Open Access

Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq1_HTML.gif is a bounded open subset and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq2_HTML.gif is its boundary. Here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq4_HTML.gif denotes the set of symmetric https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq6_HTML.gif , we say that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq7_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq8_HTML.gif ); https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq9_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq11_HTML.gif w.r.t the argument https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq12_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq13_HTML.gif is almost periodic in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ2_HTML.gif
(1.2)

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

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

(iii)for all real sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq23_HTML.gif there is a subsequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq24_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq25_HTML.gif converges uniformly on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq27_HTML.gif is almost periodicif and only if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq29_HTML.gif verifying (2.1) is called https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq31_HTML.gif is almost periodic. Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq33_HTML.gif is almost periodic. Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq34_HTML.gif converges as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq35_HTML.gif uniformly with respect to https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq38_HTML.gif :
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq39_HTML.gif is almost periodic and denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq40_HTML.gif a primitive of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq41_HTML.gif . Then https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq44_HTML.gif is almost periodic in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq45_HTML.gif uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq46_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq47_HTML.gif is continuous in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq48_HTML.gif uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq50_HTML.gif such that all interval of length https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq51_HTML.gif contain a number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq52_HTML.gif which is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq53_HTML.gif almost periodic for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq54_HTML.gif
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ6_HTML.gif
(3.1)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ7_HTML.gif
(3.2)

in (3.2) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq55_HTML.gif is an arbitrary open subset of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq57_HTML.gif be a locally compact subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq58_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq59_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ8_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq60_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq62_HTML.gif Set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ9_HTML.gif
(3.4)
and suppose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq63_HTML.gif is a local maximum of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq64_HTML.gif relative to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq65_HTML.gif Then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq66_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq67_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq68_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ11_HTML.gif
(3.6)

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

Put https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq70_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq71_HTML.gif recall that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq73_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq74_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq75_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq76_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ13_HTML.gif
(3.8)
Choosing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq77_HTML.gif one can get
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq78_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ15_HTML.gif
(3.10)

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

Assume there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq81_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq82_HTML.gif that satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq83_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq84_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq86_HTML.gif Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq88_HTML.gif

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq89_HTML.gif is a viscosity subsolution of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq90_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq91_HTML.gif and local maximum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq92_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq93_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq94_HTML.gif is a local maximum of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq95_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq96_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ19_HTML.gif
(3.14)
Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq97_HTML.gif is a strict local maximum of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq98_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq99_HTML.gif we consider the function
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ20_HTML.gif
(3.15)
for small https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq100_HTML.gif Then we know that the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq101_HTML.gif has a local maximum point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq102_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq103_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq104_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq105_HTML.gif . So at the point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq106_HTML.gif we deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ21_HTML.gif
(3.16)
As the term https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq107_HTML.gif is positive, so we obtain
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq110_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq111_HTML.gif periodic such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq113_HTML.gif Then https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq116_HTML.gif be a bounded open subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq117_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq119_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq120_HTML.gif ) be a subsolution (resp., supersolution) of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq121_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq122_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq123_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq124_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq125_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq126_HTML.gif .

Theorem 3.5 (see [1]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq127_HTML.gif be open and bounded. Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq129_HTML.gif with the same function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq130_HTML.gif . If https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq133_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq135_HTML.gif be a subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq136_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq137_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ23_HTML.gif
(3.18)
for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq138_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq139_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq140_HTML.gif be chosen so that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ24_HTML.gif
(3.19)
Then the following holds:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq141_HTML.gif are replaced by https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq143_HTML.gif be open and bounded. Assume https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq145_HTML.gif Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq147_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq148_HTML.gif respectively, l.s.c. supersolution of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq149_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq150_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq151_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq152_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ28_HTML.gif
(3.23)

where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ29_HTML.gif
(3.24)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq154_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq155_HTML.gif As we know that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq156_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq157_HTML.gif are bounded semicontinuous in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq158_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq160_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq161_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq163_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq164_HTML.gif is compact, these maxima https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq167_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ30_HTML.gif
(3.25)
which implies https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq169_HTML.gif being a subsolution and https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq171_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ32_HTML.gif
(3.27)
as we know that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ33_HTML.gif
(3.28)
hence we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ34_HTML.gif
(3.29)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq173_HTML.gif For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq174_HTML.gif consider
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ35_HTML.gif
(3.30)
if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq175_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq177_HTML.gif is nondecreasing with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq178_HTML.gif then we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq179_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq180_HTML.gif Hence we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ36_HTML.gif
(3.31)
Notice that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq181_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ37_HTML.gif
(3.32)

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

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

Theorem 3.9.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq194_HTML.gif be open and bounded. Assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq195_HTML.gif be continuous, proper, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq198_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq199_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq201_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq202_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq203_HTML.gif Then one has
https://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)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ42_HTML.gif
(3.37)
We introduce that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq204_HTML.gif By integration by parts we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq205_HTML.gif we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq206_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq207_HTML.gif then

https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq211_HTML.gif and a supersolution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq213_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq214_HTML.gif Then
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq217_HTML.gif is locally compact, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq218_HTML.gif are defined on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq219_HTML.gif and have the following properties: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq220_HTML.gif is upper semicontinuous, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq222_HTML.gif on relatively open subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq223_HTML.gif are solutions of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq224_HTML.gif Suppose, moreover, that whenever https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq225_HTML.gif is a solution of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq226_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq227_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq228_HTML.gif is a solution of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq229_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq230_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq231_HTML.gif on https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq234_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq235_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq237_HTML.gif be open and bounded. Assume https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq239_HTML.gif Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq241_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq242_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq243_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq244_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq246_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq247_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq248_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq249_HTML.gif Then one has for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq250_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ47_HTML.gif
(3.42)

Proof.

Take https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq252_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ48_HTML.gif
(3.43)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq253_HTML.gif Then the conclusion follows by passing https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq255_HTML.gif be a bounded open subset in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq256_HTML.gif . Assume https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq258_HTML.gif is almost periodic and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq260_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq261_HTML.gif of (1.1), where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ49_HTML.gif
(3.44)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq263_HTML.gif . As we know that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq264_HTML.gif , there exists a viscosity solution https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq269_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq270_HTML.gif large enough
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ50_HTML.gif
(3.45)
By passing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq271_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq272_HTML.gif and therefore
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq274_HTML.gif

When https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq276_HTML.gif be open and bounded. Assume https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq278_HTML.gif Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq279_HTML.gif is almost periodic function such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq280_HTML.gif is bounded on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq282_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq284_HTML.gif , we take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq285_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq286_HTML.gif , and observe that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq287_HTML.gif for
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq288_HTML.gif for
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ56_HTML.gif
(3.51)
In fact we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq289_HTML.gif for any https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ57_HTML.gif
(3.52)
similarly we can get https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq292_HTML.gif is bounded, thus we know https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq294_HTML.gif to a bounded uniformly continuous function https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq298_HTML.gif there is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq299_HTML.gif such that any interval of length https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq300_HTML.gif contains an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq301_HTML.gif almost period of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq302_HTML.gif . Take an interval of length https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq303_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq304_HTML.gif an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq305_HTML.gif almost period of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq307_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq308_HTML.gif one gets https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ59_HTML.gif
(3.54)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq312_HTML.gif notice that https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ60_HTML.gif
(3.55)

Theorem 3.15.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq315_HTML.gif be open and bounded. Assume https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq317_HTML.gif and (3.55) where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq319_HTML.gif of (3.48), that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq320_HTML.gif is bounded and denote by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq322_HTML.gif converges uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq323_HTML.gif towards https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq324_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq325_HTML.gif

Proof.

As https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq326_HTML.gif is almost periodic, we introduce https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq327_HTML.gif which is also almost periodic. As https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq328_HTML.gif satisfies in the viscosity sense https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq329_HTML.gif we deduce that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq330_HTML.gif satisfies in the viscosity sense
https://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
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ64_HTML.gif
(3.59)
and similarly https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq331_HTML.gif We have for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq332_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq333_HTML.gif one gets for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_IEq334_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F873526/MediaObjects/13661_2009_Article_886_Equ66_HTML.gif
(3.61)

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

References

  1. Crandall MG, Ishii H, Lions P-L: User's guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society 1992, 27(1):1-67. 10.1090/S0273-0979-1992-00266-5MATHMathSciNetView Article
  2. Bostan M, Namah G: Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis 2007, 6(2):389-410.MATHMathSciNetView Article
  3. Nunziante D: Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time-dependence. Differential and Integral Equations 1990, 3(1):77-91.MATHMathSciNet
  4. Nunziante D: Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence. Nonlinear Analysis: Theory, Methods & Applications 1992, 18(11):1033-1062. 10.1016/0362-546X(92)90194-JMATHMathSciNetView Article
  5. Ishii H: Perron's method for Hamilton-Jacobi equations. Duke Mathematical Journal 1987, 55(2):369-384. 10.1215/S0012-7094-87-05521-9MATHMathSciNetView Article
  6. Crandall MG, Ishii H: The maximum principle for semicontinuous functions. Differential and Integral Equations 1990, 3(6):1001-1014.MATHMathSciNet
  7. Corduneanu C: Almost Periodic Functions. Chelsea, New York, NY, USA; 1989.MATH
  8. Fink AM: Almost Periodic Differential Equations, Lecture Notes in Mathematics. Volume 377. Springer, Berlin, Germany; 1974:viii+336.
  9. Yoshizawa T: Stability properties in almost periodic systems of functional differential equations. In Functional Differential Equations and Bifurcation, Lecture Notes in Mathmatics. Volume 799. Springer, Berlin, Germany; 1980:385-409. 10.1007/BFb0089326View Article
  10. Barles G: Solutions de Viscosité des Équations de Hamilton-Jacobi, Mathématiques & Applications. Volume 17. Springer, Paris, France; 1994:x+194.

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.