzkov transform to a linear change in [12] in the analysis of the comparison principle for stationary Hamilton-Jacobi equations.
From a technical point of view, they are a lot of variants for such results. For example, as it is the case in [5], one may look for a variational inequality for m(t) := max x∈R n w(x, t) or for m(t) := max x∈Ω w(x, t) where Ω is a suitable domain of R n . This last form can be typically useful when one wants to couple different assumptions on H on Ω and its complementary as in [5] where the coupling with Namah-Roquejoffre type assumptions was solved in that way, the point being to control the behavior of u on ∂Ω.
For the connections between our assumptions and Namah-Roquejoffre type assumptions, we refer to Section 4.

A formal computation.
Here we explain the algebra which bridges condition (A6) + to Proposition 6 under the strong regularity assumptions that u, w ∈ C 1 (T n × [0, ∞)) and v 0 ∈ C 1 (T n ) and that for each (x, t) ∈ Q there exists an s > t such that Of course, these conditions do not hold in general. Fix any (x, t) ∈ Q and an s > t so that (6) holds. If w(x, t) ≤ 0, then (4) holds at (x, t). We thus suppose that w(x, t) > 0. Setting we have Also, by the choice of s, we get Combining (8) and (12) yields (13) H(x, q) ≥ η.
Now, in view of inequalities (7) and (13), we may use assumption (A6) + , to get Using (10), we get Using the definition of L > 0, we clearly have and therefore we get This together with (9) and (11) yields This shows under our convenient regularity assumptions that (4) holds.
Remark 1. The actual requirement to v 0 is just the subsolution property in the above computation, which is true also in the following proof of Theorem 4. Some of subsolutions of (1) may have a better property, which solutions of (1) do not have. This is the technical insight in the generalization of Theorem 4 in Section 4.

2.1.2.
Proof of Proposition 6. We begin with the following lemma.
Lemma 7. We have Proof. We just need to note that for all ( Proof of Proposition 6. Noting that u ∈ Lip (T n × [0, ∞)) and v 0 ∈ Lip (T n ) and rewriting w as we deduce that w ∈ Lip (T n × [0, ∞)). Fix any φ 0 ∈ C 1 (Q) and (x,t) ∈ Q, and assume that We intend to prove that for R = (2θ 0 + 1)L, If w(x,t) ≤ 0, then (14) clearly holds. We may thus suppose that w(x,t) > 0. We choose anŝ ≥t so that Observe that for any s =t, which guarantees thatŝ >t.
Define the function φ ∈ C 1 (Q × (0, ∞)) by Note that the function is an open ball of R 3n+2 centered at (x,x,x,t,ŝ) with its closure B contained in R 3n × (0, ∞) 2 , we use the technique of "tripling variables" and consider the function Φ on B given by Let (x α , y α , z α , t α , s α ) ∈ B be a maximum point of Φ. As usual in viscosity solutions theory, we observe that lim α→∞ (x α , y α , z α , t α , s α ) = (x,x,x,t,ŝ).
Consequently, if α is sufficiently large, then We assume henceforth that α is sufficiently large so that the above inclusion holds.

2.1.3.
Completion of the proof of Theorem 4 under (A6) + . We set Moreover, the convergence Proof. It is sufficient to prove that the convergence (23) holds uniformly in x ∈ R n . Contrary to this, we suppose that there is a sequence (x j , t j ) ∈ Q such that lim j→∞ t j = ∞ and w(x j , t j ) ≥ δ for all j ∈ N and some constant δ > 0. In view of the periodicity of w, we may assume that lim j→∞ x j = y for some y ∈ R n . Moreover, in view of the Ascoli-Arzela theorem, we may assume by passing to a subsequence of {(x j , t j )} if needed that lim j→∞ w(x, t + t j ) = f (x, t) locally uniformly in R n × (−∞, +∞), for some bounded function f ∈ Lip (T n × R). Now, note that f (y, 0) ≥ δ. By the stability of the subsolution property under uniform convergence, we see that f is a subsolution of Since f ∈ C(T n × R) and f is bounded on R n+1 , for every ε > 0 the function f (x, t) − εt 2 attains a maximum over R n+1 at a point (x ε , t ε ). Observe as usual in the viscosity solutions theory that In particular, we have lim ε→0+ εt ε = 0. In view of inequality (24), we get 2εt ε − ω H,R (0) + ψ ≤ 0, which, in the limit as ε → 0+, yields ψ ≤ 0, a contradiction. This shows that the uniform convergence (23) holds.
To accommodate the previous w to (A6) − , we modify and replace it by the new function, which we denote by the same symbol, given by where (η, θ) is chosen arbitrarily in (0, η 0 ) × (1, θ 0 ) and the constants η 0 and θ 0 are those from (A6) − .
We have the following proposition similar to Proposition 6.
Since the proof of the above proposition is very similar to that of Proposition 6, we present just an outline of it.
Outline of proof. Note that for any (x, t) ∈ R n × (T, ∞) and s ∈ [0, t − T ), Hence, in view of Lemma 9, for any (x, t) ∈ R n × (T, ∞) we have From this latter expression of w, as the functions u and v 0 are Lipschitz continuous in Q and R n , respectively, we see that w is Lipschitz continuous in R n × [T, ∞). Also, from (30) we see that for any (x, t) ∈ R n × (T, ∞), if for some 0 ≤ s ≤ t, then s ≥ t − T > 0.
To see that (29) holds, we fix any test function φ 0 ∈ C 1 (R n × (T, ∞)) and assume that w − φ 0 attains a strict maximum at a point (x,t).
Outline of proof of Theorem 4 under (A6) − . Using Proposition 10 and arguing as the proof of Lemma 8, we deduce that lim t→∞ max{w(x, t), 0} = 0 uniformly in R n .
We fix any ε > 0 and choose a constant T ε ≡ T ε,η,θ > T so that for any t ≥ T ε , Let t ≥ T ε and x ∈ R n . For any 0 ≤ s ≤ t, we have We may assume that T ε > 1, and from the above, for any 0 ≤ s ≤ 1, we have Since u ∈ Lip (T n × (0, ∞)) and it is bounded in Q, the Ascoli-Arzela theorem assures that there is a sequence {τ j } j∈N ⊂ (0, ∞) diverging to infinity such that for some function z ∈ Lip (T n × R), lim u(x, t + τ j ) = z(x, t) locally uniformly in R n+1 .
We see immediately from (32) that the function z(x, t) is nonincreasing in t for every x. Furthermore, we infer that for some function u ∞ ∈ C(T n ), lim t→∞ z(x, t) = u ∞ (x) uniformly in R n .
As exactly under (A6) + , we deduce from this that lim t→∞ u(x, t) = u ∞ (x) uniformly in R n , which completes the proof.
The following example shows that (A) + is a stronger requirement on H than (A6) + even in a neighborhood of the points (x, p) where H vanishes. In this regard, the difference between two conditions is that the term ψ(η, θ) in (A6) + depends generally on η, θ while the term ν(η)(θ − 1) in (A) + depends linearly in θ − 1.
Set r = p + θ(q − p) and note that q = λr + (1 − λ)p. Note by the choice of ψ that Hence, using (A) − , (34) and (33), we deduce that This is a contradiction, which shows that (34) holds. Now, let H ∈ C(T n × R n ) satisfy (A6) + , and we show that for each x ∈ R n the sublevel set {p ∈ R n : H(x, p) ≤ 0} is convex.
To do this, we fix any x ∈ R n and let p 1 , p 2 ∈ K := {p ∈ R n : H(x, p) ≤ 0}. We need to show that We suppose that this is not the case and will get a contradiction.
Let η 0 > 0 and θ 0 > 0 be the constants from (A6) + . Then, setting we have By the definition of λ 0 , we may select a λ ∈ (0, λ 0 ) so that and note that H(x, q) > 0. Fix an 0 < η < η 0 so that H(x, q) ≥ η, and use condition (A6) + , to get H(x, p 2 + θ(q − p 2 )) > θη > 0, and moreover, This is a contradiction. An argument similar to the above guarantees that if H ∈ C(T n × R n ) satisfies (A6) − , then the sublevel set {p ∈ R n : H(x, p) < 0} is convex for every x ∈ R n . We leave it for the interested reader to check this convexity property.
The following example of H(x, p) = H(p) explicitly shows that condition (A) − is more stringent than (A6) − . Define the functions f, g ∈ C(R) by and then H ∈ C(R) (see Fig. 3  We do not give the detail, but observing that in the py plane, for each slope m < 0, the halh line y = mp, p > 0, meets the graph y = H(p) at exactly one point, we can deduce that the function H satisfies (A6) − . On the other hand, setting p = 0 and q = 1/2 k , with k ∈ N, observing that if 1 2 ≤ λ ≤ 1, then 1/2 k+1 ≤ λq ≤ 1/2 k and that for any 1 2 ≤ λ ≤ 1, H(λq) = λH(q) − (λ − 1) 2 2 k , we may deduce that (A) − does not hold with the current function H.
Let η 0 and θ 0 be the constants from (A8) + . We may assume, by replacing θ 0 by a smaller one if needed, that θ 0 < 2.

A generalization of (A6) ±
We recall that the following conditions on the Hamiltonian H ∈ C(T n × R n ) has been introduced by Namah-Roquejoffre [18] in their study of the large time asymptotic behavior of solutions of (CP).
Assume for the moment that H ∈ C(T n ×R n ) satisfies (NR3). Then the function v(x) ≡ 0 solves in the classical sense H(x, Dv(x)) = H(x, 0) in R n .
We take this observation into account and modify conditions (A6) ± as follows. The new conditions depend on our choice of a subsolution v 0 of (1), which plays the same role as the function v 0 in the proof of Theorem 4. As we have already noted in Remark 1, the function v 0 in the proof of Theorem 4 is needed to be just a subsolution of (1) and the outcome may depend on our choice of v 0 . Now we fix a subsolution v 0 ∈ C(T n ) of (1) and choose a nonnegative function f ∈ C(T n ) so that v 0 is a subsolution of H(x, Dv 0 (x)) ≤ −f (x) in R n .