Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions

We consider the evolutionary Hamilton-Jacobi equation depending on the unknown function with the continuous initial condition on a connected closed manifold. 
Under certain assumptions on $H(x,u,p)$ with respect to $u$ and $p$, we provide an implicit variational principle. By introducing an implicitly defined solution semigroup and an admissible value set $\mathcal{C}_H$, we extend weak KAM theory to certain more general cases, in which $H$ depends on the unknown function $u$ explicitly. As an application, we show that for $0\notin \mathcal{C}_H$, as $t\rightarrow +\infty$, the viscosity solution of 
\begin{equation*} 
\begin{cases} 
\partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ 
u(x,0)=\varphi(x), 
\end{cases} 
\end{equation*} 
diverges, otherwise for $0\in \mathcal{C}_H$, it converges to a weak KAM solution of the stationary Hamilton-Jacobi equation 
\begin{equation*} 
H(x,u(x),\partial_xu(x))=0. 
\end{equation*}

1. Introduction and main results. Let M be a connected closed (compact and without boundary) C r (r ≥ 2) manifold and H : T * M × R → R be a C r (r ≥ 2) function called a Hamiltonian. For a given T > 0, we consider the following Hamilton-Jacobi equation: where (x, t) ∈ M × [0, T ] and with the initial condition: where ϕ(x) ∈ C(M, R). The characteristics of (1) satisfy the following ordinary differential equations: The equation (2) is also referred as to the contact Hamiltonian equation on the contact manifold (J 1 (M, R), du − pdx) (see [1]). In 1983, M. Crandall and P. L. Lions introduced a notion of weak solution of (1) named viscosity solution for overcoming the lack of uniqueness of the solution due to the crossing of characteristics (see [1,8,11]). During the same period, S. Aubry and J. Mather developed a seminal work so called Aubry-Mather theory on global action minimizing orbits for area-preserving twist maps (see [3,4,21,22,23,24] for instance). In 1991, J. Mather generalized Aubry-Mather theory to positive definite and superlinear Lagrangian systems with multi-degrees of freedom (see [25]).
There is a close connection between viscosity solutions and Aubry-Mather theory. Roughly speaking, the global minimizing orbits in Aubry-Mather theory can be embedded into the characteristic fields of PDEs. The similar ideas were reflected in pioneering papers [15] and [16] respectively. In [15], W. E was concerned with certain weak solutions of the Burgers equation, which are corresponding to areapreserving twist maps. In [16], A. Fathi considered the Hamilton-Jacobi equations under so called Tonelli conditions (see (H1)-(H2) below), which are corresponding to positive definite and superlinear Lagrangian systems. Since then, weak KAM theory has been well developed. A systematic introduction to weak KAM theory can be found in [18].
In this paper, we are devoted to exploring the dynamics of more general Hamilton-Jacobi equations, in which the Hamiltonian H depends on the unknown function u explicitly. Precisely speaking, we are concerned with a C r (r ≥ 2) Hamiltonian H(x, u, p) satisfying the following assumptions: (H1) Positive Definiteness. For every (x, u) ∈ M × R, the second partial derivative ∂ 2 H/∂p 2 (x, u, p) is positive definite as a quadratic form; (H2) Superlinear Growth. For every (x, u) ∈ M × R, H(x, u, p) is superlinear with respect to p; (H3) Uniform Lipschitzity. H(x, u, p) is uniformly Lipschitzian with respect to u. (H4) Monotonicity. H(x, u, p) is non-decreasing with respect to u. (H1)-(H2) are called Tonelli conditions (see [18,25]). (H4) is referred to as "proper" condition for stationary Hamilton-Jacobi equations (see [9]). There is a broad class of Hamiltonians satisfying (H1)-(H4). Obviously, our assumptions cover Tonelli Hamiltonians independent of u. Besides, it also contains more general cases, for instance, discounted Tonelli Hamiltonian λu+H(x, p) with λ > 0, which was focused from the view of weak KAM under weaker assumptions [12].
Based on (H1), it is easy to see that (H2) is equivalent to the superlinearity of H(x, u, p) above each compact sets of M ×R. Since M is compact, (H2) implies that for each x ∈ M and u ∈ I with a compact subset I ⊂ R, H(x, u, p) is superlinear with respect to p. (See [18,Theorem 1.3.14] for details.) The aim of this paper is to show the main ideas of exploring the dynamics of more general Hamilton-Jacobi equations. To avoid the digression, we do not discuss whether the assumptions (H1)-(H4) are optimal, which will be focused in future works.
To state the main results, we introduce some terminology. Let us recall the Legendre transformation on the Hamiltonian independent of u, which is formulated as L : T * M → T M via (x,ẋ) = x, ∂H ∂p (x, p) .
LetL := (L, Id), where Id denotes the identity map from R to R. ThenL denotes the diffeomorphism from T * M × R to T M × R. ByL, the Lagrangian L(x, u,ẋ) associated to H(x, u, p) can be denoted by Since the contact vector field is of class C 1 , by existence and uniqueness theorems in ODE, for each compact subset K ⊂ T * M × R, there exist a neighborhood U of K and a ε := ε(K) > 0 such that one can define a local phase flow Ψ : (−ε, ε) × U → T * M × R Ψ t (x 0 , p 0 , u 0 ) := (X(t), P (t), U (t)), where we use (X(t), U (t), P (t)) to denote the solution of (2) with the initial data (X(0), U (0), P (0)) = (x 0 , u 0 , p 0 ).

XIFENG SU, LIN WANG AND JUN YAN
where the infimum is taken among the continuous and piecewise C 1 curves γ : [0, t] → M . The infimum is attained at a C 1 curve denoted byγ. Moreover, for τ ∈ (0, t), τ → (γ(τ ),ū(τ ), p(τ )) is of class C 1 and it satisfies the characteristic equation (2) wherē By analogy of the notion of weak KAM solutions of the Hamilton-Jacobi equation independent of u (see [18]), we define another weak solution of (1) called variational solution (see Definition 2.4 below). Based on Theorem 1.1, we construct a variational solution of (1) with the continuous initial condition. Following [18], we show that the variational solution of (1) is a viscosity solution of (1). Under the assumptions (H1)-(H4), it follows from the comparison theorem that the viscosity solution of (1) with the continuous initial condition is unique (see [11]) and it is a locally semiconcave function (see [7]). Moreover, we obtain a representation formula of the viscosity solution under (H1)-(H4). Theorem 1.2. For any ϕ(x) ∈ C(M, R), the viscosity solution u(x, t) of (1) with initial condition u(x, 0) = ϕ(x) can be represented as (3). Theorem 1.1 provides a variational principle on the evolutionary Hamilton-Jacobi equation as (1), from which there exists an implicitly defined semigroup denoted by T t such that u(x, t) = T t ϕ(x), where u(x, t) satisfies (3). To fix the notion, we call T t a solution semigroup. The name of solution semigroup came from [14] by A. Douglis under more restricted assumptions. In particular, for the discounted Hamiltonians H(x, u, p) := λu +H(x, p) with λ > 0, the solution semigroup can be reduced to the Lax-Oleinik semigroup Let a ∈ R be a constant. For u ≡ a, we use c(H(x, a, p)) to denote the Mañé critical value of H(x, a, p). From [10], we have c(H(x, a, p)) = inf Let Under the assumptions (H1)-(H4), a crucial result in weak KAM theory, the uniqueness of the Mañé's critical value, does not hold any more. We call C H an admissible value set of H(x, a, p) (and L(x, a,ẋ) by Legendre transformation). It is easy to see C H = ∅. Indeed, C H is a non-empty interval (see Proposition 5.3 below). For any c ∈ C H , there exists a ∈ R such that c(H(x, a, p)) = c. Let L c := L + c. H c and T c t are the Hamiltonian and solution semigroup associated to L c . Let · ∞ be C 0 -norm. We have the following theorem. Theorem 1.3. For any ϕ(x), ψ(x) ∈ C(M, R), t ≥ 0 and c ∈ C H , the solution semigroup T c t has following properties: (i.) Monotonicity: for ϕ ≤ ψ, T c (iv.) Equi-Lipschitzity: given δ > 0, there exists a constant κ δ > 0 such that for For the autonomous systems with Lagrangian L(x,ẋ), the convergence of so called Lax-Oleinik semigroup was established in [17]. By [19], such convergence fails for the non-autonomous Lagrangian systems. A new kind of operators was found in [27] to overcome the failure of the convergence of the Lax-Oleinik semigroup for the time periodic Lagrangian systems.
Different from the previous results, the solution semigroup T c t here is associated toL(t, x,ẋ) := L(x, u(x, t),ẋ) + c, which is defined on R × T M . Consequently, it results in the lack of conservation of energy of the system and compactness of the underlying manifold. Either the conservation or the compactness are crucial for the previous results. Hence, it is necessary to find a completely new dynamical way for establishing the convergence of the solution semigroup T c t . Based on Theorem 1.1 and Theorem 1.3, we obtain the convergence of the solution semigroup T c t by considering the evolution of H c along the characteristics. Theorem 1.4. For any ϕ(x) ∈ C(M, R), we have the dichotomy: then u c ∞ is a weak KAM solution of the following stationary equation: By inspiration of [17], the large time behavior of viscosity solutions of Hamilton-Jacobi equations with Hamiltonian independent of u was explored comprehensively based on both dynamical and PDE approaches (see [13,20,26] for instance). Theorem 1.2 implies u(x, t) := T t ϕ(x) is the unique viscosity solution of The weak KAM solution of (7) is the same as the viscosity solution. As an application of Theorem 1.4, if 0 ∈ C H , we obtain the large time behavior of the viscosity solution of (8), which appears here for the first time in this generality. In particular, for the discounted Hamiltonians H(x, u, p) := λu +H(x, p) with λ > 0 and Tonelli HamiltonianH(x, p), since C H = R, then the convergence in Theorem 1.4 always holds.
Note that the characteristics of (7) reads It is easy to see that H is conservative along the characteristics. Comparably, H is dissipative along the characteristics of (8). Roughly speaking, Theorem 1.4 shows a conservative system can be viewed as the limit of certain dissipative system in the sense of viscosity. This paper is outlined as follows. In Section 2, some definitions are recalled as preliminaries. In Section 3, an implicitly variational principle is established. Moreover, Theorem 1.1 can be obtained. In Section 4, a representation of the viscosity solution is provided, which implies Theorem 1.2. In Section 5, an implicitly defined solution semigroup is introduced and some properties are detected, from which Theorem 1.3 is proved. In Section 6, both the divergence and convergence of the solution semigroup are shown. Moreover, Theorem 1.4 can be verified.

2.
Preliminaries. In this section, we recall the definitions of the weak KAM solution and the viscosity solution of (1) (see [8,11,18]) and some aspects of Mather theory for the sake of completeness.
where H is a Tonelli Hamiltonian and c[0] is the Mañé's critical value of H. We consider the weak KAM solution of negative type here.
(ii) for any x ∈ M , there exists a C 1 curve γ : (−∞, 0] → M with γ(0) = x such that for any t ≥ 0, we have A function u is called dominated by L + c[0] if it satisfies (i) in Definition 2.1. It is denoted by u ≺ L + c[0]. For Tonelli Hamiltonians, weak KAM solutions of negative type of (10) are equivalent to viscosity solutions of (10). By analogy of the definition above, for c ∈ C H , we introduce the weak KAM solution of negative type of more general Hamilton-Jacobi equation as follows: where C H is the admissible value set of H defined in (6).
(iii) A function u : V × [0, T ] → R is called a viscosity solution of (1) if it is both a subsolution and a supersolution.
In the sequel, if not otherwise stated, solutions, subsolutions and supersolutions will be always meant in the viscosity sense, hence the adjective viscosity will be omitted in the following.  (i) for each continuous piecewise (ii) for any 0 ≤ t 1 < t 2 ≤ T and x ∈ M , there exists a C 1 curve γ : The existence of the variational solutions will be verified in Section 4.

Minimal action and Peierls barrier.
In this subsection, we recall some facts on the minimal action of Lagrangian independent of u. Let L : T M → R be a Tonelli Lagrangian. These facts will be useful in the proof of the uniform boundedness of the solution semigroup (see Proposition 5.4 below). Define the where the infimum is taken among the continuous and piecewise C 1 curves γ : [0, t] → M . By Tonelli theorem (see [18,25]), the infimum in (20) can be achieved. Letγ be a continuous and piecewise C 1 curve withγ(0) = x andγ(t) = y such that the infinmum is achieved atγ.γ is called a minimal curve. By [25], the minimal curves satisfy the Euler-Lagrange equation generated by L. The quantity h t (x, y) is called a minimal action.
From the definition of h t (x, y), it follows that for each x, y, z ∈ M and each t, t > 0, we have In particular, we have whereγ is a minimal curve For autonomous Lagrangians, "liminf" can be replaced by "lim". If a function Moreover, there holds [18, Lemma 5.3.2] Throughout this paper, we shall use | · | to denote the Euclidean norm, that is 3. Variational principle. For every given continuous function ϕ on M , we define the operator A depending on ϕ as follows: where u(x, t) ∈ C(M × [0, T ], R) and C ac ([0, t], M ) denotes the set of absolutely continuous curves γ : [0, t] → M . It is easy to see that is an operator from where the infimum is taken among the absolutely continuous curves γ : [0, t] → M . Moreover, we have Tonelli existence theorem.
For any x, y ∈ M and t > 0, the infimum in (26) can be attained at an absolutely continuous curve with γ(0) = x and γ(t) = y.
Note thatL is only continuous with respect to x and t rather than C 2 . Fortunately, the loss of regularity ofL with respect to x and t does not cause any trouble in the proof of Lemma 3.1. We omit it for the consistency of the context, see [6, Page 114] for the details. As a corollary, the infimum in (25) can be also attained.
3.1. The fixed point of A. In the following, we will prove that the operator A has a unique fixed point.
where λ is a positive constant independent of x andẋ. Hence, for any given t ∈ [0, T ], it follows from Lemma 3.1 that By exchanging the position of u and v, we obtain Let It follows from (28) that for s ∈ [0, t], we have Moreover, we have the following estimates: Moreover, continuing the above procedure, we obtain which implies Therefore, for any t ∈ [0, T ], there exists N ∈ N large enough such that A N is a contraction mapping and has a fixed point. That is, for any t ∈ [0, T ] and N ∈ N large enough, there exists a u( We now show that u is a fixed point of A. Since is also a fixed point of A N . By the uniqueness of fixed point of contraction mapping, we have A[u] = u. This completes the proof of Lemma 3.2. To fix the notions, we call the curve γ achieving the infimum in (33) a minimizer of u with γ(t) = x.

Minimizers and characteristics.
In the following, we will show the relation between minimizers and characteristics of (1). More precisely, we have the following lemma.
is of class C 1 and it satisfies the characteristic equation (2) wherē In order to prove Lemma 3.3, we need the following lemma.
Proof. Since M is compact, there exists ε := ε(K, k) such that for any (x 1 , u 1 ) ∈ K and 0 < t ≤ ε, we can choose a domain of coordinate chart U such that the set {x : x = X x1,u1 (t, p), p ∈ T * x1 M, |p| ≤ k} is contained in U . Hence, we conclude that it suffices to prove the lemma for the case when M is an open subset of R n .
In the sequel of the proof, we will thus suppose that M = U is an open subset of R n , and thus T * U = U × R n .
It follows from (2) that By differentiation under the integral sign, we have Moreover, we have which together with (36) and (38) implies that ∂ 2 X ∂t∂p (t 0 , p) exists and As which together with (37) and (39) yields as t → t 0 , On the other hand, we have Combining with (H1) and the compactness of p, we obtain that there exists ε > 0 small enough such that for for any |p| ≤ k and τ ∈ (t 0 , t 0 + ε], where det denotes the Jacobian determinant and C := C(k, ε) denotes a positive constant, thus p → X x0,u0 (t, p) is injective for any |p| ≤ k and τ ∈ (t 0 , t 0 + ε]. Moreover, it is a C 1 diffeomorphism onto its image for a given (x 0 , u 0 ).
Second, we verify the uniform existence of ε for any (x 0 , u 0 ) ∈ K. Choosing a local coordinate chart in a neighborhood of (x 0 , u 0 ). Consider the map Note that the limiting passage in (40) is uniform for any (x 0 , u 0 ) ∈ K, |p| ≤ k.
(b) Local coincidence betweenγ(τ ) and X(τ ) Fix τ ∈ (t 0 , t 0 + ε] and let S τ (x) := S(x, τ ). We denote where p = ∂ x S τ (x). It is easy to see that grad L S τ (x) gives rise to a vector field on M . Claim A. Let γ be a continuous and piecewise C 1 curve with (τ, γ(τ )) ∈ Ω ε for τ ∈ [a, b] ⊂ [t 0 , t 0 + ε], we have where the equality holds if and only if γ is a trajectory of the vector field grad L S τ (x). Proof of Claim A. From the regularity of S(x, τ ), it follows that By virtue of Fenchel inequality, for each τ whereγ(τ ) exists, we have It follows from (41) that for almost every τ ∈ [a, b] By integration, it follows from (46) that We have equality in (48) if and only if the equality holds almost everywhere in the Fenchel inequality, i.e.γ(τ ) = grad L S τ (x). Since grad L S τ (x) is continuous and defined for each t ∈ [a, b]. It follows thatγ(t) can be extended by continuity to the whole interval [a, b], which means that γ is a trajectory of the vector field grad L S τ (x).
Claim B. For any τ ∈ [t 0 , t 0 + ε], there holds whereγ is a minimizer of u withγ(t 0 ) = x 0 . Proof of Claim B. By contradiction, we assume there existst ∈ [t 0 , t 0 + ε] such that It suffices to consider the case with S(γ(t),t) < u(γ(t),t), the other case is similar. Letx :=γ(t). Since S(x, t) is constructed by the method of characteristics, by Lemma 3.4, there exists a C 1 curveγ : By (L4), a simple calculation implies which contradicts the assumption S(x,t) < u(x,t). Hence, for any τ ∈ [t 0 , t 0 + ε], we have Similarly, we have the converse inequality, which verifies the claim.
From the definition of u (see (33)), it follows that which impliesγ(τ ) is a solution of the vector field grad L S τ (x). i.e.
So far, we complete the proof of Theorem 1.1.
Remark 3.5. Generally, a flow generated by H(x, u, p) may not be complete, but in this paper, we only care about the flow associated to the minimizers, which is complete necessarily from (H1)-(H4).

Representation of the viscosity solution.
In this section, we will provide a representation formula of the solution of (1). By Theorem 1.1, there exists a unique where the infimum is taken among the continuous and piecewise C 1 curves. In particular, the infimum is attained at the characteristics of (1).
Based on Definition 2.3, it is easy to see that a variational solution of (1) is a viscosity solution. Proof. Let u be a variational solution of (1). Since u(x, 0) = ϕ(x) it suffices to consider t ∈ (0, T ]. We use V to denote an open subset of M . Let ϕ : V ×[0, T ] → R be a C 1 test function such that u − ϕ has a maximum at (x 0 , t 0 ). This means ϕ(x 0 , t 0 ) − ϕ(x, t) ≤ u(x 0 , t 0 ) − u(x, t). Fix v ∈ T x0 M and for a given δ > 0, we choose a C 1 curve γ : where the second inequality is based on (i) of Definition 2.4. Hence, Let t → t 0 , we have which together with Legendre transformation implies which shows that u is a subsolution.
To complete the proof of Lemma 4.2, it remains to show that u is a supersolution. ψ : V × [0, T ] → R be a C 1 test function and u − ψ has a minimum at (x 0 , t 0 ). We Hence Moreover, we have Let t tend to t 0 , it gives rise to This finishes the proof of Lemma 4.2.
By the comparison theorem (see [5] for instance), it yields that the solution of (1) is unique under the assumptions (H1)-(H4). So far, we have obtained that there exists a unique solution u(x, t) of (1) with initial condition u(x, 0) = ϕ(x) and u(x, t) can be represented implicitly as This completes the proof of Theorem 1.2.
5. Solution semigroup. Based on Section 4, the solution of (1) can be represented as u(x, t) = T t ϕ(x). Based on the fact that u(x, t) is a viscosity solution, T t satisfies the properties of semigroup T t+t = T t • T t . To fix the notion, we call T t solution semigroup. This notion was introduced by [14] under more strict conditions on H. Under the assumptions (H1)-(H4), we will detect some further properties of the solution semigroup. Moreover, we will complete the proof of Theorem 1.3.
First of all, it is easy to obtain the following proposition about the monotonicity of T t . Proposition 5.1 (Monotonicity). For given ϕ, ψ ∈ C(M, R) and t ≥ 0, if ϕ ≤ ψ, then T t ϕ ≤ T t ψ.
By a similar argument as the one in Proposition 5.1, one can obtain the nonexpansiveness of T t . For ϕ ∈ C(M, R), we use ϕ ∞ to denote C 0 -norm of ϕ. We have the following proposition.

XIFENG SU, LIN WANG AND JUN YAN
A similar calculation as (70) implies Similarly, we have This finishes the proof of Proposition 5.1.
We use c (H(x, a, p)) to denote the Mañé critical value of H(x, a, p). It is easy to see that c(H(x, a, p)) is continuous with respect to a. Let (H(x, a, p)).
Proof. It is clear that C H is non-empty. It remains to show C H is connected. That is, if c 1 , c 2 ∈ C H , then for any c ∈ [c 1 , c 2 ], c ∈ C H . For c 1 , c 2 ∈ C H and c 1 = c 2 , one can find a 1 = a 2 ∈ R such that c 1 = c(H(x, a 1 , p)) and c 2 = c(H(x, a 2 , p)). Since c(H(x, a, p)) is continuous with respect to a, then for any c ∈ [c 1 , c 2 ], there exists at least oneā ∈ [a 1 , a 2 ] such that c = c (H(x,ā, p)), which is contained in C H .
Let L c = L + c. For the sake of simplicity, we will prove Proposition 5.4 and Proposition 5.5 by taking c 0 := c(H(x, 0, p)) = c (L(x, 0,ẋ)). It is similar to prove the cases with other elements in C H . In the following context, we consider L c0 instead of L. Without ambiguity, we still denote T t := T c0 t , i.e.
It is easy to see that the following Proposition 5.4 and 5.5 still hold for other elements in C H .
For Case (II), a similar calculation yields It follows from the compactness of M and (23) that there exists a constat K 1 independent of (x, t) such that u(x, t) ≥ K 1 for any (x, t) ∈ M × [0, +∞).
On the other hand, we show that u(x, t) is bounded from above. We assume that there exists (x, t) ∈ M × (0, +∞) such that u(x, t) > 0. Otherwise, we have u(x, t) ≤ 0 for any (x, t) ∈ M × (0, +∞), which gives the upper bound of u(x, t).
For Case (II), a similar calculation yields By the compactness of M , there exists a constant K 2 independent of (x, t) such that u(x, t) ≤ K 2 for any (x, t) ∈ M × [0, +∞). This completes the proof of Proposition 5.4.
Based on Proposition 5.4, one can obtain the equi-Lipschitzity of the familiy of functions T t ϕ(x).
The key point to prove Proposition 5.5 is a priori compactness, from which it is easy to verify Proposition 5.5 following from a similar argument as [18]. Let u(x, t) := T t ϕ(x). Lemma 5.6 (a priori Compactness). Given δ > 0, there exists a compact subset K δ such that for every minimizer γ of u and any t ≥ δ, we have (γ(t), u(γ(t), t),γ(t)) ∈ K δ .

XIFENG SU, LIN WANG AND JUN YAN
which together with the uniform boundedness of u(x, t) implies where K denote the bound of u(x, t). Since η ≤ 1, there exists a constant κ such that u(y, t) − u(x, t) ≤ κη. For the case with η > 1, letγ : [0, η] → M be a geodesic of length η, parameterized by arclength and connecting x and y. One can find a finite sequence it follows that for any i ∈ {0, . . . , n − 1}, Adding these inequalities, we obtain u(y, t) − u(x, t) ≤ κη.
By exchanging the roles of x and y, we have Note that κ is independent of t for t ≥ δ > 0. This finishes the proof of Proposition 5.5.
So far, we complete the proof of Theorem 1.3.

6.
Convergence of the solution semigroup. In this section, we will prove Theorem 1.4. First of all, we are concerned with the divergence of the solution semigroup generated by the Lagrangian L c := L + c. Proof. Fix c / ∈ C H . Without loss of generality, we assume c = 0 / ∈ C H . Let u(x, t) := T 0 t ϕ(x). It is similar to prove the cases with other elements which are not contained in C H .
For 0 / ∈ C H , it follows from Proposition 5.3 that there are two cases: (I) for any a ∈ R, 0 < c(H(x, a, p)); (II) for any a ∈ R, 0 > c (H(x, a, p)).
For Case (I), we will show that there exists x 1 ∈ M and {t n } n∈N such that u(x 1 , t n ) → −∞ as t n → +∞. By contradiction, we assume that there exists a finite k ∈ R such that for any x ∈ M and t → +∞, we have That is, there exists t 1 > 0 such that for any t ≥ t 1 and x ∈ M , (80) holds. Let c k := c(L(x, k,ẋ)), then c k > 0. Let γ t : [0, t] → M be a minimizer of u with γ t (t) = x. Letγ t : [0, t − t 1 ] → M be a minimal curve withγ t (0) = γ t (t 1 ) and γ t (t − t 1 ) = x such that where the infimum is taken among the continuous and piecewise C 1 curves. Moreover, we have . It follows from Proposition 2.5 and c k > 0 that u(x, t) tends to −∞ as t → +∞, which contradicts (80).
For Case (II), we will show that there exists x 1 ∈ M and a sequence {t n } n∈N such that u(x 1 , t n ) → +∞ as t n → +∞. By contradiction, we assume that there exists a finite k ∈ R such that for any x ∈ M and t → +∞, we have That is, there exists t 2 > 0 such that for any t ≥ t 2 and x ∈ M , (81) holds. Let . It follows from Proposition 2.5 and c k < 0 that u(x, t) tends to +∞ as t → +∞, which contradicts (81).
This verifies (i) of Theorem 1.4. In the following, we are concerned with the convergence of the solution semigroup generated by the Lagrangian L c := L + c for a given c ∈ C H . H c and T c t are the associated Hamiltonian and solution semigroup. Without ambiguity, we still use L, H and T t instead of L c , H c and T c t for the simplicity.
By the monotonicity assumption (H4), we have ∂H ∂u ≥ 0. If ∂H ∂u ≡ 0, it was proved by Fathi [17,18], which is based on the conservation of energy and some properties of Mather sets for the corresponding Hamiltonian systems. Note that dH ds = − ∂H ∂u H, the energy is not conservative generally. Besides, the Mather theory for contact Hamiltonian systems has not been established yet. In order to overcome these difficulties, we have to establish a completely new and unified dynamical method to handle the general case with ∂H ∂u ≥ 0. We will show that for any ϕ(x) ∈ C(M, R), T t ϕ(x) converges as t → +∞ to a weak KAM solution of H(x, u, ∂ x u) = 0, which will verify (ii) of Theorem 1.4. The proof will be divided into four steps. 6.1.
Step 1: weak KAM solutions of the stationary equation. In this step, we will prove the existence of weak KAM solutions of (82).
Lemma 6.2. u is a weak KAM solution of (82) if and only if T t u = u for each t ≥ 0.
For each t > 0, there exists a C 1 minimizer γ t : [−t, 0] → M with γ t (0) = x such that for any t ∈ [−t, 0], we have Based on the a priori compactness, for a given δ > 0, there exists a compact subset K δ such that for any t > δ and s ∈ [−t, 0], we have Since γ t is a minimizer, it follows from the implicit variational principle that The points (γ t (0), u(γ t (0)),γ t (0)) are contained in a compact subset, then one can find a sequence t n such that (γ tn (0), u(γ tn (0)),γ tn (0)) tends to (x, u(x), v ∞ ) as n → ∞. Fixing t ∈ (−∞, 0], the function s → Φ s (x, u(x),γ tn (0)) is defined on [t , 0] for n large enough. By the continuity of Φ s , the sequence converges uniformly on the compact interval [t , 0] to the map s → Φ s (x, u(x), v ∞ ). Moreover, we have then for any t ∈ (−∞, 0], we have which implies (ii) of Definition 2.2. Hence, u is a weak KAM solution of (82).  (1) is the viscosity solution. Using a similar argument, one can obtain that the weak KAM solution of (82) is a viscosity solution. Let u(x) be a viscosity solution of (82). Note that for each t ≥ 0, both T t u(x) and u(x) are the viscosity solutions of The uniqueness of the solution of (89) implies T t u(x) = u(x). Hence, the weak KAM solution of (82) is the same as the viscosity solution.
By virtue of Proposition 5.4, we have T t ϕ is bounded for any ϕ ∈ C(M, R). Hence, lim sup t→+∞ T t ϕ does exist, which is denoted byū(x). We have the following lemma. Proof. Due to the definition of limsup, for every > 0, there exists s 0 > 0 such that for any s ≥ s 0 , we have T s ϕ ≤ū + , which the non-expansiveness and monotonicity of T t implies T t • T s ϕ ≤ T t (ū + ) ≤ T tū + .

XIFENG SU, LIN WANG AND JUN YAN
Fixing t ≥ 0, we take limsup for the above inequality as s → +∞. Since lim sup then we obtainū ≤ T tū + .
(93) Since is arbitrary, we haveū ≤ T tū . By the monotonicity of T t , it follows from the semigroup property that T tū is nondecreasing with respect to t. Combining with boundedness of T tū , it follows that the limit lim t→+∞ T tū does exist, which is denoted by u ∞ . Then, we have Based on Proposition 5.4 and Proposition 5.5, it follows from Arzela-Ascoli theorem that u ∞ (x) ∈ C(M, R). By Lemma 6.2, u ∞ is a weak KAM solution of (82). This completes the proof of Lemma 6.4.

6.2.
Step 2: zero level set of the modified Lagrangian. Since u ∞ (x) is a weak KAM solution, then it is easy to see that u ∞ (x) is Lipschitzian. Moreover, it follows from (H1) and (H2), u ∞ is locally semiconcave (see [7]). Let D be the set of all differentiable points of u ∞ on M . Due to the Lipschitzian property of u ∞ , it follows that D has full Lebesgue measure. For x ∈ D, we have We define L(x,ẋ) = L(x, u ∞ (x),ẋ) − ∂ x u ∞ (x),ẋ , x ∈ D.
(96) Denote where ∂H ∂p denotes the partial derivative of H with respect to the third argument. We have the following lemma. Proof. By (96) and (97), we have In addition, we have By (L2), it follows from (98) that there exists K 1 > 0 large enough such that for |ẋ| > K 1 , L(x,ẋ) ≥ d > 0, where d is a constant independent of (x,ẋ).
This completes the proof of Lemma 6.6.
Proof. It follows from Lemma 6.6 that H(x, u(x), ∂ x u(x)) ≤ 0 for almost all x ∈ M . Using a covering of a curve by coordinates charts, one can assume M = U is an open convex set in R n . Note that a C 1 curve can be approximated by piecewise affine curves in the topology of uniform convergence. The following proof is similar to [18,Proposition 4.2.3]. We omit the details.
By contradiction, we assume u(x) > T t u(x). Let γ : [0, t] → M be a minimizer of T t u with γ(t) = x, i.e.
Based on the preparations above, by Proposition 5.1, we conclude that for each t ≥ t ≥ 0,ũ ≤ T tũ ≤ T t ũ. Let us recall lim tn→+∞ T tn ϕ =ũ. Up to a subsequence, we choose t n+1 −t n → +∞. Let s n := t n+1 − t n . Note that T sn • T tn u = T tn+1 u, it follows from Proposition 5.2 that which together with lim tn→+∞ T tn ϕ =ũ shows T snũ →ũ as s n → +∞. Since T tũ is non-decreasing with respect to t, thenũ is a fixed point of T t for t ≥ 0. By virtue of Lemma 4.1, we haveũ is a weak KAM solution of (82). Moreover, using Proposition 5.2 again, it follows that for t > t n , we have Since T tn ϕ →ũ as t n → +∞, we obtain lim t→+∞ T t ϕ =ũ, whereũ is a weak KAM solution of (82). This finishes the proof of Theorem 1.4. Remark 6.9. Based on the uniqueness of the limit of T t ϕ(x) as t → +∞, we know that u ∞ given by Lemma 6.4 is the same asũ given by (105).