数理解析研究所講究録
第 1542 巻 2007 年 24-32

24

Asymptotic solutions of Hamilton-Jacobi equations with
non-periodic perturbations*
大阪大学大学院基礎工学研究科

市原直幸 (Naoyuki Ichihaxa)\dagger

Graduate School of Engineering Science,
Osaka University

概要.

We study the long time behavior of viscosity solutions to some Cauchy
problem for Hamilton-Jacobi equations. We deal with Hamiltonians and initial
data that consist of the principal part which is periodic and a non-periodic
perturbation term. We also discuss a generalization of the results.

1

Introduction and Known results.

We are concerned with the large time behavior of continuous viscosity solutions to
Hamilton-Jacobi equations of the form

(1)$\{\begin{array}{ll}u_{t}+H(x, Du)=0 in \mathbb{R}^{n}\cross(0, +\infty),u(\cdot, 0)=u_{0}(\cdot)\in UC(\mathbb{R}^{n}) on \mathbb{R}^{n},\end{array}$where the Hamiltonian $H=H(x,p)$ is always assumed to satisfy the folowing:(H1) $H\in BUC(\mathbb{R}^{n}\cross B(0, R))$ for all $R>0$ , where $B(O, R)$ $:=\{x\in \mathbb{R}^{n};|x|\leq R\}$ .
(H2) $H$ is coercive, i.e., $\lim_{farrow+\infty}\inf\{H(x,p).;x\in \mathbb{R}^{n}, |p|\geq r\}=+\infty$ .
(H3) $H(x,p)$ is strictly convex in for every.
Our objective is to show that the unique continuous viscosity solution $u(x, t)$ of (1)has the asymptotic behavior of the form$p$$u(x, t)+ct-\phi(x)arrow 0$$x\in \mathbb{R}^{n}$

uniformly on compact subsets of$\mathbb{R}^{n}$,(2)’Joint work with Hitoshi Ishii.
$t_{JSPS}$
Post-Doctoral Research Fellow. E-mail: naoyuki@sigmath.es.osaka-u.ac.jp. Supported inpart by the JSPS Research Fellowship for Young Scientists.
2000 Mathematics Subject Classification: Primary $35B40$ ; Secondary $35F25,35B15$ .
Keywords: Hamilton-Jacobi equations, long time behavior, weak KAM theory, perturbations.

25. Remark
where $(c, \phi)$ is the pair of some real number and continuous function onhere that if the convergence (2) holds, then $(c, \phi)$ should satisfy the following timeindependent Hamilton-Jacobi equation (additive eigenvalue $proble\dot{m}$ ):$\mathbb{R}^{n}$$H(x, D\phi)-c=0$in$\mathbb{R}^{n}$.

(3)

In particular, the function $\phi(x)-ct$ is a viscosity solution of $\phi_{t}+H(x, D\phi)=0$ in, and it characterizes the large time asymptotic behavior of $u(x, t)$ . We
shall call such function the asymptotic solution of the Cauchy problem (1). Unfortunately, as far as as ymptotic problems in the whole Euclidean space are concerned,
the above three conditions $(H1)-(H3)$ are insufficient to guarantee such solutions.
for
Our aim is, therefore, to find some reasonable sufficient conditions on $H$ andthe existence of asymptotic solutions of (1).
and define theThis problem can be restated as folows. Letoracting on $UC(M)$ by
Lax-Oleinik semigroup$\mathbb{R}^{n}\cross(0, +\infty)$$u_{0}$$M=\mathbb{T}^{n}$$\mathbb{R}^{n}$$(T_{t})_{t\geq 0}$$(T_{t}u_{0})(x):= \inf\{\int_{-t}^{0}L(\gamma(s),\ddot{\gamma}(s))ds+u_{0}(\gamma(0))|\gamma\in AC([-t, 0], \mathbb{R}^{n})$,$7(0)=x\}$ ,(4)where $L(x, \xi)$ $;= \sup_{p\in R^{n}}(\xi\cdot p-H(x,p))$ and $AC([-t, 0], \mathbb{R}^{n})$ stands for the totality. We would like toof absolutely continuous functions on $[-t, 0]$ with values inknow if $T_{t}u_{0}+ct$ has the limit in the topology of $C(M)$ as $tarrow\infty$ for someand $u0\in UC(M)$ .
(or $M$ is
The first attempt to attack such problem in the case wherea smooth compact manifold) was made by Fathi $[5, 6]$ . He proves (2) under some
additional assumptions on $H$ . Recently, basing on the so-called Aubry-Mather theory, Davini-Siconolfi [4] improve his results and show the convergence result without
assuming any condition except for $(H1)-(H3)$ .
Similar results are obtained by Namah-Roquejoffre [12] and Barles-Souganidis [3].
Their proof is based on the theory of partial differential equations and viscosity
solutions. It is worth noting that the latter admits some class of Hamiltonians thatare not convex.
$\mathbb{R}^{n}$$c\in \mathbb{R}$$M=\mathbb{T}^{n}$

Concerning asymptotic problems in non-compact regions, Fujita-Ishii-Loreti [7]and Ishii [10] treat the case $M=R^{n}$ . The main assumption of [10] in addition to$(H1)-(H3)$ is(H4)$\exists\phi_{i}\in C^{0+1}(\mathbb{R}^{n}),$$\exists\sigma_{i}\in C(\mathbb{R}^{n})$with $i=0,1$ such that for $i=0,1$ ,$H(x, D\phi_{i}(x))\leq-\sigma_{i}(x)$$\lim\sigma_{i}(x)=\infty$

国\rightarrow \infty,ae.

$x\in \mathbb{R}^{n}$,$\lim(\phi_{0}-\phi_{1})(x)=\infty$$|x|arrow\infty$,

26
and the class of initial data is taken as$\Phi_{0}$$:= \{v\in C(\mathbb{R}^{n}) ; \inf_{\mathbb{R}^{n}}(v-\phi_{0})>-\infty\}$.

The proof is based on some dynamical approach associated with the variational
formula (4) as well as some techniques on the theory of viscosity solution. An
important feature by virtue of assumption (A4) is that any extremal curvein
the right-hand side of (4) stays in a compact subset offor all $t>0$ . Roughly
speaking, this fact corresponds to the compactness of the Aubry set, a uniquenessset for (3).
$\gamma(\cdot)$$\mathbb{R}^{n}$Assumption and the Main theorem.

2

In this note, we try to find another type of conditions on the Hamiltonian so that
there exist asymptotic solutions of (1) for some class of initial data, Notice that
this note is based on the paper [9], and a part of the results presented in this note
has been announced in [8]. We are especially interested in the case where extreme
curves may diverge (i.e. $|\gamma(t)|arrow\infty$) as goes to the infinity. $Si_{1}nilar$ situations
are also investigated by Barles-Roquejoffre [2].
Now, we state our standing assumption.
$t$

(A1)

$H(x,p)=h(x,p)-f(x)$ for some $h\in C(R^{n}\cross R^{n})$ satisfying $(H1)-(H3)$ and$f\in C(\mathbb{R}^{n})$

(A2)
(A3)

.

is -periodic for all$f\geq 0$ and $supp(f)$ is compact.
$h(\cdot,p)$$\mathbb{Z}^{n}$$p\in \mathbb{R}^{n}$.

The main theorem of this note is the following:
Theorem 2.1 (c.f. Theorem 2.4 of [8]). Let $H$ satish (A ) $-(A3)$ . Suppose moreoverthat additive eigenvalue problem $(S)$ has a solution in the dassfor some. Then, for any initial function belonging to$l$$BUC(\mathbb{R}^{n})$$c$$u_{0}$$\Phi_{0}$$:=\{u_{0}\in BUC(\mathbb{R}^{n});\exists\hat{u}_{0}\in C(\mathbb{R}^{n}):\mathbb{Z}^{n}\cdot pe\dot{n}odic$$\hat{u}_{0}\leq u_{0}$in$\mathbb{R}^{n}$andsuch that$\lim_{|x|arrow\infty}(u_{0}-\hat{u}_{0})(x)=0$},

there exists an asymptotic solution $\phi(x)-ct$ of (1).
We give here one of the simplest but most typical examples of Hamiltonian satisfying $(A1)-(A3)$ .

27
Example 1. Let $n=1$ , and define$h(x_{J}.p)$$h\in C(\mathbb{R}\cross \mathbb{R})$by$:=|p-1|^{2}-1-V(x)$ ,where
is a non-negative and -periodic function such that $\min_{R}V=0$
and $\int_{0}^{1}\sqrt{V(x)}dx<1$ . Remark that the equation $h(x, D\phi)=0$ in has bounded
solutions $(c.f. [11])$ .
Now, let $f\in C(R)$ be such that $f\geq 0$ and supp $f\subset B(O, 1)$ , and define $H$ by$V\in C(\mathbb{R})$$\mathbb{Z}$$\mathbb{R}$$H(x,p)$ $:=h(x,p)-f(x)$.

Clearly, $H$ satisfies $(A1)-(A3)$ . We shall prove that the equation$H(x, D\phi)=0$

has a solution in the classwith $H$ can be calculated as$BUC(\mathbb{R}^{n})$

in

(5)

$\mathbb{R}$. Observe first that the Lagrangian associated$L$$L(x, \xi)=\frac{1}{4}|\xi+2|^{2}+f(x)\geq 0$

For any$t>0$ ,

$x\in \mathbb{R}$, we define $\gamma_{x}\in AC((-\infty, 0$]) by$\gamma_{x}(s)$.

$:=x-2s$ . Then, for every$\int_{-t}^{0}L(\gamma_{x}(s),\dot{\gamma}_{x}(s))ds=\int_{-t}^{0}f(\gamma_{x}(s))ds\leq\max f$.

If we set$d(x,y)$ $:= \inf\{\int_{-t}^{0}L(\gamma(s),\dot{\gamma}(s))ds|t>0,$then,$d(\cdot, y)$is a viscosity solution of (5) in$\gamma\in AC([-t, 0]),$$\mathbb{R}\backslash \{y\}$$\gamma(-t)=y,$$\gamma(0)=x\}$,. Moreover, for any $a>0$ , we see.

$0 \leq d(x, x+a)\leq\int_{-\frac{\alpha}{2}}^{0}L(\gamma_{x}(s),\dot{\gamma}_{x}(s))ds\leq\max f$Thus, by the Ascoli-Arzela theorem, we conclude that there exists $\phi\in BUC(\mathbb{R}^{n})$such that $d(x, x+j_{k})arrow\phi$ inasfor some diverging sequence$karrow\infty$$Th\inftyrem$. In view of stability, is indeed a viscosity solution of (5). Hence,
2.1 is valid with $c=0$ .
$C(\mathbb{R})$$\{j_{k}\}_{k\in N}\subset N$$\phi$

3

Proof of Theorem 2.1.

This section is devoted to the proof of Theorem 2.1. Note first that in order to prove$Th\infty rem2.1$ , we need only to study thecase where $c=0$ by replacing, if necessary,

28$H-c$ and $u(x, t)+ct$ with $H$ and $u(x, t)$ , respectively. Therefore, from now on, wealways assume that $c=0$ .
(resp.
Let us denote bypersolutions) of$S_{H}^{+}$$S_{H}^{-}$) the totality of viscosity subsolutions (resp. su-in$H(x, D\phi)=0$

We set$S_{H}$$Lip(\mathbb{R}^{n})$.

Let$:=S_{H}^{-}\cap S_{H}^{+}$$\phi\in S_{H}$$\mathbb{R}^{n}$.

(6). It is known that, under (H1) and (H2), we have

. Then, for any$(x, t)\in \mathbb{R}^{n}\cross[0, \infty)$, we have$\phi(x)=\inf\{\int_{-t}^{0}L(\gamma(s),\dot{\gamma}(s))ds+\phi(\gamma(-t))|\gamma\in AC([-t, 0]),$ $\gamma(0)=x\}$

Fora givenp $>0$ , we denote bysatisfying $\gamma(0)=x$ and$\mathcal{E}_{\rho}((-\infty, 0$Lemma 3.1. For any
and a modulus$\lambda>1$$\mathcal{E}_{\rho}((-\infty, 0$(7)] $;x;\phi$ )$\neq\emptyset$

for all $t>0$ .

(8).

] $;x;\phi$ ), there exists
and,such that for $eve\eta\tau$ and $t>0$ satisfy ing$\phi\in S_{H_{f}}x\in \mathbb{R}^{n},$$\omega_{1}$.

] $;x;\phi$ ) the set of curves $\gamma\in AC((-\infty, 0$])$\phi(x)>\int_{-t}^{0}L(\gamma(s),\dot{\gamma}(s))ds+\phi(\gamma(-t))-\rho$

It is not difficult to check that

$S_{H}^{-}\subset$$\rho>0$$\gamma\in \mathcal{E}_{\rho}((-\infty, 0$$t\geq\lambda\tau$$u(x, t)- \phi(x)\leq u(\gamma(-t), \tau)-\phi(\gamma(-t))+\rho+\frac{t\tau}{t-\tau}\omega_{1}(\frac{\tau}{t-\tau})$.

(9)

Proof. This lemma has essentially been proved in [4] and [10]. So, we omit toreproduce the proof (see also {9]).

$\square$be the unique solution of Cauchy problem (1) satisfying$u($ . , $0)=u_{0}\in\Phi_{0}$ (see Appendix in [9] for the solvability of Cauchy problem (1)).
is bounded, we can take $A>0$ so
Let be any bounded solution of (6). Since. Remark also that $\phi+A$ andthat $\phi(x)-A\leq u_{0}(x)\leq\phi(x)+A$ for$\phi-A$ are solutions of (1) with initial data $\phi+A$ and $\phi-A$ , respectively. Then, thestandard comparison theorem for (1) infers that $\phi(x)-A\leq u(x, t)\leq\phi(x)+A$ for).
all $(x, t)\in \mathbb{R}^{n}\cross[0, +\infty)$ . In particular, is bounded on$u^{-}\in BUC(\mathbb{R}^{n})$ by
We next define

Proof of Theorem 2.1. Let$u$$\phi$$u_{0}$$aUx\in \mathbb{R}^{n}$$\mathbb{R}^{n}\cross[0, +\infty$$u$$u^{+},$$u^{+}(x)$$:= \lim_{tarrow+}\sup_{\infty}u(x, t)$,$u^{-}(x)$$:=1 \inf_{tarrow+\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}u(x, t)$.

are sub- andandNote that $hom$ the general theory of viscosity solution,
supersolutions of (6), respectively. Moreover, the convexity of $H(x, \cdot)$ implies thatis a bounded solution of (6).
is a subsolution of (6) (see [1]). In particular,$u^{+}$$u^{-}$$u^{-}$$u^{-}$

29
We now show that $u^{+}\leq u^{-}$ inand choose a diverging. Fix any
sequence
so that $u^{+}(y)= \lim_{jarrow\infty}u(y, t_{j})$ . Take any $\rho>0$ ,, and set $y_{j}=\gamma(-t_{j})$ for $j\in N$ .
$R^{n}$$y\in \mathbb{R}^{n}$$\{t_{j}\}_{j\in N}\subset(0, \infty)$$\gamma\in \mathcal{E}_{\rho}((-\infty, 0];y;u^{-})$

Case 1: $|y_{j}|arrow\infty$ as $jarrow$ oo.
In this case, since $h(x,p)$ is -periodic in , we may as sume by taking a subsequenceof
if necessary that there exists
by modsuch thatandfor some
and using (A3), we
as $jarrow\infty$ . Setting$\mathbb{Z}^{n}$$x$$\{y_{j}\}$$\mathbb{Z}^{d}$$\{\theta_{j}\}_{j\in N}\subset[0,1)^{n}$$\theta_{j}arrow\theta$$\theta\in[0,1]^{\dot{n}}$$y_{j}\equiv\theta_{j}$$\xi_{j}$$:=\theta-\theta_{j}$seein$H(\cdot+y_{j}, \cdot)arrow h(\cdot+\theta, \cdot)$$H(x+y_{j}+\xi_{j},p)\leq h(x+\theta,p)$

Similarly, by the definition ofthat$\Phi_{0}$for allin$u_{0}(x+y_{j}+\xi_{j})\geq\hat{u}_{0}(x+\theta)$

Case 2: $\sup_{j}|y_{j}|<\infty$ .
In this case, there exists$\xi_{j}:=z-y_{j}$ , we have$z\in \mathbb{R}^{n}$$\mathbb{Z}^{n}$-periodicas$C(\mathbb{R}^{n})$for all$jarrow\infty$,N.

$(x,p,j)\in \mathbb{R}^{n}\cross \mathbb{R}^{n}x$, there exists a$u_{0}(\cdot+y_{j})arrow\hat{u}_{0}(\cdot+\theta)$as$C(\mathbb{R}^{n}\cross \mathbb{R}^{n})$$\hat{u}_{0}\in BUC(\mathbb{R}^{n})$$jarrow\infty$

such

,

$(x,j)\in \mathbb{R}^{n}\cross \mathbb{R}^{n}\cross$N.

such that $y_{j}arrow z$ as $jarrow\infty$ . Thus, by setting$H(\cdot+y_{j}, \cdot)arrow H(\cdot+z, \cdot)$$H(x+y_{j}+\xi_{j},p)\leq H(x+z,p)$in$C(\mathbb{R}^{n}\cross \mathbb{R}^{n})$for allas $jarrow\infty$ ,$(x,p,j)\in \mathbb{R}^{n}\cross \mathbb{R}^{n}\cross N$

,

and
in

$u_{0}(\cdot+y_{j})arrow u_{0}(\cdot+z)$$uo(x+y_{j}+\xi_{j})\geq u_{0}(x+z)$for$C(\mathbb{R}^{n})$as$jarrow\infty$,$aU(x,j)\in \mathbb{R}^{n}\cross \mathbb{R}^{n}\cross$N.

Summarizing these two cases, we may assume that there exist functions$v0\in BUC(\mathbb{R}^{n})$ and a sequence
converging to zero such that

$G\in C(\mathbb{R}^{n}\cross$$\mathbb{R}^{n}),$$\{\xi_{j}\}_{j\in N}\subset \mathbb{R}^{n}$$H(\cdot+y_{j}, \cdot)arrow G$

in$C(\mathbb{R}^{n}\cross \mathbb{R}^{n})$$H(x+y_{j}+\xi_{j},p)\leq G(x,p)$for$u_{0}(\cdot+y_{j}+\xi_{j})arrow v_{0}$in$u_{0}(x+y_{j}+\xi_{j})\geq v_{0}(x)$for allas$jarrow\infty$,$a^{g}(x,p,j)\in \mathbb{R}^{n}\cross \mathbb{R}^{n}\cross N$

and$C(\mathbb{R}^{n})$as$jarrow\infty$,$(x,j)\in \mathbb{R}^{n}\cross N$

.

,

30
We now consider the Cauchy problem(10)$\{\begin{array}{ll}v_{t}+G(x, Dv)=0 in \mathbb{R}^{n}\cross(0, +\infty),v(\cdot, 0)=v_{0}(\cdot) on \mathbb{R}^{n},\end{array}$and let $v(x, t)$ be the unique viscosity solution of (10). We denote by
set of all sub- and supersolutions of$G(x, D\phi)=0$

respectively. Set
particular, the function$S_{G}$$:=S_{G}^{-}\cap S_{G}^{+}$

in$\mathbb{R}^{n}$$S_{G}^{+}$the,. Then, we can check that$v^{-}(x)$$S_{G}^{-},$$\emptyset\neq S_{G}\subset BUC(\mathbb{R}^{n})$. In$:= \lim_{tarrow+}\inf_{\infty}v(x, t)$is $wen$-defined and moreover $v^{-}\in S_{G}$ .
Now we apply Proposition 3.1 by taking $\phi:=u^{-}$ to get(11)$u(y,t_{j})-u^{-}(y) \leq u(y_{j}, \tau)-u^{-}(y_{j})+\rho+\frac{t_{j}\tau}{t_{j}-\tau}\omega_{1}(\frac{\tau}{t_{j}-\tau})$for every $\tau>0$ and sufficiently large $j\in N$ . On the other hand, by comparison andstability, we see$u(x+y_{j}+\xi_{j}, t)arrow v(x, t)$$u(x+y_{j}+\xi_{js}t)\geq v(x, t)$inas$C(\mathbb{R}^{n}\cross[0, \infty))$$jarrow\infty$,

for all $(x, t,j)\in \mathbb{R}^{n}\cross[0, \infty)\cross N$ .

In particular, the latter implies $u^{-}(y_{j})\geq v^{-}(-\xi_{j})$ for all $j\in N$ . Thus, in view of(11), we have$u(y, t_{j})-u^{-}(y) \leq.u(y_{j}, \tau)-v^{-}(-\xi_{j})+\rho+\frac{t_{j}\tau}{t_{j}-\tau}\omega_{1}(\frac{\tau}{t_{j}-\tau})$

.

(12)

Sending $jarrow\infty$ in (12), we have$u^{+}(y)-u^{-}(y)\leq v(0, \tau)-v^{-}(0)+\rho$Letting$\tau=\tau_{j}arrow\infty$

along a sequence$\{\tau_{j}\}$such that.

$\lim_{jarrow\infty}v(0, \tau_{j})=\lim\inf v(O, t)\iotaarrow\infty$

weare arbitrary, we conclude
finally obtain $u^{+}(y)\leq u^{-}(y)+\rho$ . Since $\rho>0$ and, and the proof of Theorem 2.1 has been completed.
inthat$y\in \mathbb{R}^{n}$$u^{+}\leq u^{-}$

4$\mathbb{R}^{\mathfrak{n}}$$\square$Final remarks.

Theorem 1 can be generalized considerably. We first introduce the notion of semiperiodicity,and obliquely semi-almost period.icity (see [9] for details).

31
Definition 1. A function
is called lower (resp. upper) semi-periodic if, a sequence
for any sequence, there exist a subsequenceofsuch that $\phi(\cdot+z_{j})arrow$converging to zero, and a functioninas $jarrow\infty$ and $\phi(x+z_{j}+\xi_{j})\geq\psi(x)$ (resp. $\phi(x+z_{j}+\xi_{j})\leq\psi(x)$ ).
for all$\phi\in C(\mathbb{R}^{n})$$\{z_{j}\}_{j\in N}$$\{y_{j}\}_{j\in N}\subset \mathbb{R}^{n}$$\psi\in C(\mathbb{R}^{n})$$\{\xi_{j}\}_{j\in N}\subset \mathbb{R}^{n}$$\psi(\cdot)$$\{y_{j}\}$$C(\mathbb{R}^{n})$$(x,j)\in \mathbb{R}^{n}\cross N$

Definition 2. A function
is called obliquely lower (resp. upper) semialmost periodic if for any sequence
and any $\epsilon>0$ , there exist a
subsequencesuch that $\phi(\cdot+z_{j})-$ofand a function$\phi(z_{j})arrow\psi(\cdot)$in
as $jarrow\infty$ and $\phi(x+z_{j})-\phi(z_{j})+\epsilon>\psi(x)$ (resp.
$\phi(x+z_{j})-\phi(z_{j})-\epsilon<\psi(x))$ for all.
$\phi\in C(\mathbb{R}^{n})$$\{y_{j}\}_{j\in N}\subset \mathbb{R}^{n}$$\{z_{j}\}_{j\in N}$$\psi\in C(\mathbb{R}^{n})$$\{y_{j}\}$$C(\mathbb{R}^{n})$$(x,j)\in \mathbb{R}^{n}\cross N$

Theorem 4.1 (Theorem 2.2 of [9]). Let $H$ be a Hamiltonian satisfying $(Hl)-(HS)$and$(H5)$$(H\theta)$

that,is upper semi-periodic for all
Theoe exzst a constantand functions
for some $C_{0}>0$ and$H(\cdot,p)$$p\in \mathbb{R}^{n}$$c\in \mathbb{R}$$\phi_{0}\leq u_{0}\leq\phi_{0}+C_{0}$$u_{0}^{-}(x)$$u_{0}^{-}\leq\psi 0$$:= \sup$Then, for any initia datum$l$$\Phi_{0}:=${$u_{0}$

{

and
such
, whereis defined by$\psi 0\in S_{H-c}^{+}$$\phi_{0}\in S_{H-c}^{-}$$\phi(x)|\phi\in S_{H-c}^{-}$,$\phi\leq u_{0}$$u_{0}^{-}$

in$\mathbb{R}^{n}$}.

belonging to$v\in UC(\mathbb{R}^{n});v$

is obliquely lower semi-almost$per\dot{\tau}odic$},

there enists an asymptotic solution of (1).
Remark. It seems that almost periodicity for $H$ might be sufficient to guarantee
the existence of as ymptotic solutions. In fact, there are a few examples that answer this question affirmatively. However, a complete research will be left in future
investigation.

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Hamilton-Jacobi equations with convex Hamiltonians. Comm. Partial. Diff.
Equ. $15(12):1713- 1742$ .
[2] Barles, G., Roquejoffre J.-M. (2006). Ergodic type problems and large time

behavior of unbounded solutions of Hamilton-Jacobi equations. Comm. Partial
Differential Equations 31:1209-1225.

32[3] Barles, G., Souganidis, P.E. (2000). On the large time behavior of solutions ofHamilton-Jacobi equations. SIAM J. Math. Math. Anal. $31(4):925- 939$ .
[4] Davini, A., Siconolfi, A. (2005). A generalized dynamical approach to the large

time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal.
$38(2):478- 502$.

[5] Fathi, A. (1997). Th\’eor\‘eme KAM faible et th\’eorie de Mather pour les syst\‘emes
lagrangiens. C.R. Acad. Sci. Paris S\’er. I $324(9):1043- 1046$ .

[6] Fathi, A. (1998). Sur la convergence du semi-groupe de Lax-Oleinik, C.R. Acad.
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equations in Euclidean space. Indiana Univ. Math. J. $55(5):1671- 1700$ .
$n$[8] Ichihara, N. (2006). Asymptotic solutions of Hamilton-Jacobi equations with
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[9] Ichihara, N., Ishii, H. (2006). Asymptotic solutions of Hamilton-Jacobi equations for semi-periodic Hamiltonians, preprint.
[10] Ishii, H. (2006). Asymptotic solutions for large time of Hamilton-Jacobi equa-tions in Euclidean$n$space. To appear in Ann. Inst. H. Poincar\’e Anal. Non

Lin\’eaire.

[11] Lions, P.-L., Papanicolaou, G., Varadhan, S.R.S. (1986). Homogenization ofHamilton-Jacobi equations. Unpublished preprint.
[12] Namah, G., Roquejoffie, J.-M, (1999). Remarks on the long time behaviourof the solutions of Hamilton-Jacobi equations. Comm. Partial. Diff. Equ. $24(5-$$6):883- 893$.