Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations

  • Shilin Zhang1 and

    Affiliated with

    • Daxiong Piao1Email author

      Affiliated with

      Boundary Value Problems20092009:873526

      DOI: 10.1155/2009/873526

      Received: 26 March 2009

      Accepted: 9 June 2009

      Published: 14 July 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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ1_HTML.gif
      (1.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq1_HTML.gif is a bounded open subset and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq2_HTML.gif is its boundary. Here http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq3_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq4_HTML.gif denotes the set of symmetric http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq5_HTML.gif matrices equipped with its usual order (i.e., for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq6_HTML.gif , we say that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq7_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq8_HTML.gif ); http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq10_HTML.gif denote the gradient and Hessian matrix, respectively, of the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq11_HTML.gif w.r.t the argument http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq12_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq13_HTML.gif is almost periodic in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq14_HTML.gif . 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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ2_HTML.gif
      (1.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq15_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq16_HTML.gif be a continuous function. The following conditions are equivalent:

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq17_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq18_HTML.gif satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ3_HTML.gif
      (2.1)

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq19_HTML.gif there is a trigonometric polynomial http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq20_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq21_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq22_HTML.gif

      (iii)for all real sequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq23_HTML.gif there is a subsequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq24_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq25_HTML.gif converges uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq26_HTML.gif

      Definition 2.2.

      One saysthat a continuous function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq27_HTML.gif is almost periodicif and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq28_HTML.gif satisfies one of the three conditions of Proposition 2.1.

      A number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq29_HTML.gif verifying (2.1) is called http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq30_HTML.gif almost period. By using Proposition 2.1 we get the following property of almost periodic functions.

      Proposition 2.3.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq31_HTML.gif is almost periodic. Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq32_HTML.gif is bounded uniformly continuous function.

      Proposition 2.4.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq33_HTML.gif is almost periodic. Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq34_HTML.gif converges as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq35_HTML.gif uniformly with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq36_HTML.gif Moreover the limit does not depend on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq37_HTML.gif and it is called the average of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq38_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ4_HTML.gif
      (2.2)

      Proposition 2.5.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq39_HTML.gif is almost periodic and denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq40_HTML.gif a primitive of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq41_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq42_HTML.gif is almost periodic if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq43_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq44_HTML.gif is almost periodic in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq45_HTML.gif uniformly with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq46_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq47_HTML.gif is continuous in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq48_HTML.gif uniformly with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq50_HTML.gif such that all interval of length http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq51_HTML.gif contain a number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq52_HTML.gif which is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq53_HTML.gif almost periodic for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq54_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ5_HTML.gif
      (2.3)

      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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ6_HTML.gif
      (3.1)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ7_HTML.gif
      (3.2)

      in (3.2) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq55_HTML.gif is an arbitrary open subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq56_HTML.gif .

      In [1], Crandall et al. proved such a theorem.

      Theorem 3.1 (see [1]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq57_HTML.gif be a locally compact subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq58_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq59_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ8_HTML.gif
      (3.3)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq61_HTML.gif be twice continuously differentiable in a neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq62_HTML.gif Set
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ9_HTML.gif
      (3.4)
      and suppose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq63_HTML.gif is a local maximum of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq64_HTML.gif relative to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq65_HTML.gif Then for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq66_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq67_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ10_HTML.gif
      (3.5)
      and the block diagonal matrix with entries http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq68_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ11_HTML.gif
      (3.6)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq69_HTML.gif

      Put http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq70_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq71_HTML.gif recall that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq72_HTML.gif then, from Theorem 3.1, at a local maximum http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq73_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq74_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ12_HTML.gif
      (3.7)
      We conclude that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq75_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq76_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ13_HTML.gif
      (3.8)
      Choosing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq77_HTML.gif one can get
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ14_HTML.gif
      (3.9)

      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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq78_HTML.gif , that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ15_HTML.gif
      (3.10)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq79_HTML.gif . Then we will say that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq80_HTML.gif is proper.

      Assume there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq81_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ16_HTML.gif
      (3.11)
      and there is a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq82_HTML.gif that satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq83_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ17_HTML.gif
      (3.12)

      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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq85_HTML.gif is a viscosity subsolution (resp., supersolution) of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq86_HTML.gif Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq87_HTML.gif is a viscosity subsolution (resp., supersolution) of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq88_HTML.gif

      Proof.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq89_HTML.gif is a viscosity subsolution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq90_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq91_HTML.gif and local maximum http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq92_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq93_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ18_HTML.gif
      (3.13)
      Now we prove that if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq94_HTML.gif is a local maximum of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq95_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq96_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ19_HTML.gif
      (3.14)
      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq97_HTML.gif is a strict local maximum of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq98_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq99_HTML.gif we consider the function
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ20_HTML.gif
      (3.15)
      for small http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq100_HTML.gif Then we know that the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq101_HTML.gif has a local maximum point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq102_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq103_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq104_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq105_HTML.gif . So at the point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq106_HTML.gif we deduce that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ21_HTML.gif
      (3.16)
      As the term http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq107_HTML.gif is positive, so we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ22_HTML.gif
      (3.17)

      The results following upon letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq108_HTML.gif This process can be easily applied to the viscosity supersolution case.

      By time periodicity one gets the following.

      Proposition 3.3.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq109_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq110_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq111_HTML.gif periodic such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq112_HTML.gif is a viscosity subsolution (resp., supersolution) of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq113_HTML.gif Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq114_HTML.gif is a viscosity subsolution (resp., supersolution) of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq115_HTML.gif

      Crandall et al. have proved the following two comparison results.

      Theorem 3.4 (see [6]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq116_HTML.gif be a bounded open subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq117_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq118_HTML.gif be proper and satisfy (3.11), (3.12). Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq119_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq120_HTML.gif ) be a subsolution (resp., supersolution) of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq121_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq122_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq123_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq124_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq125_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq126_HTML.gif .

      Theorem 3.5 (see [1]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq127_HTML.gif be open and bounded. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq128_HTML.gif be continuous, proper, and satisfy (3.12) for each fixed http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq129_HTML.gif with the same function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq130_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq131_HTML.gif is a subsolution of (1.2) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq132_HTML.gif is a supersolution of (1.2), then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq133_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq134_HTML.gif

      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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq135_HTML.gif be a subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq136_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq137_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ23_HTML.gif
      (3.18)
      for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq138_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq139_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq140_HTML.gif be chosen so that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ24_HTML.gif
      (3.19)
      Then the following holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ25_HTML.gif
      (3.20)

      Remark 3.7.

      In Proposition 3.6, when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq141_HTML.gif are replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq142_HTML.gif , respectively, we can get the following results:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ26_HTML.gif
      (3.21)

      Now we have the following.

      Theorem 3.8.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq143_HTML.gif be open and bounded. Assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq144_HTML.gif be continuous, proper, and satisfy (3.11), (3.12) for each fixed http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq145_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq146_HTML.gif be bounded u.s.c. subsolution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq147_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq148_HTML.gif respectively, l.s.c. supersolution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq149_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq150_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq151_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ27_HTML.gif
      (3.22)
      Then one has for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq152_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ28_HTML.gif
      (3.23)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq153_HTML.gif

      Proof.

      Let us consider the function given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ29_HTML.gif
      (3.24)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq154_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq155_HTML.gif As we know that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq156_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq157_HTML.gif are bounded semicontinuous in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq158_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq159_HTML.gif is open and bounded, we can find http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq160_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq161_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq162_HTML.gif here without loss of generality, we can assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq163_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq164_HTML.gif is compact, these maxima http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq165_HTML.gif converge to a point of the form http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq166_HTML.gif from Remark 3.7. From Theorem 3.1 and its following discussion, there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq167_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ30_HTML.gif
      (3.25)
      which implies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq168_HTML.gif At the maximum point, from the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq169_HTML.gif being a subsolution and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq170_HTML.gif being a supersolution we arrive at the following:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ31_HTML.gif
      (3.26)
      by the proper condition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq171_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ32_HTML.gif
      (3.27)
      as we know that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq172_HTML.gif satisfying (3.12) then we deduce that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ33_HTML.gif
      (3.28)
      hence we get
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ34_HTML.gif
      (3.29)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq173_HTML.gif For any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq174_HTML.gif consider
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ35_HTML.gif
      (3.30)
      if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq175_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq176_HTML.gif otherwise. From hypothesis (3.11) we deduce that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq177_HTML.gif is nondecreasing with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq178_HTML.gif then we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq179_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq180_HTML.gif Hence we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ36_HTML.gif
      (3.31)
      Notice that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq181_HTML.gif we get
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ37_HTML.gif
      (3.32)

      Replacing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq182_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq183_HTML.gif in the expression of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq184_HTML.gif we know that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq185_HTML.gif is integrable and denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq186_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq187_HTML.gif After integration one gets

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ38_HTML.gif
      (3.33)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq188_HTML.gif Now taking http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq189_HTML.gif instead of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq190_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq191_HTML.gif and letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq192_HTML.gif we can get
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ39_HTML.gif
      (3.34)
      Finally we deduce that for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq193_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ40_HTML.gif
      (3.35)

      Theorem 3.9.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq194_HTML.gif be open and bounded. Assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq195_HTML.gif be continuous, proper, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq196_HTML.gif periodic, and satisfy (3.11), (3.12). Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq197_HTML.gif be a bounded time periodic viscosity u.s.c. subsolution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq198_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq199_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq200_HTML.gif a bounded time periodic viscosity l.s.c. supersolution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq201_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq202_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq203_HTML.gif Then one has
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ41_HTML.gif
      (3.36)

      Proof.

      As the proof of Theorem 3.8, we get equation (3.34)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ42_HTML.gif
      (3.37)
      We introduce that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq204_HTML.gif By integration by parts we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ43_HTML.gif
      (3.38)
      We deduce that for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq205_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ44_HTML.gif
      (3.39)

      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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq206_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq207_HTML.gif then

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ45_HTML.gif
      (3.40)

      Theorem 3.10 (Perron's method).

      Let comparison hold for (3.2); that is, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq208_HTML.gif is a subsolution of (3.2) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq209_HTML.gif is a supersolution of (3.2), then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq210_HTML.gif Suppose also that there is a subsolution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq211_HTML.gif and a supersolution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq212_HTML.gif of (3.2) that satisfies the boundary condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq213_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq214_HTML.gif Then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ46_HTML.gif
      (3.41)

      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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq215_HTML.gif in (3.2) in some part of the proof in Theorem 3.10 was just open in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq216_HTML.gif . In order to generalize this and formulate the version of Theorem 3.10 we will need later, we now make some remarks. Suppose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq217_HTML.gif is locally compact, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq218_HTML.gif are defined on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq219_HTML.gif and have the following properties: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq220_HTML.gif is upper semicontinuous, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq221_HTML.gif is lower semicontinuous, and classical solutions (twice continuously differentiable solutions in the pointwise sense) of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq222_HTML.gif on relatively open subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq223_HTML.gif are solutions of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq224_HTML.gif Suppose, moreover, that whenever http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq225_HTML.gif is a solution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq226_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq227_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq228_HTML.gif is a solution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq229_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq230_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq231_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq232_HTML.gif . Then we conclude that the existence of such a subsolution and supersolution guarantees that there is a unique function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq233_HTML.gif , obtained by the Perron's construction, that is a solution of both http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq234_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq235_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq236_HTML.gif .

      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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq237_HTML.gif be open and bounded. Assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq238_HTML.gif be continuous, proper, and satisfy (3.11), (3.12) for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq239_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq240_HTML.gif be a bounded u.s.c. viscosity subsolution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq241_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq242_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq243_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq244_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq245_HTML.gif a bounded l.s.c. viscosity supersolution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq246_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq247_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq248_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq249_HTML.gif Then one has for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq250_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ47_HTML.gif
      (3.42)

      Proof.

      Take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq251_HTML.gif and by using Theorem 3.8 write for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq252_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ48_HTML.gif
      (3.43)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq253_HTML.gif Then the conclusion follows by passing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq254_HTML.gif

      Now we concentrate on the existence part.

      Theorem 3.13.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq255_HTML.gif be a bounded open subset in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq256_HTML.gif . Assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq257_HTML.gif be continuous, proper, and satisfy (3.11), (3.12). Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq258_HTML.gif is almost periodic and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq259_HTML.gif Then there is a time almost periodic viscosity solution in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq260_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq261_HTML.gif of (1.1), where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq262_HTML.gif is a constant.

      Proof.

      Here we consider the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ49_HTML.gif
      (3.44)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq263_HTML.gif . As we know that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq264_HTML.gif , there exists a viscosity solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq265_HTML.gif of (3.44) from Theorem 3.5 and Remark 3.11. Then we will prove that for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq266_HTML.gif converges to a almost periodic viscosity solution of (1.1). As we already know that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq267_HTML.gif we can deduce by Theorem 3.5 that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq268_HTML.gif Similar to the proof of Proposition  6.6 in paper [2], using Theorem 3.8, we get for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq269_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq270_HTML.gif large enough
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ50_HTML.gif
      (3.45)
      By passing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq271_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq272_HTML.gif and therefore
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ51_HTML.gif
      (3.46)

      Since we already know that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq273_HTML.gif by time almost periodicity we deduce also that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq274_HTML.gif

      When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq275_HTML.gif does not satisfy the hypothesis (3.11), we study the time almost periodic viscosity solutions of
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ52_HTML.gif
      (3.47)
      We introduce also the stationary equation
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ53_HTML.gif
      (3.48)

      Then we can prove our main theorem as follows.

      Theorem 3.14.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq276_HTML.gif be open and bounded. Assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq277_HTML.gif be continuous, proper, and satisfy (3.12) for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq278_HTML.gif Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq279_HTML.gif is almost periodic function such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq280_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq281_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq282_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq283_HTML.gif .Assume that (3.48) has a bounded viscosity solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq284_HTML.gif , we take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq285_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq286_HTML.gif , and observe that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ54_HTML.gif
      (3.49)
      Then by using Perron's Method from Theorem 3.10 and Remark 3.11 we can construct the family of solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq287_HTML.gif for
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ55_HTML.gif
      (3.50)
      and the family of time almost periodic solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq288_HTML.gif for
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ56_HTML.gif
      (3.51)
      In fact we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq289_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq290_HTML.gif and by using Theorem 3.9 we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ57_HTML.gif
      (3.52)
      similarly we can get http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq291_HTML.gif From the above two inequalities we know that the family http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq292_HTML.gif is bounded, thus we know http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq293_HTML.gif Therefore we can extract a subsequence which converges uniformly on compact sets of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq294_HTML.gif to a bounded uniformly continuous function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq295_HTML.gif of (3.47). Next we will check that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq296_HTML.gif is almost periodic. By the hypotheses and Proposition 2.5 we deduce that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq297_HTML.gif is almost periodic and thus, for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq298_HTML.gif there is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq299_HTML.gif such that any interval of length http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq300_HTML.gif contains an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq301_HTML.gif almost period of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq302_HTML.gif . Take an interval of length http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq303_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq304_HTML.gif an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq305_HTML.gif almost period of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq306_HTML.gif in this interval. We have for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq307_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ58_HTML.gif
      (3.53)

      After passing to the limit for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq308_HTML.gif one gets http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq309_HTML.gif Hence we prove the almost periodic of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq310_HTML.gif .

      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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ59_HTML.gif
      (3.54)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq311_HTML.gif is almost a periodic function. For all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq312_HTML.gif notice that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq313_HTML.gif is almost periodic and has the same average as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq314_HTML.gif . Now suppose that such a hypothesis exists
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ60_HTML.gif
      (3.55)

      Theorem 3.15.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq315_HTML.gif be open and bounded. Assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq316_HTML.gif be continuous, proper, and satisfy (3.12) for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq317_HTML.gif and (3.55) where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq318_HTML.gif is almost periodic function. Suppose also that there is a bounded l.s.c viscosity supersolution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq319_HTML.gif of (3.48), that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq320_HTML.gif is bounded and denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq321_HTML.gif the minimal stationary, respectively, time almost periodic l.s.c. viscosity supersolution of (3.48), respectively, (3.54). Then the sequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq322_HTML.gif converges uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq323_HTML.gif towards http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq324_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq325_HTML.gif

      Proof.

      As http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq326_HTML.gif is almost periodic, we introduce http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq327_HTML.gif which is also almost periodic. As http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq328_HTML.gif satisfies in the viscosity sense http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq329_HTML.gif we deduce that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq330_HTML.gif satisfies in the viscosity sense
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ61_HTML.gif
      (3.56)
      which can be rewrote as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ62_HTML.gif
      (3.57)
      Recall also that we have in the viscosity sense
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ63_HTML.gif
      (3.58)
      By using Theorem 3.9 we deduce that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ64_HTML.gif
      (3.59)
      and similarly http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq331_HTML.gif We have for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq332_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ65_HTML.gif
      (3.60)
      and after passing to the limit for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq333_HTML.gif one gets for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq334_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ66_HTML.gif
      (3.61)

      Finally we deduce that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq335_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq336_HTML.gif

      Declarations

      Acknowledgment

      The authors appreciate referee's careful reading and valuable suggestions. Partially supported by National Science Foundation of China (Grant no. 10371010).

      Authors’ Affiliations

      (1)
      School of Mathematical Sciences, Ocean University of China

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      Copyright

      © S. Zhang and D. Piao. 2009

      This article is published under license to BioMed Central Ltd. 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.