Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations
© S. Zhang and D. Piao. 2009
Received: 26 March 2009
Accepted: 9 June 2009
Published: 14 July 2009
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.
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. . Bostan and Namah  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  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  except to parabolic case.
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  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  or Fink .
Let be a continuous function. The following conditions are equivalent:
(ii) there is a trigonometric polynomial where such that
(iii)for all real sequence there is a subsequence such that converges uniformly on
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.
Assume that is almost periodic. Then is bounded uniformly continuous function.
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  extended almost periodic functions to so called uniformly almost periodic functions.
Definition 2.6 ().
3. Almost Periodic Viscosity Solutions
In this section we get some results for almost periodic viscosity solutions.
in (3.2) is an arbitrary open subset of .
In , Crandall et al. proved such a theorem.
Theorem 3.1 (see ).
To prove the existence and uniqueness of viscosity solutions, let us see the following main hypotheses first.
where . Then we will say that is proper.
Now we can easily prove the following result. There is a similar result for first-order Hamilton-Jacobi equations in the book of Barles .
Assume that and is a viscosity subsolution (resp., supersolution) of Then is a viscosity subsolution (resp., supersolution) of
The results following upon letting This process can be easily applied to the viscosity supersolution case.
By time periodicity one gets the following.
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 ).
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 ).
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  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 ).
Proposition 3.6 (see ).
Now we have the following.
Replacing by in the expression of we know that is integrable and denote by the function After integration one gets
Similar to the proof of Corollary 2.2 in paper , we can reach the conclusion.
Theorem 3.10 (Perron's method).
is a solution of (3.2).
From paper , 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.
where Then the conclusion follows by passing
Now we concentrate on the existence part.
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.
Since we already know that by time almost periodicity we deduce also that
Then we can prove our main theorem as follows.
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).
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 , it can be easily proved from Theorems 3.8, 3.9, and Remark 3.11.
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
Finally we deduce that for all
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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