In this section we get some results for almost periodic viscosity solutions.

We consider the following two equations to get some results used for the existence and uniqueness of almost periodic viscosity solutions. That is, the Dirichlet problems of the form

in (3.2)
is an arbitrary open subset of
.

In [1], Crandall et al. proved such a theorem.

Theorem 3.1 (see [1]).

Let

be a locally compact subset of

for

and

be twice continuously differentiable in a neighborhood of

Set

and suppose

is a local maximum of

relative to

Then for each

there exists

such that

and the block diagonal matrix with entries

satisfies

where

Put

where

recall that

then, from Theorem 3.1, at a local maximum

of

, we have

We conclude that for each

there exists

such that

Choosing

one can get

To prove the existence and uniqueness of viscosity solutions, let us see the following main hypotheses first.

As in Crandall et al. [

1], we present a fundamental monotonicity condition of

, that is,

where
. Then we will say that
is *proper*.

Assume there exists

such that

and there is a function

that satisfies

such that

Now we can easily prove the following result. There is a similar result for first-order Hamilton-Jacobi equations in the book of Barles [10].

Lemma 3.2.

Assume that
and
is a viscosity subsolution (resp., supersolution) of
Then
is a viscosity subsolution (resp., supersolution) of

Proof.

Since

is a viscosity subsolution of

if

and local maximum

of

, we have

Now we prove that if

is a local maximum of

in

, then

Suppose that

is a strict local maximum of

in

we consider the function

for small

Then we know that the function

has a local maximum point

such that

and

when

. So at the point

we deduce that

As the term

is positive, so we obtain

The results following upon letting
This process can be easily applied to the viscosity supersolution case.

By time periodicity one gets the following.

Proposition 3.3.

Assume that
and
are
periodic such that
is a viscosity subsolution (resp., supersolution) of
Then
is a viscosity subsolution (resp., supersolution) of

Crandall et al. have proved the following two comparison results.

Theorem 3.4 (see [6]).

Let
be a bounded open subset of
,
be proper and satisfy (3.11), (3.12). Let
(resp.,
) be a subsolution (resp., supersolution) of
in
and
on
. Then
in
.

Theorem 3.5 (see [1]).

Let
be open and bounded. Let
be continuous, proper, and satisfy (3.12) for each fixed
with the same function
. If
is a subsolution of (1.2) and
is a supersolution of (1.2), then
on

We generalize the comparison result in article [2] for first-order Hamilton-Jacobi equations, and get two theorems for second-order parabolic equations. Let us see a proposition we will need in the proof of the comparison result (see [1]).

Proposition 3.6 (see [1]).

for

Let

and

be chosen so that

Then the following holds:

Remark 3.7.

In Proposition 3.6, when

are replaced by

, respectively, we can get the following results:

Now we have the following.

Theorem 3.8.

Let

be open and bounded. Assume

be continuous, proper, and satisfy (3.11), (3.12) for each fixed

Let

be bounded u.s.c. subsolution of

in

respectively, l.s.c. supersolution of

in

where

Then one has for all

where

Proof.

Let us consider the function given by

where

, and

As we know that

and

are bounded semicontinuous in

and

is open and bounded, we can find

for

such that

here without loss of generality, we can assume that

Since

is compact, these maxima

converge to a point of the form

from Remark 3.7. From Theorem 3.1 and its following discussion, there exists

such that

which implies

At the maximum point, from the definition of

being a subsolution and

being a supersolution we arrive at the following:

by the

*proper* condition of

, we have

as we know that

satisfying (3.12) then we deduce that

where

For any

consider

if

and

otherwise. From hypothesis (3.11) we deduce that

is nondecreasing with respect to

then we have

for all

Hence we have

Notice that

we get

Replacing
by
in the expression of
we know that
is integrable and denote by
the function
After integration one gets

Now taking

instead of

for any

and letting

we can get

Finally we deduce that for all

Theorem 3.9.

Let

be open and bounded. Assume

be continuous, proper,

periodic, and satisfy (3.11), (3.12). Let

be a bounded time periodic viscosity u.s.c. subsolution of

in

and

a bounded time periodic viscosity l.s.c. supersolution of

in

where

Then one has

Proof.

As the proof of Theorem 3.8, we get equation (

3.34)

We introduce that

By integration by parts we have

We deduce that for all

we have

Similar to the proof of Corollary 2.2 in paper [2], we can reach the conclusion.

In order to prove the existence of viscosity solution, we recall the the Perron's method as follows (see [1, 5]). To discuss the method, we assume if
where
then

Theorem 3.10 (Perron's method).

Let comparison hold for (3.2); that is, if

is a subsolution of (3.2) and

is a supersolution of (3.2), then

Suppose also that there is a subsolution

and a supersolution

of (3.2) that satisfies the boundary condition

for

Then

is a solution of (3.2).

From paper [1], we have the following remarks as a supplement to Theorem 3.10.

Remark 3.11 s.

Notice that the subset
in (3.2) in some part of the proof in Theorem 3.10 was just open in
. In order to generalize this and formulate the version of Theorem 3.10 we will need later, we now make some remarks. Suppose
is locally compact,
are defined on
and have the following properties:
is upper semicontinuous,
is lower semicontinuous, and classical solutions (twice continuously differentiable solutions in the pointwise sense) of
on relatively open subset of
are solutions of
Suppose, moreover, that whenever
is a solution of
on
and
is a solution of
on
we have
on
. Then we conclude that the existence of such a subsolution and supersolution guarantees that there is a unique function
, obtained by the Perron's construction, that is a solution of both
and
on
.

Now we will prove the uniqueness and existence of almost periodic viscosity solutions. For the uniqueness we have the following result.

Theorem 3.12.

Let

be open and bounded. Assume

be continuous, proper, and satisfy (3.11), (3.12) for

Let

be a bounded u.s.c. viscosity subsolution of

, and

a bounded l.s.c. viscosity supersolution of

,

where

Then one has for all

Proof.

Take

and by using Theorem 3.8 write for all

where
Then the conclusion follows by passing

Now we concentrate on the existence part.

Theorem 3.13.

Let
be a bounded open subset in
. Assume
be continuous, proper, and satisfy (3.11), (3.12). Assume that
is almost periodic and
Then there is a time almost periodic viscosity solution in
of (1.1), where
is a constant.

Proof.

Here we consider the problem

for all

. As we know that

, there exists a viscosity solution

of (3.44) from Theorem 3.5 and Remark 3.11. Then we will prove that for all

converges to a almost periodic viscosity solution of (1.1). As we already know that

we can deduce by Theorem 3.5 that

Similar to the proof of Proposition 6.6 in paper [

2], using Theorem 3.8, we get for

and

large enough

By passing

we have

and therefore

Since we already know that
by time almost periodicity we deduce also that

When

does not satisfy the hypothesis (3.11), we study the time almost periodic viscosity solutions of

We introduce also the stationary equation

Then we can prove our main theorem as follows.

Theorem 3.14.

Let
be open and bounded. Assume
be continuous, proper, and satisfy (3.12) for
Assume that
is almost periodic function such that
is bounded on
Then there is a bounded time almost periodic viscosity solution of (3.47) and if only if there is a bounded viscosity solution of (3.48).

Proof.

Let

, then

.Assume that (3.48) has a bounded viscosity solution

, we take

for

, and observe that

Then by using Perron's Method from Theorem 3.10 and Remark 3.11 we can construct the family of solutions

for

and the family of time almost periodic solutions

for

In fact we have

for any

and by using Theorem 3.9 we have

similarly we can get

From the above two inequalities we know that the family

is bounded, thus we know

Therefore we can extract a subsequence which converges uniformly on compact sets of

to a bounded uniformly continuous function

of (3.47). Next we will check that

is almost periodic. By the hypotheses and Proposition 2.5 we deduce that

is almost periodic and thus, for all

there is

such that any interval of length

contains an

almost period of

. Take an interval of length

and

an

almost period of

in this interval. We have for all

After passing to the limit for
one gets
Hence we prove the almost periodic of
.

The converse is similar to Theorem 4.1 in paper [2], it can be easily proved from Theorems 3.8, 3.9, and Remark 3.11.

Now we discuss asymptotic behavior of time almost periodic viscosity solutions for large frequencies, and there is a similar description for Hamilton-Jacobi equations in paper [

2]. Let us see the following equation:

where

is almost a periodic function. For all

notice that

is almost periodic and has the same average as

. Now suppose that such a hypothesis exists

Theorem 3.15.

Let
be open and bounded. Assume
be continuous, proper, and satisfy (3.12) for
and (3.55) where
is almost periodic function. Suppose also that there is a bounded l.s.c viscosity supersolution
of (3.48), that
is bounded and denote by
the minimal stationary, respectively, time almost periodic l.s.c. viscosity supersolution of (3.48), respectively, (3.54). Then the sequence
converges uniformly on
towards
and

Proof.

As

is almost periodic, we introduce

which is also almost periodic. As

satisfies in the viscosity sense

we deduce that

satisfies in the viscosity sense

Recall also that we have in the viscosity sense

By using Theorem 3.9 we deduce that

and similarly

We have for all

and after passing to the limit for

one gets for all

Finally we deduce that
for all