Boundary Value Problems
Volume 2009 (2009), Article ID 873526, 15 pages
doi:10.1155/2009/873526
Research Article

Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations

Shilin Zhang and Daxiong Piao

School of Mathematical Sciences, Ocean University of China, Qingdao 266071, China

Received 26 March 2009; Accepted 9 June 2009

Academic Editor: Zhitao Zhang

Copyright © 2009 Shilin Zhang and Daxiong Piao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We 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 𝜕 𝑡 𝑢 + 𝐻 𝑥 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 = 𝑓 ( 𝑡 ) , ( 𝑥 , 𝑡 ) ∈ Ω × ℝ , 𝑢 ( 𝑥 , 𝑡 ) = 0 , ( 𝑥 , 𝑡 ) ∈ 𝜕 Ω × ℝ , ( 1 . 1 ) where Ω ∈ ℝ 𝑁 is a bounded open subset and 𝜕 Ω is its boundary. Here 𝐻 ∶ ℝ 𝑁 × ℝ × ℝ 𝑁 × 𝒮 ( 𝑁 ) → ℝ and 𝒮 ( 𝑁 ) denotes the set of symmetric 𝑁 × 𝑁 matrices equipped with its usual order (i.e., for 𝑋 , 𝑌 ∈ 𝒮 ( 𝑁 ) , we say that 𝑋 ≤ 𝑌 if and only if 𝑝 𝑡 𝑋 𝑝 ≤ 𝑝 𝑡 𝑌 𝑝 , ( ∀ 𝑝 ∈ ℝ 𝑁 ) ); 𝐷 𝑢 and 𝐷 2 𝑢 denote the gradient and Hessian matrix, respectively, of the function 𝑢 w.r.t the argument 𝑥 . 𝑓 is almost periodic in 𝑡 . 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 𝜕 𝑡 𝑢 + 𝐻 𝑥 , 𝑡 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 𝑢 = 0 , i n Ω × ( 0 , 𝑇 ) , 𝑢 ( 𝑥 , 𝑡 ) = 0 , f o r 𝑥 ∈ 𝜕 Ω , 0 ≤ 𝑡 < 𝑇 , ( 𝑥 , 0 ) = 𝑢 0 ( 𝑥 ) , f o r 𝑥 ∈ Ω , ( 1 . 2 ) where 𝑢 0 ( 𝑥 ) ∈ 𝐶 ( Ω ) 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 𝑓 ∶ ℝ → ℝ be a continuous function. The following conditions are equivalent: (i) ∀ 𝜀 > 0 , ∃ 𝑙 ( 𝜀 ) > 0 such that ∀ 𝑎 ∈ ℝ , ∃ 𝜏 ∈ [ 𝑎 , 𝑎 + 𝑙 ( 𝜀 ) ) satisfying | | | | 𝑓 ( 𝑡 + 𝜏 ) − 𝑓 ( 𝑡 ) < 𝜀 , ∀ 𝑡 ∈ ℝ ; ( 2 . 1 ) (ii) ∀ 𝜀 > 0 , there is a trigonometric polynomial 𝑇 𝜀 ( 𝑡 ) = Σ 𝑛 𝑘 = 1 { 𝑎 𝑘 ⋅ c o s ( 𝜆 𝑘 𝑡 ) + 𝑏 𝑘 ⋅ s i n ( 𝜆 𝑘 𝑡 ) } where 𝑎 𝑘 , 𝑏 𝑘 , 𝜆 𝑘 ∈ ℝ , 1 ≤ 𝑘 ≤ 𝑛 such that | 𝑓 ( 𝑡 ) − 𝑇 𝜀 ( 𝑡 ) | < 𝜀 , ∀ 𝑡 ∈ ℝ ; (iii)for all real sequence ( ℎ 𝑛 ) 𝑛 there is a subsequence ( ℎ 𝑛 𝑘 ) 𝑘 such that ( 𝑓 ( ⋅ + ℎ 𝑛 𝑘 ) ) 𝑘 converges uniformly on ℝ .

Definition 2.2. One saysthat a continuous function 𝑓 is almost periodicif and only if 𝑓 satisfies one of the three conditions of Proposition 2.1.

A number 𝜏 verifying (2.1) is called 𝜀 almost period. By using Proposition 2.1 we get the following property of almost periodic functions.

Proposition 2.3. Assume that 𝑓 ∶ ℝ → ℝ is almost periodic. Then 𝑓 is bounded uniformly continuous function.

Proposition 2.4. Assume that 𝑓 ∶ ℝ → ℝ is almost periodic. Then ∫ ( 1 / 𝑇 ) 𝑎 𝑎 + 𝑇 𝑓 ( 𝑡 ) 𝑑 𝑡 converges as 𝑇 → + ∞ uniformly with respect to 𝑎 ∈ 𝑅 . Moreover the limit does not depend on 𝑎 and it is called the average of 𝑓 : ∃ ⟨ 𝑓 ⟩ ∶ = l i m 𝑇 → + ∞ 1 𝑇 𝑎 𝑎 + 𝑇 𝑓 ( 𝑡 ) 𝑑 𝑡 , u n i f o r m l y w . r . t . 𝑎 ∈ 𝑅 . ( 2 . 2 )

Proposition 2.5. Assume that 𝑓 ∶ ℝ → ℝ is almost periodic and denote by 𝐹 a primitive of 𝑓 . Then 𝐹 is almost periodic if and only if 𝐹 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 𝑢 ∶ Ω × ℝ → ℝ is almost periodic in 𝑡 uniformly with respect to 𝑥 if 𝑢 is continuous in 𝑡 uniformly with respect to 𝑥 and ∀ 𝜀 > 0 , ∃ 𝑙 ( 𝜀 ) > 0 such that all interval of length 𝑙 ( 𝜀 ) contain a number 𝜏 which is 𝜀 almost periodic for 𝑢 ( 𝑥 , ⋅ ) , ∀ 𝑥 ∈ Ω | | | | 𝑢 ( 𝑥 , 𝑡 + 𝜏 ) − 𝑢 ( 𝑥 , 𝑡 ) < 𝜀 , ∀ ( 𝑥 , 𝑡 ) ∈ Ω × ℝ . ( 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 𝜕 𝑡 𝑢 + 𝐻 𝑥 , 𝑡 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 𝐻 = 0 , i n Ω × ( 0 , 𝑇 ) , 𝑢 ( 𝑥 , 𝑡 ) = 0 , f o r 𝑥 ∈ 𝜕 Ω , 0 ≤ 𝑡 < 𝑇 , ( 3 . 1 ) 𝑥 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 = 0 , i n Ω , 𝑢 = 0 , o n 𝜕 Ω , ( 3 . 2 ) in (3.2) Ω is an arbitrary open subset of ℝ 𝑁 .

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

Theorem 3.1 (see [1]). Let 𝒪 𝑖 be a locally compact subset of ℝ 𝑁 𝑖 for 𝑖 = 1 , … , 𝑘 , 𝒪 = 𝒪 1 × ⋯ × 𝒪 𝑘 , ( 3 . 3 ) 𝑢 𝑖 ∈ U S C ( 𝒪 𝑖 ) , and 𝜑 be twice continuously differentiable in a neighborhood of 𝒪 . Set 𝑤 ( 𝑥 ) = 𝑢 1 𝑥 1 + ⋯ + 𝑢 𝑘 𝑥 𝑘 𝑥 f o r 𝑥 = 1 , … , 𝑥 𝑘 ∈ 𝒪 , ( 3 . 4 ) and suppose ̂ 𝑥 = ( ̂ 𝑥 1 , … , ̂ 𝑥 𝑘 ) ∈ 𝒪 is a local maximum of 𝑤 − 𝜑 relative to 𝒪 . Then for each 𝜀 > 0 there exists 𝑋 𝑖 ∈ 𝒮 ( 𝑁 𝑖 ) such that 𝐷 𝑥 𝑖 𝜑 ( ̂ 𝑥 ) , 𝑋 𝑖 ∈ 𝐽 𝒪 2 , + 𝑖 𝑢 𝑖 ̂ 𝑥 𝑖 f o r 𝑖 = 1 , … , 𝑘 , ( 3 . 5 ) and the block diagonal matrix with entries 𝑋 𝑖 satisfies − 1 𝜀 𝑋 + ‖ 𝐴 ‖ 𝐼 ≤ 1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 𝑋 𝑘 ≤ 𝐴 + 𝐴 2 , ( 3 . 6 ) where 𝐴 = 𝐷 2 𝜑 ( ̂ 𝑥 ) ∈ 𝒮 ( 𝑁 ) , 𝑁 = 𝑁 1 + ⋯ + 𝑁 𝑘 .

Put 𝑘 = 2 , 𝒪 1 = 𝒪 2 = Ω , 𝑢 1 = 𝑢 , 𝑢 2 = − 𝑣 , 𝜑 ( 𝑥 , 𝑦 ) = ( 𝛼 / 2 ) | 𝑥 − 𝑦 | 2 , where 𝛼 > 0 , recall that 𝐽 Ω 2 , − 𝑣 = − 𝐽 Ω 2 , + ( − 𝑣 ) , then, from Theorem 3.1, at a local maximum ( ̂ 𝑥 , ̂ 𝑦 ) of 𝑢 ( 𝑥 ) − 𝑣 ( 𝑦 ) − 𝜑 ( 𝑥 , 𝑦 ) , we have 𝐷 𝑥 𝜑 ( ̂ 𝑥 , ̂ 𝑦 ) = − 𝐷 𝑦 𝜑 ( ̂ 𝑥 , ̂ 𝑦 ) = 𝛼 ( ̂ 𝑥 − ̂ 𝑦 ) , 𝐴 = 𝛼 𝐼 − 𝐼 − 𝐼 𝐼 , 𝐴 2 = 2 𝛼 𝐴 , ‖ 𝐴 ‖ = 2 𝛼 . ( 3 . 7 ) We conclude that for each 𝜀 > 0 , there exists 𝑋 , 𝑌 ∈ 𝒮 ( 𝑁 ) such that ( 𝛼 ( ̂ 𝑥 − ̂ 𝑦 ) , 𝑋 ) ∈ 𝐽 Ω 2 , + 𝑢 ( ̂ 𝑥 ) , ( 𝛼 ( ̂ 𝑥 − ̂ 𝑦 ) , 𝑌 ) ∈ 𝐽 Ω 2 , − − 1 𝑣 ( ̂ 𝑦 ) , 𝜀 ≤ . + 2 𝛼 𝐼 0 0 𝐼 𝑋 0 0 − 𝑌 ≤ 𝛼 ( 1 + 2 𝜀 𝛼 ) 𝐼 − 𝐼 − 𝐼 𝐼 ( 3 . 8 )

Choosing 𝜀 = 1 / 𝛼 one can get ≤ − 3 𝛼 𝐼 0 0 𝐼 𝑋 0 0 − 𝑌 ≤ 3 𝛼 𝐼 − 𝐼 − 𝐼 𝐼 . ( 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 𝐻 , that is, 𝐻 ( 𝑥 , 𝑟 , 𝑝 , 𝑋 ) ≤ 𝐻 ( 𝑥 , 𝑠 , 𝑝 , 𝑌 ) w h e n e v e r 𝑟 ≤ 𝑠 , 𝑌 ≤ 𝑋 , ( 3 . 1 0 ) where 𝑟 , 𝑠 ∈ ℝ , 𝑥 ∈ Ω , 𝑝 ∈ ℝ 𝑁 , 𝑋 , 𝑌 ∈ 𝒮 ( 𝑁 ) . Then we will say that 𝐻 is proper.

Assume there exists 𝛾 > 0 such that 𝛾 ( 𝑟 − 𝑠 ) ≤ 𝐻 ( 𝑥 , 𝑟 , 𝑝 , 𝑋 ) − 𝐻 ( 𝑥 , 𝑠 , 𝑝 , 𝑋 ) , f o r 𝑟 ≥ 𝑠 , ( 𝑥 , 𝑝 , 𝑋 ) ∈ Ω × ℝ 𝑁 × 𝒮 ( 𝑁 ) , ( 3 . 1 1 ) and there is a function 𝜔 ∶ [ 0 , ∞ ] → [ 0 , ∞ ] that satisfies 𝜔 ( 0 + ) = 0 such that 𝛼 | | | | 𝐻 ( 𝑦 , 𝑟 , 𝛼 ( 𝑥 − 𝑦 ) , 𝑌 ) − 𝐻 ( 𝑥 , 𝑟 , 𝛼 ( 𝑥 − 𝑦 ) , 𝑋 ) ≤ 𝜔 𝑥 − 𝑦 2 + | | | | 𝑥 − 𝑦 w h e n e v e r 𝑥 , 𝑦 ∈ Ω , 𝑟 ∈ ℝ , 𝑋 , 𝑌 ∈ 𝒮 ( 𝑁 ) , a n d ( 3 . 9 ) h o l d s . ( 3 . 1 2 )

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 𝐻 ∈ 𝐶 ( Ω × ( 0 , 𝑇 ] × ℝ × ℝ 𝑁 × 𝑆 ( 𝑁 ) ) and 𝑢 ∈ 𝐶 ( Ω × ( 0 , 𝑇 ] ) is a viscosity subsolution (resp., supersolution) of 𝜕 𝑡 𝑢 + 𝐻 ( 𝑥 , 𝑡 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 ) = 0 , ( 𝑥 , 𝑡 ) ∈ Ω × ( 0 , 𝑇 ) . Then 𝑢 is a viscosity subsolution (resp., supersolution) of 𝜕 𝑡 𝑢 + 𝐻 ( 𝑥 , 𝑡 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 ) = 0 , ( 𝑥 , 𝑡 ) ∈ Ω × ( 0 , 𝑇 ] .

Proof. Since 𝑢 ∈ 𝐶 ( Ω × ( 0 , 𝑇 ] ) is a viscosity subsolution of 𝜕 𝑡 𝑢 + 𝐻 ( 𝑥 , 𝑡 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 ) = 0 , ( 𝑥 , 𝑡 ) ∈ Ω × ( 0 , 𝑇 ) , if ∀ 𝜑 ∈ 𝐶 2 ( Ω × ( 0 , 𝑇 ] ) and local maximum ̂ ( ̂ 𝑥 , 𝑡 ) ∈ ( Ω × ( 0 , 𝑇 ) ) of 𝑢 − 𝜑 , we have 𝜕 𝑡 𝜑 ̂ 𝑡 ̂ ̂ 𝑡 ̂ 𝑡 ̂ 𝑥 , + 𝐻 ̂ 𝑥 , 𝑡 , 𝑢 ̂ 𝑥 , , 𝐷 𝜑 ̂ 𝑥 , , 𝐷 2 𝜑 ̂ 𝑡 ̂ 𝑥 , ≤ 0 . ( 3 . 1 3 ) Now we prove that if ( 𝑥 0 , 𝑇 ) is a local maximum of 𝑢 − 𝜑 in Ω × ( 0 , 𝑇 ] , then 𝜕 𝑡 𝜑 𝑥 0 𝑥 , 𝑇 + 𝐻 0 𝑥 , 𝑇 , 𝑢 0 𝑥 , 𝑇 , 𝐷 𝜑 0 , 𝑇 , 𝐷 2 𝜑 𝑥 0 , 𝑇 ≤ 0 . ( 3 . 1 4 ) Suppose that ( 𝑥 0 , 𝑇 ) is a strict local maximum of 𝑢 − 𝜑 in Ω × ( 0 , 𝑇 ] , we consider the function 𝜓 𝜀 ( 𝑥 , 𝑡 ) = 𝑢 ( 𝑥 , 𝑡 ) − 𝜑 ( 𝑥 , 𝑡 ) − 𝜀 ( 𝑇 − 𝑡 ) − 1 ( 3 . 1 5 ) for small 𝜀 > 0 . Then we know that the function 𝜓 𝜀 ( 𝑥 , 𝑡 ) has a local maximum point ( 𝑥 𝜀 , 𝑡 𝜀 ) such that 𝑡 𝜀 < 𝑇 and ( 𝑥 𝜀 , 𝑡 𝜀 ) → ( 𝑥 0 , 𝑇 ) when 𝜀 → 0 . So at the point ( 𝑥 𝜀 , 𝑡 𝜀 ) we deduce that 𝜕 𝑡 𝜑 𝑥 𝜀 , 𝑡 𝜀 + 𝜀 𝑇 − 𝑡 𝜀 2 𝑥 + 𝐻 𝜀 , 𝑡 𝜀 𝑥 , 𝑢 𝜀 , 𝑡 𝜀 𝑥 , 𝐷 𝜑 𝜀 , 𝑡 𝜀 , 𝐷 2 𝜑 𝑥 𝜀 , 𝑡 𝜀 ≤ 0 . ( 3 . 1 6 ) As the term 𝜀 / ( 𝑇 − 𝑡 𝜀 ) 2 is positive, so we obtain 𝜕 𝑡 𝜑 𝑥 𝜀 , 𝑡 𝜀 𝑥 + 𝐻 𝜀 , 𝑡 𝜀 𝑥 , 𝑢 𝜀 , 𝑡 𝜀 𝑥 , 𝐷 𝜑 𝜀 , 𝑡 𝜀 , 𝐷 2 𝜑 𝑥 𝜀 , 𝑡 𝜀 ≤ 0 . ( 3 . 1 7 ) The results following upon letting 𝜀 → 0 . This process can be easily applied to the viscosity supersolution case.

By time periodicity one gets the following.

Proposition 3.3. Assume that 𝐻 ∈ 𝐶 ( Ω × ℝ × ℝ × ℝ 𝑁 × 𝑆 ( 𝑁 ) ) and 𝑢 ∈ 𝐶 ( Ω × ℝ ) are 𝑇 periodic such that 𝑢 is a viscosity subsolution (resp., supersolution) of 𝜕 𝑡 𝑢 + 𝐻 ( 𝑥 , 𝑡 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 ) = 0 , ( 𝑥 , 𝑡 ) ∈ Ω × ( 0 , 𝑇 ) . Then 𝑢 is a viscosity subsolution (resp., supersolution) of 𝜕 𝑡 𝑢 + 𝐻 ( 𝑥 , 𝑡 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 ) = 0 , ( 𝑥 , 𝑡 ) ∈ Ω × ℝ .

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

Theorem 3.4 (see [6]). Let Ω be a bounded open subset of ℝ 𝑁 , 𝐹 ∈ 𝐶 ( Ω × ℝ × ℝ 𝑁 × 𝒮 ( 𝑁 ) ) be proper and satisfy (3.11), (3.12). Let 𝑢 ∈ 𝑈 𝑆 𝐶 ( Ω ) (resp., 𝑣 ∈ L S C ( Ω ) ) be a subsolution (resp., supersolution) of 𝐹 = 0 in Ω and 𝑢 ≤ 𝑣 on 𝜕 Ω . Then 𝑢 ≤ 𝑣 in Ω .

Theorem 3.5 (see [1]). Let Ω ∈ ℝ 𝑁 be open and bounded. Let 𝐻 ∈ 𝐶 ( Ω × [ 0 , 𝑇 ] × ℝ × ℝ 𝑁 × 𝒮 ( 𝑁 ) ) be continuous, proper, and satisfy (3.12) for each fixed 𝑡 ∈ [ 0 , 𝑇 ) , with the same function 𝜔 . If 𝑢 is a subsolution of (1.2) and 𝑣 is a supersolution of (1.2), then 𝑢 ≤ 𝑣 on [ 0 , 𝑇 ) × Ω .

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 𝒪 be a subset of ℝ 𝑀 , Φ ∈ U S C ( 𝒪 ) , Ψ ∈ L S C ( 𝒪 ) , Ψ ≥ 0 , and 𝑀 𝛼 = s u p 𝒪 ( Φ ( 𝑥 ) − 𝛼 Ψ ( 𝑥 ) ) ( 3 . 1 8 ) for 𝛼 > 0 . Let − ∞ < l i m 𝛼 → ∞ 𝑀 𝛼 < ∞ and 𝑥 𝛼 ∈ 𝒪 be chosen so that l i m 𝛼 → ∞ 𝑀 𝛼 − Φ 𝑥 𝛼 𝑥 − 𝛼 Ψ 𝛼 = 0 . ( 3 . 1 9 ) Then the following holds: ( i ) l i m 𝛼 → ∞ 𝑥 𝛼 Ψ 𝛼 = 0 , ( i i ) Ψ ( ̂ 𝑥 ) = 0 , l i m 𝛼 → ∞ 𝑀 𝛼 = Φ ( ̂ 𝑥 ) = s u p { Ψ ( 𝑥 ) = 0 } Φ ( 𝑥 ) w h e n e v e r ̂ 𝑥 ∈ 𝒪 i s a l i m i t p o i n t o f 𝑥 𝛼 a s 𝛼 ⟶ ∞ . ( 3 . 2 0 )

Remark 3.7. In Proposition 3.6, when 𝑀 , 𝒪 , 𝑥 , Φ ( 𝑥 ) , Ψ ( 𝑥 ) are replaced by 2 𝑁 , 𝒪 × 𝒪 , ( 𝑥 , 𝑦 ) , 𝑢 ( 𝑥 ) − 𝑣 ( 𝑦 ) , ( 1 / 2 ) | 𝑥 − 𝑦 | 2 , respectively, we can get the following results: ( i ) l i m 𝛼 → ∞ 𝛼 | | 𝑥 𝛼 − 𝑦 𝛼 | | 2 = 0 , ( i i ) Ψ ( ̂ 𝑥 ) = 0 , l i m 𝛼 → ∞ 𝑀 𝛼 = 𝑢 ( ̂ 𝑥 ) − 𝑣 ( ̂ 𝑥 ) = s u p 𝒪 ( 𝑢 ( 𝑥 ) − 𝑣 ( 𝑥 ) ) w h e n e v e r ̂ 𝑥 ∈ 𝒪 i s a l i m i t p o i n t o f 𝑥 𝛼 a s 𝛼 ⟶ ∞ . ( 3 . 2 1 ) Now we have the following.

Theorem 3.8. Let Ω ∈ ℝ 𝑁 be open and bounded. Assume 𝐻 ∈ 𝐶 ( Ω × [ 0 , 𝑇 ] × ℝ × ℝ 𝑁 × 𝒮 ( 𝑁 ) ) be continuous, proper, and satisfy (3.11), (3.12) for each fixed 𝑡 ∈ [ 0 , 𝑇 ) . Let 𝑢 , 𝑣 be bounded u.s.c. subsolution of 𝜕 𝑡 𝑢 + 𝐻 ( 𝑥 , 𝑡 , 𝑢 , 𝐷 𝑢 , 𝐷 2 𝑢 ) = 𝑓 ( 𝑥 , 𝑡 ) in Ω × ( 0 , 𝑇 ) , 𝑢 ( 𝑥 , 𝑡 ) = 0 f o r 𝑥 ∈ 𝜕 Ω a n d 0 ≤ 𝑡 < 𝑇 , respectively, l.s.c. supersolution of 𝜕 𝑡 𝑣 + 𝐻 ( 𝑥 , 𝑡 , 𝑣 , 𝐷 𝑣 , 𝐷 2 𝑣 ) = 𝑔 ( 𝑥 , 𝑡 ) in Ω × ( 0 , 𝑇 ) , 𝑣 ( 𝑥 , 𝑡 ) = 0 f o r 𝑥 ∈ 𝜕 Ω a n d 0 ≤ 𝑡 < 𝑇 where 𝑓 , 𝑔 ∈ B U C ( Ω × [ 0 , 𝑇 ] ) . l i m 𝑡 ↘ 0 ( 𝑢 ( 𝑥 , 𝑡 ) − 𝑢 ( 𝑥 , 0 ) ) + = l i m 𝑡 ↘ 0 ( 𝑣 ( 𝑥 , 𝑡 ) − 𝑣 ( 𝑥 , 0 ) ) − = 0 , u n i f o r m l y f o r 𝑥 ∈ Ω , 𝑢 ( ⋅ , 0 ) ∈ B U C Ω o r 𝑣 ( ⋅ , 0 ) ∈ B U C Ω . ( 3 . 2 2 ) Then one has for all 𝑡 ∈ [ 0 , 𝑇 ] 𝑒 𝛾 𝑡 ‖ ( 𝑢 ( ⋅ , 𝑡 ) − 𝑣 ( ⋅ , 𝑡 ) ) ‖ 𝐿 ∞ Ω ≤ ‖ ‖ ( 𝑢 ( ⋅ , 0 ) − 𝑣 ( ⋅ , 0 ) ) + ‖ ‖ 𝐿 ∞ Ω + 𝑡 0 𝑒 𝛾 𝑠 ‖ ( 𝑓 ( ⋅ , 𝑠 ) − 𝑔 ( ⋅ , 𝑠 ) ) ‖ 𝐿 ∞ Ω 𝑑 𝑠 , ( 3 . 2 3 ) where 𝛾 = 𝛾 𝑅 0 , 𝑅 0 = m a x ( ‖ 𝑢 ‖ 𝐿 ∞ ( Ω × ( 0 , 𝑇 ) ) , ‖ 𝑣 ‖ 𝐿 ∞ ( Ω × ( 0 , 𝑇 ) ) ) .

Proof. Let us consider the function given by 𝑤 𝛼 ( 𝑥 , 𝑦 , 𝑡 ) = 𝑢 ( 𝑥 , 𝑡 ) − 𝑣 ( 𝑦 , 𝑡 ) − 𝜑 ( 𝑥 , 𝑦 , 𝑡 ) , ( 3 . 2 4 ) where 𝜑 ( 𝑥 , 𝑦 , 𝑡 ) = ( 𝛼 / 2 ) ( | 𝑥 − 𝑦 | 2 + 𝜙 ( 𝑡 ) ) , and 𝜙 ( 𝑡 ) ∈ 𝐶 1 ( [ 0 , 𝑇 ] ) . As we know that 𝑢 and 𝑣 are bounded semicontinuous in Ω × [ 0 , 𝑇 ] and Ω ∈ ℝ 𝑁 is open and bounded, we can find ( ̂ 𝑥 ( 𝑡 𝛼 ) , ̂ 𝑦 ( 𝑡 𝛼 ) ) ∈ Ω × Ω , for 𝑡 𝛼 ∈ [ 0 , 𝑇 ] such that 𝑀 𝛼 ( 𝑡 𝛼 ) ∶ = s u p Ω × Ω ( 𝑢 ( 𝑥 , 𝑡 𝛼 ) − 𝑣 ( 𝑦 , 𝑡 𝛼 ) − 𝜑 ( 𝑥 , 𝑦 , 𝑡 𝛼 ) ) = 𝑢 ( ̂ 𝑥 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) − 𝑣 ( ̂ 𝑦 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) − 𝜑 ( ̂ 𝑥 ( 𝑡 𝛼 ) , ̂ 𝑦 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) , here without loss of generality, we can assume that 𝑀 𝛼 ( 𝑡 𝛼 ) = 0 . Since Ω × Ω × [ 0 , 𝑇 ] is compact, these maxima ( ̂ 𝑥 ( 𝑡 𝛼 ) , ̂ 𝑦 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) converge to a point of the form ( 𝑧 ( 𝑡 ) , 𝑧 ( 𝑡 ) , 𝑡 ) from Remark 3.7. From Theorem 3.1 and its following discussion, there exists 𝑋 𝛼 , 𝑌 𝛼 ∈ 𝒮 ( 𝑁 ) such that 𝛼 𝑡 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 ∈ 𝐽 Ω 2 , + 𝑢 𝑡 ̂ 𝑥 𝛼 , 𝑡 𝛼 , 𝛼 𝑡 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑌 𝛼 ∈ 𝐽 Ω 2 , − 𝑣 𝑡 ̂ 𝑦 𝛼 , 𝑡 𝛼 , ≤ 𝑋 − 3 𝛼 𝐼 0 0 𝐼 𝛼 0 0 − 𝑌 𝛼 , ≤ 3 𝛼 𝐼 − 𝐼 − 𝐼 𝐼 ( 3 . 2 5 ) which implies 𝑋 𝛼 ≤ 𝑌 𝛼 . At the maximum point, from the definition of 𝑢 being a subsolution and 𝑣 being a supersolution we arrive at the following: 𝜕 𝑡 𝛼 𝜑 𝑡 ̂ 𝑥 𝛼 𝑡 , ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 + 𝐻 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝑢 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 − 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑌 𝛼 𝑡 ≤ 𝑓 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 − 𝑔 ̂ 𝑦 𝛼 , 𝑡 𝛼 , ( 3 . 2 6 ) by the proper condition of 𝐻 , we have 𝐻 𝑡 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑌 𝛼 𝑡 ≤ 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 , ( 3 . 2 7 ) as we know that 𝐻 satisfying (3.12) then we deduce that 𝐻 𝑡 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝑢 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 − 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 = 𝐻 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝑢 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 − 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑢 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 + 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑢 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 − 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 ≥ 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑢 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 − 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝛼 | | − 𝜔 ̂ 𝑥 ( 𝑡 𝛼 ) − ̂ 𝑦 ( 𝑡 𝛼 ) | | 2 + | | 𝑡 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 | | , ( 3 . 2 8 ) hence we get 𝜕 𝑡 𝛼 𝜑 𝑡 ̂ 𝑥 𝛼 𝑡 , ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 + 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑢 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 − 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝛼 | | − 𝜔 ̂ 𝑥 ( 𝑡 𝛼 ) − ̂ 𝑦 ( 𝑡 𝛼 ) | | 2 + | | 𝑡 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 | | 𝑡 ≤ ℎ 𝛼 , ( 3 . 2 9 ) where ℎ ( 𝑡 𝛼 ) = 𝑓 ( ̂ 𝑥 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) − 𝑔 ( ̂ 𝑦 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) , ∀ 𝑡 𝛼 ∈ [ 0 , 𝑇 ] . For any 𝑡 𝛼 ∈ [ 0 , 𝑇 ] consider 𝑟 𝑡 𝛼 = 1 𝑢 𝑡 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 − 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝐻 𝑡 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑢 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 − 𝛾 𝑢 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 − 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 + 𝛾 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 , ( 3 . 3 0 ) if 𝑢 ( ̂ 𝑥 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) ≠ 𝑣 ( ̂ 𝑦 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) , and 𝑟 ( 𝑡 𝛼 ) = 0 otherwise. From hypothesis (3.11) we deduce that 𝐻 ( 𝑥 , 𝑡 , 𝑧 , 𝑝 , 𝑋 ) − 𝛾 ⋅ 𝑧 is nondecreasing with respect to 𝑧 , then we have 𝑟 ( 𝑡 𝛼 ) ≥ 0 for all 𝑡 𝛼 ∈ [ 0 , 𝑇 ] . Hence we have 𝐻 𝑡 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑢 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 𝑡 − 𝐻 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 𝑡 , 𝛼 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 , 𝑋 𝛼 = 𝑡 𝛾 + 𝑟 𝛼 𝑢 𝑡 ̂ 𝑥 𝛼 , 𝑡 𝛼 𝑡 − 𝑣 ̂ 𝑦 𝛼 , 𝑡 𝛼 , ∀ 𝑡 𝛼 ∈ [ ] . 0 , 𝑇 ( 3 . 3 1 ) Notice that 𝑢 ( ̂ 𝑥 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) − 𝑣 ( ̂ 𝑦 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) = 𝜑 ( ̂ 𝑥 ( 𝑡 𝛼 ) , ̂ 𝑦 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) , we get 𝜕 𝑡 𝛼 𝜑 𝑡 ̂ 𝑥 𝛼 𝑡 , ̂ 𝑦 𝛼 , 𝑡 𝛼 + 𝑡 𝛾 + 𝑟 𝛼 𝜑 𝑡 ̂ 𝑥 𝛼 𝑡 , ̂ 𝑦 𝛼 , 𝑡 𝛼 𝛼 | | − 𝜔 ̂ 𝑥 ( 𝑡 𝛼 ) − ̂ 𝑦 ( 𝑡 𝛼 ) | | 2 + | | 𝑡 ̂ 𝑥 𝛼 𝑡 − ̂ 𝑦 𝛼 | | 𝑡 ≤ ℎ 𝛼 . ( 3 . 3 2 )
Replacing 𝑢 ( ̂ 𝑥 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) − 𝑣 ( ̂ 𝑦 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) by 𝜑 ( ̂ 𝑥 ( 𝑡 𝛼 ) , ̂ 𝑦 ( 𝑡 𝛼 ) , 𝑡 𝛼 ) in the expression of 𝑟 ( 𝑡 𝛼 ) we know that 𝑟 ( ⋅ ) is integrable and denote by 𝐴 ( 𝑡 𝛼 ) the function 𝐴 ( 𝑡 𝛼 ∫ ) = 𝑡 𝛼 0 { 𝛾 + 𝑟 ( 𝜎 ) } 𝑑 𝜎 , 𝑡 𝛼 ∈ [ 0 , 𝑇 ] . After integration one gets
𝜑 𝑡 𝛼