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