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Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations
Boundary Value Problems volume 2009, Article number: 873526 (2009)
Abstract
We generalize the comparison result 2007 on Hamilton-Jacobi equations to nonlinear parabolic equations, then by using Perron's method to study the existence and uniqueness of time almost periodic viscosity solutions of nonlinear parabolic equations under usual hypotheses.
1. Introduction
In this paper we will study the time almost periodic viscosity solutions of nonlinear parabolic equations of the form

where is a bounded open subset and
is its boundary. Here
and
denotes the set of symmetric
matrices equipped with its usual order (i.e., for
, we say that
if and only if
);
and
denote the gradient and Hessian matrix, respectively, of the function
w.r.t the argument
.
is almost periodic in
. Most notations and notions of this paper relevant to viscosity solutions are borrowed from the celebrated paper of Crandall et al. [1]. Bostan and Namah [2] have studied the time periodic and almost periodic viscosity solutions of first-order Hamilton-Jacobi equations. Nunziante considered the existence and uniqueness of viscosity solutions of parabolic equations with discontinuous time dependence in [3, 4], but the time almost periodic viscosity solutions of parabolic equations have not been studied yet as far as we know. We are going to use Perron's Method to study the existence of time almost periodic viscosity solutions of (1.1). Perron's Method was introduced by Ishii [5] in the proof of existence of viscosity solutions of first-order Hamilton-Jacobi equations, Crandall et al. had applications of Perron's Method to second-order partial differential equations in [1] except to parabolic case.
To study the existence and uniqueness of viscosity solutions of (1.1), we will use some results on the Cauchy-Dirichlet problem of the form

where is given. Crandall et al. studied the comparison result of the Cauchy-Dirichlet problem in [1], and it follows the maximum principle of Crandall and Ishii [6].
This paper is structured as follows. In Section 2, we present the definition and some properties of almost periodic functions. In Section 3, first we list some hypotheses and some results that will be used for existence and uniqueness of viscosity solutions, here we give an improvement of comparison result in paper [2] to fit for second-order parabolic equations; then we prove the uniqueness and existence of time almost periodic viscosity solutions. In the end, we concentrate on the asymptotic behavior of time almost periodic solutions for large frequencies.
2. Almost Periodic Functions
In this section we recall the definition and some fundamental properties of almost periodic functions. For more details on the theory of almost periodic functions and its application one can refer to Corduneanu [7] or Fink [8].
Proposition 2.1.
Let be a continuous function. The following conditions are equivalent:
(i) such that
satisfying

(ii) there is a trigonometric polynomial
where
such that
(iii)for all real sequence there is a subsequence
such that
converges uniformly on
Definition 2.2.
One saysthat a continuous function is almost periodicif and only if
satisfies one of the three conditions of Proposition 2.1.
A number verifying (2.1) is called
almost period. By using Proposition 2.1 we get the following property of almost periodic functions.
Proposition 2.3.
Assume that is almost periodic. Then
is bounded uniformly continuous function.
Proposition 2.4.
Assume that is almost periodic. Then
converges as
uniformly with respect to
Moreover the limit does not depend on
and it is called the average of
:

Proposition 2.5.
Assume that is almost periodic and denote by
a primitive of
. Then
is almost periodic if and only if
is bounded.
For the goal of applications to the differential equations, Yoshizawa [9] extended almost periodic functions to so called uniformly almost periodic functions.
Definition 2.6 ([9]).
One says that is almost periodic in
uniformly with respect to
if
is continuous in
uniformly with respect to
and
such that all interval of length
contain a number
which is
almost periodic for

3. Almost Periodic Viscosity Solutions
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
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Acknowledgment
The authors appreciate referee's careful reading and valuable suggestions. Partially supported by National Science Foundation of China (Grant no. 10371010).
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Zhang, S., Piao, D. Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations. Bound Value Probl 2009, 873526 (2009). https://doi.org/10.1155/2009/873526
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DOI: https://doi.org/10.1155/2009/873526