Table of Contents
ISRN Mathematical Analysis
Volume 2011, Article ID 415358, 13 pages
http://dx.doi.org/10.5402/2011/415358
Research Article

Time Remotely Almost Periodic Viscosity Solutions of Hamilton-Jacobi Equations

1School of Mathematics, Shandong University, Jinan 250100, China
2School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Received 6 December 2010; Accepted 6 January 2011

Academic Editor: A. Carpio

Copyright © 2011 Shilin Zhang and Daxiong Piao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study some properties of the remotely almost periodic functions. This paper studies viscosity solutions of general Hamilton-Jacobi equations in the time remotely almost periodic case. Existence and uniqueness results are presented under usual hypotheses.

1. Introduction

In this paper we consider the viscosity solutions of first-order Hamilton-Jacobi equations of the form𝜕𝑡𝑢+𝐻(𝑥,𝑢,𝐷𝑢)=𝑓(𝑡),(𝑥,𝑡)𝑁×.(1.1) This problem was studied in [1] in the time periodic and almost periodic cases. And papers by Crandall and Lions (see [25]) proved the uniqueness and stability of viscosity solutions for a large class of equations, in particular for the initial value problem𝜕𝑡𝑢+𝐻(𝑥,𝑡,𝑢,𝐷𝑢)=0,(𝑥,𝑡)𝑁×][,0,𝑇𝑢(𝑥,0)=𝑢0(𝑥),𝑥𝑁(1.2) and also for the stationary problem𝐻(𝑥,𝑢,𝐷𝑢)=0,𝑥𝑁.(1.3) These results were extended by several papers, for example [6, 7].

Now in this paper we study this problem in a more regular condition, that is, in the time remotely almost periodic case. That is, we will look for such viscosity solutions when the Hamiltonian 𝐻 and 𝑓 are continuous functions 𝑓 is remotely almost periodic in 𝑡. The definition of remotely almost periodic was introduced by Sarason in 1984 in [8]. And Zhang and Yang in [9] and Zhang and Jiang in [10] gave such functions' applications.

This paper is structured as follows. In Section 2, we study a new type of almost periodic function—remotely almost periodic function. We present the definitions and prove some properties of such functions. Section 3 proves the uniqueness and existence of time remotely almost periodic viscosity solutions. In Section 3.1, we list some usual hypotheses used for the existence and uniqueness results and present two properties of viscosity solutions. In Section 3.2, we get some theorems for the uniqueness and existence of time remotely almost periodic viscosity solutions. And for the proof of the theorem we give two lemmas which play an important part. In Section 3.3, we concentrate on the asymptotic behaviour of time remotely almost periodic solutions for large frequencies.

In this paper, there are some abbreviations, like BUC, u.s.c, l.s.c, they stand for bounded uniformly continuous, upper semicontinuous, and lower semicontinuous, respectively. For the definition of viscosity subsolution and supersolution the reader can refer to [11].

2. Remotely Almost Periodic Function

It is well known that Bohr almost periodic function space is a Banach space, in which the distance is the supremum of the function. In [8], the author uses the superior limit as the distance in the space and defines a new type of almost periodic function, that is, remotely almost periodic function.

Definition 2.1. Let 𝑓 be a bounded uniformly continuous function on =(,+). We say that 𝑓 is remotely almost periodic if and only if for all 𝜀>0𝑇(𝑓,𝜀)=𝜏limsup|𝑡|||||𝑓(𝑡+𝜏)𝑓(𝑡)<𝜀(2.1) is relatively dense on . The number 𝜏𝑇(𝑓,𝜀) is called 𝜀 remotely almost period.
And RAP() denotes all these functions.

Definition 2.2. Let 𝑓 be a bounded uniformly continuous function on . We say that 𝑓 oscillates slowly if and only if for every 𝜏lim|𝑡|||||𝑓(𝑡+𝜏)𝑓(𝑡)=0.(2.2) And SO() denotes all these functions.

Next we will prove two propositions.

Proposition 2.3. Assume that 𝑓(𝑡) is remotely almost periodic and denote by 𝐹(𝑡)=𝑡0𝑓(𝑠)𝑑𝑠 a primitive of 𝑓(𝑡). Then 𝐹(𝑡) is remotely almost periodic if and only if 𝐹(𝑡) is bounded.

Proof. When 𝐹(𝑡) is remotely almost periodic, 𝐹(𝑡) is certainly bounded. For the converse, let 𝐹(𝑡) be bounded, without losing general, and assume that 𝐹(𝑡) is a real function. For any 𝜀>0, there exists 𝑡0>0 large enough; we have 𝐺=sup|𝑡|>𝑡0𝐹(𝑡)>𝑔=inf|𝑡|>𝑡0𝐹(𝑡);(2.3) take fixed 𝑡1 and 𝑡2, |𝑡1|>𝑡0, |𝑡2|>𝑡0, and assume that 𝑡1<𝑡2, satisfying 𝐹𝑡1𝜀<𝑔+6𝑡,𝐹2𝜀>𝐺6.(2.4) Assume that 𝑙=𝑙(𝜀1) is an interval length of 𝑇(𝑓,𝜀1), where 𝜀1=𝜀/6𝑑, 𝑑=|𝑡1𝑡2|. For every 𝛼𝑅, take 𝜏𝑇(𝑓,𝜀1)[𝛼𝑡1,𝛼𝑡1+𝑙].
As we already know that 𝑓(𝑡) is remotely almost periodic, then we have limsup|𝑡|||||𝑓(𝑡+𝜏)𝑓(𝑡)<𝜀1;(2.5) that is, for 𝜀1>0, there exists 𝑡0>0, and when |𝑡|>𝑡0, there is ||||𝑓(𝑡+𝜏)𝑓(𝑡)<𝜀1,𝜀1<𝑓(𝑡+𝜏)𝑓(𝑡)<𝜀1.(2.6) Now take 𝑠𝑖=𝑡𝑖+𝜏(𝑖=1,2),𝐿=𝑙+𝑑. So 𝑠1,𝑠2[𝛼,𝛼+𝐿], and 𝐹𝑠2𝑠𝐹1𝑡=𝐹2𝑡𝐹1𝑡2𝑡1𝑓(𝑡)𝑑𝑡+𝑡2𝑡+𝜏1+𝜏𝑡𝑓(𝑡)𝑑𝑡=𝐹2𝑡𝐹1+𝑡2𝑡1[]𝜀𝑓(𝑡+𝜏)𝑓(𝑡)𝑑𝑡>𝐺𝑔3𝜀1𝜀𝑑=𝐺𝑔2,(2.7) that is, 𝐹𝑠1+𝑠𝑔𝐺𝐹2<𝜀2;(2.8) as the formulas in two brackets of previous inequality are both nonnegative, so there are two numbers 𝑠1 and 𝑠2 in any interval of length 𝐿 satisfying simultaneously 𝐹𝑠1𝜀<𝑔+2𝑠,𝐹2𝜀>𝐺2.(2.9) Now take 𝜀2=𝜀/2𝐿, and we will prove that when 𝜏𝑇(𝑓,𝜀2), there is 𝜏𝑇(𝑓,𝜀). In fact for every 𝑡𝑅, we can choose 𝑠1 and 𝑠2 in the interval [𝑡,𝑡+𝐿] satisfying 𝐹(𝑠1)<𝑔+(𝜀/2) and 𝐹(𝑠2)>𝐺(𝜀/2). Hence for 𝜏𝑇(𝑓,𝜀2), there are, respectively, limsup|𝑡|(𝐹(𝑡+𝜏)𝐹(𝑡))=limsup|𝑡|𝐹𝑠1𝑠+𝜏𝐹1+𝑠1𝑡𝑓(𝑡)𝑑𝑡𝑠1+𝜏𝑡+𝜏𝑓𝜀(𝑡)𝑑𝑡>𝑔𝑔+2𝜀2𝐿=𝜀,limsup|𝑡|(𝐹(𝑡+𝜏)𝐹(𝑡))=limsup|𝑡|𝐹𝑠2𝑠+𝜏𝐹2+𝑠2𝑡𝑓(𝑡)𝑑𝑡𝑠2+𝜏𝑡+𝜏𝜀𝑓(𝑡)𝑑𝑡<𝐺𝐺2+𝜀2𝐿=𝜀.(2.10) So for 𝜏𝑇(𝑓,𝜀2), we have 𝜏𝑇(𝑓,𝜀); hence 𝐹(𝑡) is remotely almost periodic.

Proposition 2.4. Assume that 𝑓(𝑡) is remotely almost periodic. Then (1/𝑇)𝑎𝑎+𝑇𝑓(𝑡)𝑑𝑡 converges as 𝑇+ uniformly with respect to 𝑎𝑅. Moreover the limit does not depend on 𝑎, and it is called the average of 𝑓𝑓=lim𝑇+1𝑇𝑎𝑎+𝑇𝑓(𝑡)𝑑𝑡,uniformlywithrespectto𝑎𝑅.(2.11)

Proof. As 𝑓(𝑡)RAP(𝑅), then 𝑓(𝑡) is bounded, and for all𝜀>0, for all 𝜏𝑇(𝑓,𝜀), there exists 𝑠0>0, when |𝑡|>𝑠0,|𝑓(𝑡+𝜏)𝑓(𝑡)|<𝜀. Let 𝐺=sup𝑡𝑅|𝑓(𝑡)|, take 𝜀>0, and assume that 𝑙=𝑙(𝜀/4) is an interval length of 𝑇(𝑓,𝜀/4). Take 𝜏𝑇(𝑓,𝜀/4)[𝑎,𝑎+𝑙]; then for any 𝑎,𝑠𝑅||||𝑎𝑎+𝑠𝑓(𝑡)𝑑𝑡𝑠0||||=||||𝑓(𝑡)𝑑t𝜏𝜏+𝑠𝑠0+𝑎+𝑠𝜏+𝑠+𝜏𝑎||||𝑓(𝑡)𝑑𝑡𝑠0||||𝑓(𝑡+𝜏)𝑓(𝑡)𝑑𝑡+𝑎+𝑠𝜏+𝑠||||𝑓(𝑡)𝑑𝑡+𝜏0||||=𝑓(𝑡)𝑑𝑡𝑠00||||𝑓(𝑡+𝜏)𝑓(𝑡)𝑑𝑡+𝑠𝑠0||||+𝑓(𝑡+𝜏)𝑓(𝑡)𝑑𝑡𝑎+𝑠𝜏+𝑠||𝑓||(𝑡)𝑑𝑡+𝜏0||𝑓||(𝑡)𝑑𝑡sup𝑠0,𝑠||||𝑓(𝑡+𝜏)𝑓(𝑡)𝑠𝑠0+2𝐺𝑙+𝑠0<𝜀4𝑠𝑠0+2𝐺𝑙+𝑠0,(2.12) so ||||1𝑇𝑎𝑎+𝑇1𝑓(𝑡)𝑑𝑡𝑇𝑇0||||𝜀𝑓(𝑡)𝑑𝑡4𝑇𝑇𝑇0+2𝐺𝑙+𝑇0𝑇||||1,(2.13)𝑛𝑇0𝑛𝑇1𝑓(𝑡)𝑑𝑡𝑇𝑇0||||=1𝑓(𝑡)𝑑𝑡𝑛|||||𝑛𝑘=11𝑇𝑘𝑇(𝑘1)𝑇𝑓(𝑡)𝑑𝑡𝑇0|||||𝜀𝑓(𝑡)𝑑𝑡4𝑇𝑇𝑇0+2𝐺𝑙+𝑇0𝑇.(2.14) By passing 𝑛+ in (2.14), we get ||||1𝑓𝑇𝑇0||||𝜀𝑓(𝑡)𝑑𝑡4𝑇𝑇𝑇0+2𝐺𝑙+𝑇0𝑇.(2.15) Using triangle inequality from (2.13) and (2.15) we deduce ||||1𝑇𝑎𝑎+𝑇||||𝜀𝑓(𝑡)𝑑𝑡𝑓2𝑇𝑇𝑇0+4𝐺𝑙+𝑇0𝑇<𝜀,(2.16) if only 𝑇>(8𝐺(𝑙+𝑇0)/𝜀)𝑇0. That is, when 𝑇,(1/𝑇)𝑎𝑎+𝑇𝑓(𝑡)𝑑𝑡 converges at 𝑓 uniformly with respect to 𝑎𝑅. Moreover notice the identical equation 1𝑇𝑎𝑎+𝑇1𝑓(𝑡)𝑑𝑡=𝑇𝑇0𝑓(𝑡+𝑎)𝑑𝑡.(2.17) This means that the limit does not depend on 𝑎.

3. Remotely Almost Periodic Viscosity Solutions

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

Definition 3.1. One says that 𝑢𝑁× is remotely almost periodic in 𝑡 uniformly with respect to 𝑥 if 𝑢 is bounded and uniformly continuous in 𝑡 uniformly with respect to 𝑥 and for all 𝜀>0, and there exists𝑙(𝜀)>0 such that all intervals of length 𝑙(𝜀) contain a number 𝜏 which is 𝜀 remotely almost periodic for 𝑢(𝑥,), for all𝑥𝑁limsup|𝑡|||||𝑢(𝑥,𝑡+𝜏)𝑢(𝑥,𝑡)<𝜀,(𝑥,𝑡)𝑁×.(3.1)

3.1. Some Hypotheses and Theorems

In this section we list some usual hypotheses used for the uniqueness and existence results and present two properties of viscosity solutions.

First let us list some hypotheses in the stationary case:0<𝑅<+,𝛾𝑅>0𝐻(𝑥,𝑢,𝑝)𝐻(𝑥,𝑣,𝑝)𝛾𝑅(𝑢𝑣),𝑥𝑁,𝑅𝑣𝑢𝑅,𝑝𝑁,(3.2)𝑅>0,𝑚𝑅,lim𝑧0𝑚𝑅||||(𝑧)=0𝐻(𝑥,𝑢,𝑝)𝐻(𝑦,𝑢,𝑝)𝑚𝑅||||||𝑝||,𝑥𝑦1+𝑥,𝑦𝑁,𝑅𝑢𝑅,𝑝𝑁,(3.3)0<𝑅<+,lim||𝑝||+𝐻(𝑥,𝑢,𝑝)=+,uniformlyfor(𝑥,𝑢)𝑁×[]𝑅,𝑅,(3.4)0<𝑅<+,𝐻isuniformlycontinuouson𝑁×[]×𝑅,𝑅𝐵𝑅,(3.5)𝑀>0𝐻(𝑥,𝑀,0)0𝐻(𝑥,𝑀,0),𝑥𝑁.(3.6) From [1] we know that hypotheses (3.2), (3.3) or (3.4), (3.5), (3.6) ensure the existence of a unique solution for the stationary equation (1.3). And more regularly (3.2) can be replaced by𝐻(𝑥,𝑢,𝑝)𝐻(𝑥,𝑣,𝑝)0,𝑥𝑁,𝑣𝑢,𝑝𝑁(3.7) (which comes to taking 𝛾𝑅=0 in (3.2)).

When the Hamiltonian is time dependent the corresponding assumptions are0<𝑅<+,𝛾𝑅>0𝐻(𝑥,𝑡,𝑢,𝑝)𝐻(𝑥,𝑡,𝑣,𝑝)𝛾𝑅(𝑢𝑣),𝑥𝑁,0𝑡𝑇,𝑅𝑣𝑢𝑅,𝑝𝑁,(3.8)𝑅>0,𝑚𝑅||||𝐻(𝑥,𝑡,𝑢,𝑝)𝐻(𝑦,𝑡,𝑢,𝑝)𝑚𝑅||||||𝑝||,𝑥𝑦1+𝑥,𝑦𝑁[],𝑡0,𝑇,𝑅𝑢𝑅,𝑝𝑁,wherelim𝑧0𝑚𝑅(𝑧)=0,(3.9)0<𝑅<+,𝐻isuniformlycontinuouson𝑁×[]×[]×0,𝑇𝑅,𝑅𝐵𝑅,(3.10)𝑀>0𝐻(𝑥,𝑡,𝑀,0)0𝐻(𝑥,𝑡,𝑀,0),𝑥𝑁[].,𝑡0,𝑇(3.11) Now we present two results of viscosity solutions (see [1, 6, 7]).

Theorem 3.2. Assume that (3.8), (3.9), (3.10), and (3.11) hold (with 𝛾𝑅, for all 𝑅>0). Then for every 𝑢0BUC(𝑁) there is a unique viscosity solution 𝑢BUC(𝑁×[0,𝑇]) of (1.2), for all 𝑇>0.

Theorem 3.3. Let 𝑢 be a bounded time periodic viscosity u.s.c. subsolution of 𝜕𝑡𝑢+𝐻(𝑥,𝑡,𝑢,𝐷𝑢)=𝑓(𝑥,𝑡) in 𝑁× and 𝑣 a bounded time periodic viscosity l.s.c. supersolution of 𝜕𝑡𝑣+𝐻(𝑥,𝑡,𝑣,𝐷𝑣)=𝑔(𝑥,𝑡) in 𝑁×, where 𝑓,𝑔BUC(𝑁×) and 𝐻 are 𝑇 periodic such that (3.8), (3.9), and (3.10) hold. Then one has sup𝑥𝑁(𝑢(𝑥,𝑡)𝑣(𝑥,𝑡))sup𝑠𝑡tssup𝑥𝑁(𝑓(𝑥,𝜎)𝑔(𝑥,𝜎))𝑑𝜎.(3.12) Moreover, the hypothesis (3.9) can be replaced by 𝑢𝑊1,(𝑁×) or 𝑣𝑊1,(𝑁×).

3.2. Uniqueness and Existence of Time Remotely Almost Periodic Viscosity Solutions

In this section we establish uniqueness and existence results for time remotely almost periodic viscosity solutions. For the uniqueness we have the more general result.

Proposition 3.4. Let 𝑢 a bounded u.s.c. viscosity subsolution of 𝜕𝑡𝑢+𝐻(𝑥,𝑡,𝑢,𝐷𝑢)=𝑓(𝑥,𝑡), in 𝑁× and 𝑣 a bounded l.s.c. viscosity supersolution of 𝜕𝑡𝑣+𝐻(𝑥,𝑡,𝑣,𝐷𝑣)=𝑔(𝑥,𝑡), in 𝑁× where 𝑓,𝑔BUC(𝑁×) and (3.8), (3.9), (3.10) hold uniformly for 𝑡. Then one has for all 𝑡sup𝑥𝑁(𝑢(𝑥,𝑡)𝑣(𝑥,𝑡))+𝑒𝛾𝑡𝑡sup𝑥𝑁(𝑓(𝑥,𝜎)𝑔(𝑥,𝜎))+𝑑𝜎.(3.13) Moreover hypotheses (3.9) can be replaced by 𝑢𝑊1,(𝑁×) or 𝑣𝑊1,(𝑁×).

The proof of this proposition is similar to Proposition 6.5 in [1]. Hence we do not prove it here.

Before we concentrate on the existence part, let us see two important lemmas first. Now take (𝑡)=𝑡𝑒𝛾(𝜎𝑡)𝑓(𝜎)𝑑𝜎, where 𝛾>0 is a constant, 𝑡.

Lemma 3.5. If 𝑓(𝑡)SO(), then (𝑡)SO().

Proof. As 𝑓(𝑡)SO(), so for every 𝜏lim|𝑡|||||𝑓(𝑡+𝜏)𝑓(𝑡)=0.(3.14) Now for every 𝜏||||=||||(𝑡+𝜏)(𝑡)𝑡+𝜏𝑒𝛾(𝜎𝑡𝜏)𝑓(𝜎)𝑑𝜎𝑡𝑒𝛾(𝜎𝑡)||||=||||𝑓(𝜎)𝑑𝜎0𝑒𝛾𝜎𝑓(𝑡+𝜎+𝜏)𝑑𝜎0𝑒𝛾𝜎||||=||||𝑓(𝑡+𝜎)𝑑𝜎0𝑒𝛾𝜎[𝑓]||||(𝑡+𝜎+𝜏)𝑓(𝑡+𝜎)𝑑𝜎0𝑒𝛾𝜎||||𝑓(𝑡+𝜎+𝜏)𝑓(𝑡+𝜎)𝑑𝜎sup𝜎||||1𝑓(𝑡+𝜎+𝜏)𝑓(𝑡+𝜎)𝛾,(3.15) hence lim|𝑡|||||(𝑡+𝜏)(𝑡)limsup|𝑡|||||1𝑓(𝑡+𝜎+𝜏)𝑓(𝑡+𝜎)𝛾=0.(3.16) Since we already know that 𝑓(𝑡)BUC(), we deduce also that (𝑡)BUC(). That is, (𝑡)SO().

Lemma 3.6. If 𝑓(𝑡)RAP(), then (𝑡)RAP().

Proof. The main result in [8] proved that 𝑓(𝑡)RAP() is the closed subalgebra in 𝐶() created by AP() and SO(). Hence, if 𝑓(𝑡)RAP(), for every 𝜀>0, take 𝜀1=𝛾𝜀, there exists 𝑔1,𝑔2𝐴𝑃() and 𝜑1,𝜑2SO(); hence 𝑔𝑓1+𝜑1+𝑔2𝜑2<𝜀14.(3.17) If 𝜑2=0, consider a number 𝜏 which is an 𝜀1/2 remotely almost period of 𝑔1: ||||=||||(𝑡+𝜏)(𝑡)0𝑒𝛾𝜎[]||||𝑓(𝑡+𝜎+𝜏)𝑓(𝑡+𝜎)𝑑𝜎0𝑒𝛾𝜎||||𝑓(𝑡+𝜎+𝜏)𝑓(𝑡+𝜎)𝑑𝜎0𝑒𝛾𝜎||𝑔𝑓(𝑡+𝜎+𝜏)1(𝑡+𝜎+𝜏)+𝜑1||+(𝑡+𝜎+𝜏)𝑑𝜎0𝑒𝛾𝜎||𝑔1(𝑡+𝜎+𝜏)𝑔1||+(𝑡+𝜎)𝑑𝜎0𝑒𝛾𝜎||𝜑1(𝑡+𝜎+𝜏)𝜑1||+(𝑡+𝜎)𝑑𝜎0𝑒𝛾𝜎||𝑔𝑓(𝑡+𝜎)1(𝑡+𝜎)+𝜑1||<𝜀(𝑡+𝜎)𝑑𝜎1+𝜀4𝛾1+2𝛾0𝑒𝛾𝜎||𝜑1(𝑡+𝜎+𝜏)𝜑1||𝜀(𝑡+𝜎)𝑑𝜎+1.4𝛾(3.18) By using Lemma 3.5 we deduce limsup|𝑡|||||<𝜀(𝑡+𝜏)(𝑡)1+𝜀4𝛾1+𝜀2𝛾14𝛾=𝜀.(3.19) Thus this proves that any 𝜀1/2 remotely almost period of 𝑔1 is an 𝜀 remotely almost period of .
If 𝜑20, assume that 𝛿=min{𝜀1/4,𝜀1/(4𝜑2)}, and take number 𝜏 which is a common 𝛿 remotely almost period of 𝑔1 and 𝑔2. We will prove that 𝜏 is an 𝜀1/2 remotely almost period of (𝑔1+𝜑1+𝑔2𝜑2), and an 𝜀 remotely almost period of : ||𝑔2(𝑡+𝜎+𝜏)𝜑2(𝑡+𝜎+𝜏)𝑔2(𝑡+𝜎)𝜑2||||𝑔(𝑡+𝜎)2(𝑡+𝜎+𝜏)𝜑2(𝑡+𝜎+𝜏)𝑔2(𝑡+𝜎+𝜏)𝜑2||+||𝑔(𝑡+𝜎)2(𝑡+𝜎+𝜏)𝜑2(𝑡+𝜎)𝑔2(𝑡+𝜎)𝜑2||𝑔(𝑡+𝜎)2||𝜑2(𝑡+𝜎+𝜏)𝜑2||+𝜑(𝑡+𝜎)2||𝑔2(𝑡+𝜎+𝜏)𝑔2||.(𝑡+𝜎)(3.20) We have limsup|𝑡|||𝑔2(𝑡+𝜎+𝜏)𝜑2(𝑡+𝜎+𝜏)𝑔2(𝑡+𝜎)𝜑2||<𝜀(𝑡+𝜎)14.(3.21) Hence limsup|𝑡|||𝑔1(𝑡+𝜎+𝜏)+𝜑1(𝑡+𝜎+𝜏)+𝑔2(𝑡+𝜎+𝜏)𝜑2𝑔(𝑡+𝜎+𝜏)1(𝑡+𝜎)+𝜑1(𝑡+𝜎)+𝑔2(𝑡+𝜎)𝜑2||(𝑡+𝜎)limsup|𝑡|||𝑔1(𝑡+𝜎+𝜏)𝑔1||(𝑡+𝜎)+limsup|𝑡|||𝜑1(𝑡+𝜎+𝜏)𝜑1||(𝑡+𝜎)+limsup|𝑡|||𝑔2(𝑡+𝜎+𝜏)𝜑2(𝑡+𝜎+𝜏)𝑔2(𝑡+𝜎)𝜑2||<𝜀(𝑡+𝜎)14+𝜀14=𝜀12,||||||𝑔𝑓(𝑡+𝜎+𝜏)𝑓(𝑡+𝜎)𝑓(𝑡+𝜎+𝜏)1(𝑡+𝜎+𝜏)+𝜑1(𝑡+𝜎+𝜏)+𝑔2(𝑡+𝜎+𝜏)𝜑2||+||𝑔(𝑡+𝜎+𝜏)1(𝑡+𝜎+𝜏)+𝜑1(𝑡+𝜎+𝜏)+𝑔2(𝑡+𝜎+𝜏)𝜑2𝑔(𝑡+𝜎+𝜏)1(𝑡+𝜎)+𝜑1(𝑡+𝜎)+𝑔2(𝑡+𝜎)𝜑2||+||𝑔(𝑡+𝜎)1(𝑡+𝜎)+𝜑1(𝑡+𝜎)+𝑔2(𝑡+𝜎)𝜑2||.(𝑡+𝜎)𝑓(𝑡+𝜎)(3.22) So we have limsup|𝑡|||||(𝑡+𝜏)(𝑡)=limsup|𝑡|||||0𝑒𝛾𝜎[𝑓]||||(𝑡+𝜎+𝜏)𝑓(𝑡+𝜎)𝑑𝜎limsup|𝑡|||𝑓||1(𝑡+𝜎+𝜏)𝑓(𝑡+𝜎)𝛾<𝜀14+𝜀12+𝜀141𝛾=𝜀.(3.23) Since we already know that 𝑓(𝑡)BUC(), we deduce also that (𝑡)BUC(). So this proves that (𝑡)RAP().

Now we concentrate on the existence part.

Proposition 3.7. Assume that 𝑓 is remotely almost periodic and that the Hamiltonian 𝐻=𝐻(𝑥,𝑧,𝑝) satisfying the hypotheses (3.2), (3.3), (3.5), and there exists𝑀>0 such that 𝐻(𝑥,𝑀,0)𝑓(𝑡)𝐻(𝑥,𝑀,0), for all (𝑥,𝑡)𝑁×. Then there is a time remotely almost periodic viscosity solution in BUC(𝑁×) of 𝜕𝑡𝑢+𝐻(𝑥,𝑢,𝐷𝑢)=𝑓(𝑡), in 𝑁×.

Proof. We consider the unique viscosity solution of the problem 𝜕𝑡𝑢𝑛+𝐻𝑥,𝑢𝑛,𝐷𝑢𝑛=𝑓(𝑡),(𝑥,𝑡)𝑁×][,𝑢𝑛,+𝑛(𝑥,𝑛)=0,𝑥𝑁(3.24) for all 𝑛1. Such a solution exists by Theorem 3.2. Next we will prove that for all 𝑡, (𝑢𝑛(𝑡))𝑛𝑡 converges to a remotely almost periodic viscosity solution of 𝜕𝑡𝑢+𝐻(𝑥,𝑢,𝐷𝑢)=𝑓(𝑡), in 𝑁×. Similar to the proof of Proposition 6.6 in [1], we obtain by fixing 𝑡 and 𝑛 large enough ||𝑢𝑛(𝑥,𝑡)𝑢𝑛(||𝑥,𝑡+𝜏)2𝑀𝑒𝛾(𝑡𝑡𝑛)+𝑒𝛾𝑡𝑡𝑡𝑛𝑒𝛾𝜎||||𝑓(𝜎+𝜏)𝑓(𝜎)𝑑𝜎.(3.25) By passing 𝑛+ we have 𝑡𝑛, and therefore ||||𝑢(𝑥,𝑡)𝑢(𝑥,𝑡+𝜏)𝑡𝑒𝛾(𝑡𝜎)||||𝑓(𝜎+𝜏)𝑓(𝜎)𝑑𝜎.(3.26) As 𝑓 is remotely almost periodic, using Lemma 3.6 we deduce limsup|𝑡|||||𝑢(𝑥,𝑡)𝑢(𝑥,𝑡+𝜏)limsup|𝑡|𝑡𝑒𝛾(𝜎𝑡)||||𝑓(𝜎+𝜏)𝑓(𝜎)𝑑𝜎<𝜀.(3.27) Since we already know that 𝑢BUC(𝑁×[𝑎,𝑏]), for all 𝑎,𝑏,𝑎𝑏, by time remotely almost periodicity we deduce also that 𝑢BUC(𝑁×).

Now we will study the time remotely almost periodic viscosity solutions of𝜕𝑡𝑢+𝐻(𝑥,𝑢,𝐷𝑢)=𝑓(𝑡),(𝑥,𝑡)𝑁×,(3.28) for Hamiltonians satisfying (3.7). We introduce also the stationary equation1𝐻(𝑥,𝑢,𝐷𝑢)=𝑓=𝑇𝑇0𝑓(𝑡)𝑑𝑡,𝑥𝑁.(3.29) We have the following theorem for the existence of time remotely almost periodic viscosity solution.

Theorem 3.8. Assume that Hamiltonian H=𝐻(𝑥,𝑧,𝑝) satisfies hypotheses (3.7), (3.4), (3.5), sup{|𝐻(𝑥,0,0)|𝑥}=𝐶<+ and 𝑓 is a time remotely almost periodic function such that 𝐹(𝑡)=𝑡0{𝑓(𝜎)𝑓}𝑑𝜎 is bounded on . Then there is a bounded Lipschitz time remotely almost periodic viscosity solution of (3.28) if and only if there is a bounded viscosity solution of (3.29).

Proof. Assume that 𝑉 is a bounded viscosity of (3.29). We deduce that 𝑉 is a Lipschitz function as the Hamiltonian satisfies (3.4). For any 𝛼>0, take 𝑀𝛼=𝑉𝐿(𝑁)+(1/𝛼)(𝐶+𝑓𝐿()). By Propositions 3.4 and 3.7 we can construct the family of time remotely almost periodic solutions 𝑣𝛼 for 𝛼𝑣𝛼𝑉(𝑥)+𝜕𝑡𝑣𝛼+𝐻𝑥,𝑣𝛼,𝐷𝑣𝛼=𝑓(𝑡),(𝑥,𝑡)𝑁×.(3.30) Similar to Theorems 4.1 and 6.1 in [1], we can extract a sequence which converges uniformly on compact sets of 𝑁× towards a bounded Lipschitz solution 𝑣 of (3.28). Next we will prove that 𝑣 is remotely almost periodic. By the hypotheses and Proposition 2.3 we deduce that 𝐹 is remotely almost periodic, and thus, for all 𝜀>0, there is 𝑙(𝜀/2) such that any interval of length 𝑙(𝜀/2) contains an 𝜀/2 remotely almost period of 𝐹. Take an interval of length 𝑙(𝜀/2) and 𝜏 an 𝜀/2 remotely almost period of 𝐹 in this interval. We have for all 𝛼>0,(𝑥,𝑡)𝑁×||𝑣𝛼(𝑥,𝑡+𝜏)𝑣𝛼||||||(𝑥,𝑡)sup𝑠𝑡𝑡𝑠||||=||||{𝑓(𝜎+𝜏)𝑓(𝜎)}𝑑𝜎sup𝑠𝑡𝑡+𝜏𝑠+𝜏(𝑓(𝜎)𝑓)𝑑𝜎𝑡𝑠(||||=||||𝑓(𝜎)𝑓)𝑑𝜎sup𝑠𝑡||||{(𝐹(𝑡+𝜏)𝐹(𝑡))(𝐹(𝑠+𝜏)𝐹(𝑠))}2sup𝑡||𝐹||.(𝑡+𝜏)𝐹(𝑡)(3.31) After passing to the limit for 𝛼0 one gets |𝑣(𝑥,𝑡+𝜏)𝑣(𝑥,𝑡)|2sup𝑡|𝐹(𝑡+𝜏)𝐹(𝑡)|, and hence limsup|𝑡|||||𝑣(𝑥,𝑡+𝜏)𝑣(𝑥,𝑡)2limsup|𝑡|||||𝐹(𝑡+𝜏)𝐹(𝑡)𝜀.(3.32) By using the uniform continuity of 𝐹, we can prove exactly in the same manner that 𝑣 is continuous in 𝑡 uniformly with respect to 𝑥. The converse implication follows similarlyTheorem 4.1 in [1]; here we do not prove it.

3.3. Asymptotic Behaviour for Large Frequencies

In this section we study the asymptotic behaviour of time remotely almost periodic viscosity solutions of𝜕𝑡𝑢𝑛+𝐻𝑥,𝑢𝑛,𝐷𝑢𝑛=𝑓𝑛(𝑡),(𝑥,𝑡)𝑁×,(3.33) where 𝑓 is a remotely almost periodic function. For all 𝑛1 notice that 𝑓𝑛(𝑡)=𝑓(𝑛𝑡), for all 𝑡 is remotely almost periodic and has the same average as 𝑓. Now suppose that such a hypothesis exists𝑀>0suchthat𝐻(𝑥,𝑀,0)𝑓(𝑡),(𝑥,𝑡)𝑁×.(3.34)

Theorem 3.9. Let 𝐻=𝐻(𝑥,𝑧,𝑝) be a Hamiltonian satisfying (3.7), (3.3), (3.5), (3.34) where 𝑓 is remotely almost periodic function. Suppose also that there is a bounded l.s.c viscosity supersolution 𝑉𝑀 of (3.29), that 𝑡𝐹(𝑡)=𝑡0{𝑓(𝑠)𝑓}𝑑𝑠 is bounded, and denote by 𝑉 the minimal stationary l.s.c. viscosity supersolution of (3.29), 𝑣𝑛 the time remotely almost periodic l.s.c. viscosity supersolution of (3.33). Then the sequence (𝑣𝑛)𝑛 converges uniformly on 𝑁× towards 𝑉 and 𝑣𝑛𝑉𝐿(𝑁×)(2/𝑛)𝐹𝐿(), for all 𝑛1.

Proof. As 𝑣𝑛=sup𝛼>0𝑣𝑛,𝛼 is remotely almost periodic, we introduce 𝑤𝑛,𝛼(𝑥,𝑡)=𝑣𝑛,𝛼(𝑥,𝑡/𝑛), (𝑥,𝑡)𝑁×, which is also remotely almost periodic. Similar to Theorem 5.1 in [1] and by using Theorem 3.3 we deduce that 𝑤𝑛,𝛼(𝑥,𝑡)𝑉𝛼(𝑥)sup𝑠𝑡1𝑛𝑡𝑠(2𝑓(𝜎)𝑓)𝑑𝜎𝑛𝐹𝐿(),(3.35) and similarly 𝑉𝛼(𝑥)𝑤𝑛,𝛼(𝑥,𝑡)(2/𝑛)𝐹𝐿(), for all 𝑛1. We have for all 𝑛1||𝑤𝑛,𝛼(𝑥,𝑡)𝑉𝛼||2(𝑥)𝑛𝐹𝐿(),(3.36) and after passing to the limit for 𝛼0 one gets for all (𝑥,𝑡)𝑁×||𝑤𝑛||2(𝑥,𝑡)𝑉(𝑥)𝑛𝐹𝐿().(3.37) Finally we deduce that 𝑣𝑛𝑉𝐿(𝑁×)(2/𝑛)𝐹𝐿(𝑁×) for all 𝑛1.

Acknowledgment

This work was supported by National Science Foundation of China (Grant no. 11001152).

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