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]).
Let
be a subset of
,
and
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
hence we get
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
which can be rewrote as
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 