The Initial and Neumann Boundary Value Problem for a Class Parabolic Monge-Ampère Equation

Abstract

We consider the existence, uniqueness, and asymptotic behavior of a classical solution to the initial and Neumann boundary value problem for a class nonlinear parabolic equation of Monge-Ampère type. We show that such solution exists for all times and is unique. It converges eventually to a solution that satisfies a Neumann type problem for nonlinear elliptic equation of Monge-Ampère type.

1. Introduction

Historically, the study of Monge-Ampère is motivated by the following two problems: Minkowski and Weyl problems. One is of prescribing curvature type, and the other is of embedding type. The development of Monge-Ampère theory in PDE is closely related to that of fully nonlinear equations. Generally speaking, there are two ways to tackle the problems. One is via continuity method involving some appropriate a priori estimates, and the other is weak solution theory. Monge-Ampère equations have many applications. In recent years new applications have been found in affine geometry and optimal transportation problem.

Many scholars have studied this kind of equations (see, e.g., [1–5] and the references given therein). Their main work is directed at the first or the third boundary value problem. But concerning Neumann boundary value problem, there is lack of research. In this paper, we consider the existence, uniqueness, and asymptotic behavior of a classical solution to the initial and Neumann boundary value problem for a class parabolic equation of Monge-Ampère type as follows: 𝑢̇=det^1/𝑛𝐷²_𝑥𝑢−𝑔_𝜎()(],𝑢𝑥,𝑢inΩ×0,𝑇_𝜈()[],𝑢|=𝜑𝑥,𝑢on𝜕Ω×0,𝑇_𝑡=0=𝑢₀inΩ, (1) where 𝑢̇=𝜕𝑢/𝜕𝑡 and Ω is a bounded, uniformly convex domain in 𝑅^𝑛 with the boundary 𝜕Ω∈𝐶^4+𝛼. 𝜈 denotes the unit inner normal on 𝜕Ω which has been extended on 𝑄_𝑇 to become a properly smooth vector field independent of 𝑡. For some 𝑇₀, 𝑇₀∈(0,𝑇), when 𝑡∈(0,𝑇₀], 𝑔_𝜎(𝑥,𝑢)=𝑔₁(𝑥,𝑢), and when 𝑡∈(𝑇₀,𝑇], 𝑔_𝜎(𝑥,𝑢)=𝑔₂(𝑥,𝑢). The function 𝑔_𝜎∈𝐶^{2+𝛼,2+𝛼}(Ω×𝑅), 𝜎=1,2. For each 𝑥∈Ω, lim_𝑡→𝑇⁺₀𝑔₂(𝑥,𝑢(𝑥,𝑡))=𝑔₁(𝑥,𝑢(𝑥,𝑇₀)). Here 𝜑∈𝐶^{3+𝛼,3+𝛼}(Ω×𝑅) and 𝑢₀∈𝐶^4+𝛼(Ω). The initial value 𝑢₀ is a strictly convex function on Ω. In the sequel we assume for simplicity that 0∈Ω.

To guarantee the existence of the classical solutions for (1) and convergence to a solution with prescribed curvature, we have to assume several structure conditions analogous to [6]. These are 𝜑_𝑧≡𝜕𝜑(𝑥,𝑧)𝜕𝑧≥𝑐_𝜑𝑔>0,_𝜎>0,𝑔_𝜎𝑧≡𝜕𝑔_𝜎(𝑥,𝑧)𝜕𝑧≥0,for𝜎=1,2,det^1/𝑛𝐷²_𝑥𝑢₀−𝑔𝑥,𝑢₀≥0. (2)(3)(4)

Moreover, we will always assume the following compatibility conditions to be fulfilled on 𝜕Ω×{𝑡=0}: 𝑢_0𝜈=𝜑𝑥,𝑢₀,det^1/𝑛𝐷²_𝑥𝑢₀−𝑔₁𝑥,𝑢_0𝜈=𝜑_𝑧𝑥,𝑢₀det^1/𝑛𝐷²_𝑥𝑢₀−𝑔₁𝑥,𝑢₀. (5)

Elliptic equations of Monge-Ampère type have been explored in [7–10] by using the continuity method. Some of the techniques used there will be applied in our paper as well. For the parabolic case, Schnürer and Smoczyk [6] consider the flow of a strictly convex hypersurface driven by the Gauss curvature. For the Neumann boundary value problem and for the second boundary value problem, they show that such a flow exists for all times and converges eventually to a solution of the prescribed Gauss curvature equation. Zhou and Lian [11] proved the existence and uniqueness of classical solutions to the third initial and boundary value problem for equation of parabolic Monge-Ampère type of the form −𝜕𝑢/𝜕𝑡+det^1/𝑛(𝐷²_𝑥𝑢)=𝑓(𝑥,𝑡). In this paper we will consider more general case than [11] under the structure and compatibility conditions analogous to [6] and extend some results in [7] from elliptic case to parabolic case.

The organization of this paper is as follows. In Section 2, we will review some notations, definitions, and results. In Section 3, we will obtain the uniqueness of the strictly convex classical solutions by the comparison principle. In Section 4, we shall prove uniform estimates for |𝑢̇|. This will be used in Section 5 to derive 𝐶⁰-estimates. 𝐶¹-estimates then follow from [7]. In Section 6, we shall derive 𝐶²-estimates and the 𝐶^{2+𝛽,1+𝛽/2}-estimates. In Section 7, we will give the proof of Theorem 1.

Our main result is as follows.

Theorem 1 (the main theorem). Assume that Ω is a bounded, uniformly convex domain in 𝑅^𝑛 with the boundary 𝜕Ω∈𝐶^4+𝛼. 𝜈 denotes the unit inner normal on 𝜕Ω which has been extended on 𝑄_𝑇 to become a properly smooth vector field independent of t. Let 𝑔_𝜎∈𝐶^{2+𝛼,2+𝛼}(Ω×𝑅), 𝜎=1,2, and 𝜑∈𝐶^{3+𝛼,3+𝛼}(Ω×𝑅) that satisfy (2)-(3). Let 𝑢₀∈𝐶^4+𝛼(Ω) be a strictly convex function that satisfies (4). Moreover, the compatibility conditions (5) are fulfilled. Then there exists a unique strictly convex solution of (1) in 𝐾^4+𝛼 for some 𝛼∈(0,1), where 𝐾^4+𝛼∶=𝑣(𝑥,𝑡)∣𝑣(𝑥,𝑡)∈𝐶^2,1𝑄_𝑇∩𝐶^1,0𝑄_𝑇()[𝑎𝑛𝑑𝑣⋅,𝑡𝑖𝑠𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦𝑐𝑜𝑛𝑣𝑒𝑥𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑡𝑖𝑚𝑒𝑡∈0,𝑇]∩𝐶^{4+𝛼,2+𝛼/2}𝑄_𝑇,𝑄_𝑇=Ω×(0,𝑇]. (6) As 𝑡→∞, the functions 𝑢|_𝑡 converge to a limit function 𝑢^∞ such that 𝑢^∞ is of class 𝐶⁴(Ω) and satisfies the Neumann boundary value problem det^1/𝑛𝐷²_𝑥𝑢^∞=𝑔₂𝑥,𝑢^∞𝑢inΩ_∞𝜈(𝑥)=𝜑𝑥,𝑢^∞on𝜕Ω, (7) where 𝜈 is the inward ponting unit normal of 𝜕Ω.

Proof. Uniqueness of the strictly convex classical solution is given by Theorem 5. From the estimates obtained in Sections 4–6, we get the existence and the asymptotic behavior of the classical solution in Section 7.

2. Review of Some Notations, Definitions, and Results

We first review some notations and definitions as follows: 𝑅^𝑛 is the 𝑛-dimensional Euclidean space with 𝑛≥2; Ω is a bounded, uniformly convex domain in 𝑅^𝑛, and 𝜕Ω denotes the boundary of Ω; 𝑄_𝑇=Ω×(0,𝑇], and 𝜕_𝑃𝑄_𝑇 denotes the parabolic boundary of 𝑄_𝑇, 𝜕_𝑃𝑄_𝑇=𝑄_𝑇−𝑄_𝑇; 𝑢̇=𝜕𝑢/𝜕𝑡, 𝑢̈=𝜕²𝑢/𝜕𝑡²; 𝑢_𝑖=𝐷_𝑖𝑢=𝜕𝑢/𝜕𝑥_𝑖, 𝐷𝑢=(𝐷₁𝑢,…,𝐷_𝑛𝑢); |𝐷𝑢|²∶=∑^𝑛_𝑖=1|𝐷_𝑖𝑢|², 𝐷_𝑖𝑗∶=𝜕²/𝜕𝑥_𝑖𝑥_𝑗; (𝑢^𝑖𝑗) denotes the inverse of (𝑢_𝑖𝑗); tr(𝐷²_𝑥𝑢) denotes the trace of the Hessian matrix (𝑢_𝑖𝑗); det(𝐷²_𝑥𝑢) denotes the determinant of the Hessian matrix (𝑢_𝑖𝑗); 𝐶^𝑙,𝑘(𝑄_𝑇)∶={𝑢(𝑥,𝑡)∣𝐷^𝑖_𝑥𝑢 and 𝐷^𝑗_𝑡𝑢 are all continuous in 𝑄_𝑇(0≤𝑖≤𝑙,0≤𝑗≤𝑘)}; 𝐶^{2𝑙+𝛼,𝑙+𝛼/2}∶={𝑢;𝐷^𝛽𝐷^𝑟_𝑡𝑢∈𝐶^𝛼,𝛼/2(𝑄_𝑇), for all 𝛽 and 𝑟 that satisfy |𝛽|+2𝑟≤2𝑙} with the norm |𝑢|_{2𝑙+𝛼,𝑙+𝛼/2;𝑄𝑇}=_{𝛽+2𝑟≤2𝑙}𝐷^𝛽𝐷^𝑟_𝑡𝑢_{𝛼,𝛼/2;𝑄𝑇}. (8)

Indices 𝑧 and 𝑝_𝑖 denote partial derivatives with respect to the argument used for the function 𝑢 and for its gradient, respectively. This paper adopts the Einstein summation convention and sums over repeated Latin indices from 1 to 𝑛. For example, 𝑢_𝑖𝑣^𝑖 means ∑^𝑛_𝑖=1𝑢_𝑖𝑣^𝑖. We will say “a constant 𝐶 under control” or “a controllable constant 𝐶” if the constant 𝐶 (independent of 𝑇) depends only on the known or estimated quantities, for example, the 𝐶⁴ normal of 𝑢₀ and 𝑛-the dimension of 𝑅^𝑛. We point out that the inequalities remain valid when 𝐶 is enlarged.

Now, we state existence results.

Lemma 2 (see [11]). Let 𝐹(𝐷²_𝑥𝑢)=det^1/𝑛(𝐷²_𝑥𝑢), 𝐹^𝑖𝑗(𝐷²_𝑥𝑢)=𝜕𝐹(𝐷²_𝑥𝑢)/𝜕𝑢_𝑖𝑗; then 𝐹(𝐷²_𝑥𝑢) is a concave function, (𝐹^𝑖𝑗(𝐷²_𝑥𝑢)) is a positive matrix, and tr(𝐹^𝑖𝑗(𝐷²_𝑥𝑢))=∑^𝑛_𝑖=1𝐹^𝑖𝑖(𝐷²_𝑥𝑢)≥1.

Lemma 3 (see [12]). If 𝑓∈𝐶²([0,1]), then there exists a constant 𝑀 which is independent of 𝑓, such that 𝑓^′≤𝑀𝑓𝑓+𝑓^′′, (9) where ||𝑓||=max{|𝑓(𝑥)|∶0≤𝑥≤1}.

3. Comparison Principle and Uniqueness

This section is concerned with the uniqueness of the strictly convex classical solution for (1). First of all, we will prove a comparison principle as follows.

Lemma 4. Assume that 𝑢,𝑣∈𝐶^2,1(𝑄_𝑇) and 𝑢(⋅,𝑡), 𝑣(⋅,𝑡) are all convex for every time 𝑡∈(0,𝑇]. For some 𝑇₀, 𝑇₀∈(0,𝑇), when 𝑡∈(0,𝑇₀], 𝑔_𝜎(𝑥,𝑢)=𝑔₁(𝑥,𝑢), and when 𝑡∈(𝑇₀,𝑇], 𝑔_𝜎(𝑥,𝑢)=𝑔₂(𝑥,𝑢). Let 𝑔_𝜎∈𝐶^2,2(Ω×𝑅), 𝜎=1,2, and (𝑔_𝜎)_𝑧=𝜕𝑔_𝜎(𝑥,𝑧)/𝜕𝑧≥0. Moreover, assume that(1)−𝑢̇+det^1/𝑛(𝐷²_𝑥𝑢)−𝑔_𝜎(𝑥,𝑢)≥−𝑣̇+det^1/𝑛(𝐷²_𝑥𝑣)−𝑔_𝜎(𝑥,𝑣) in Ω×(0,𝑇],(2)if𝑢>𝑣, then 𝑢_𝜈>𝑣_𝜈 on 𝜕Ω×[0,𝑇],(3)𝑢≤𝑣 on Ω×{𝑡=0},where 𝜈 is the inward pointing unit normal of 𝜕Ω; then 𝑢≤𝑣 in 𝑄_𝑇.

Proof. Consider −𝑢̇+det^1/𝑛𝐷²_𝑥𝑢−𝑔_𝜎(𝑥,𝑢)−−𝑣̇+det^1/𝑛𝐷²_𝑥𝑣−𝑔_𝜎()(̇𝑣)̇+𝑥,𝑣=−𝑢−₁₀𝜕det^1/𝑛𝑠𝐷²_𝑥𝑢+(1−𝑠)𝐷²_𝑥𝑣𝜕𝑠𝑢_𝑖𝑗+(1−𝑠)𝑣_𝑖𝑗𝑑𝑠(𝑢−𝑣)_𝑖𝑗−¹₀𝜕𝑔_𝜎(𝑥,𝑠𝑢+(1−𝑠)𝑣)𝜕(()𝑣)()(̇𝑣)̇𝑠𝑢+1−𝑠𝑑𝑠𝑢−𝑣=−𝑢−+𝑎^𝑖𝑗(𝑢−𝑣)_𝑖𝑗−𝑏(𝑢−𝑣), (10)
where 𝑎^𝑖𝑗=∫¹₀(𝜕det^1/𝑛[𝑠𝐷²_𝑥𝑢+(1−𝑠)𝐷²_𝑥𝑣]/𝜕(𝑠𝑢_𝑖𝑗+(1−𝑠)𝑣_𝑖𝑗))𝑑𝑠, 𝑏=∫¹₀(𝜕𝑔_𝜎(𝑥,𝑠𝑢+(1−𝑠)𝑣)/𝜕(𝑠𝑢+(1−𝑠)𝑣))𝑑𝑠.
From the assumptions and Lemma 2, we obtain that (𝑎^𝑖𝑗) is a positive matrix and 𝑏≥0.
Combining (10) with condition (1), we infer that −(𝑢̇−𝑣)̇+𝑎^𝑖𝑗(𝑢−𝑣)_𝑖𝑗−𝑏(𝑢−𝑣)≥0inΩ×(0,𝑇]; (11) an application of the weak parabolic maximum principle gives max_𝑄𝑇(𝑢−𝑣)≤max_{𝜕𝑃𝑄𝑇}(𝑢−𝑣)⁺. In addition, from condition (2), 𝑢−𝑣 cannot admit a positive maximum on 𝜕Ω×[0,𝑇]. And from condition (3), 𝑢≤𝑣 on Ω×{𝑡=0}. So we obtain that 𝑢≤𝑣 in 𝑄_𝑇.

Theorem 5. Under the assumptions of Theorem 1, there exists a unique classical solution of (1).

Proof. Assume that 𝑢,𝑣∈𝐶^2,1(𝑄_𝑇) are two classical solutions of (1). Then we have 𝑢̇=det^1/𝑛𝐷²_𝑥𝑢−𝑔_𝜎(𝑥,𝑢)inΩ×(0,𝑇],𝑢=𝑢₀(𝑥)onΩ×{𝑡=0}; (12) meanwhile, 𝑣̇=det^1/𝑛𝐷²_𝑥𝑣−𝑔_𝜎(𝑥,𝑣)inΩ×(0,𝑇],𝑣=𝑢₀(𝑥)onΩ×{𝑡=0}. (13) Thus, −𝑢̇+det^1/𝑛𝐷²_𝑥𝑢−𝑔_𝜎()̇𝑥,𝑢=−𝑣+det^1/𝑛𝐷²_𝑥𝑣−𝑔_𝜎()(],{}.𝑥,𝑣inΩ×0,𝑇𝑢=𝑣onΩ×𝑡=0 (14) It follows that conditions (1) and (3) in Lemma 4 are satisfied.
From 𝑢_𝜈=𝜑(𝑥,𝑢) on 𝜕Ω×[0,𝑇] and the structure condition (2), we obtain that condition (2) in Lemma 4 is satisfied.
Since 𝑔_𝜎∈𝐶^{2+𝛼,2+𝛼}(Ω×𝑅) and the structure condition (3) is satisfied, we obtained from Lemma 4 that 𝑢=𝑣 for all (𝑥,𝑡)∈𝑄_𝑇.

4. 𝑢̇-Estimates

The proof of the 𝑢̇-estimates can be carried out as in [6]. For a constant 𝜆 we define the function 𝑟=𝑒^𝜆𝑡(𝑢̇)²; thus 𝑟̇=𝜆𝑒^𝜆𝑡(𝑢̇)²+2𝑒^𝜆𝑡𝑢̇𝑢̈=𝜆𝑟+2𝑒^𝜆𝑡𝑢̇𝑑𝑑𝑡det^1/𝑛𝐷²_𝑥𝑢−𝑔_𝜎()2𝑥,𝑢=𝜆𝑟+𝑛𝑒^𝜆𝑡𝑢̇𝑢̇_𝑖𝑗𝑢^𝑖𝑗⋅det^1/𝑛𝐷²_𝑥𝑢−2𝑟𝑔_𝜎𝑧1=𝜆𝑟+𝑛𝑟_𝑖𝑗−2𝑒^𝜆𝑡𝑢̇_𝑖𝑢̇_𝑗𝑢^𝑖𝑗⋅det^1/𝑛𝐷²_𝑥𝑢−2𝑟𝑔_𝜎𝑧=1𝑛det^1/𝑛𝐷²_𝑥𝑢𝑢^𝑖𝑗𝑟_𝑖𝑗−2𝑛𝑒^𝜆𝑡det^1/𝑛𝐷²_𝑥𝑢𝑢^𝑖𝑗𝑢̇_𝑖𝑢̇_𝑗+𝜆−2𝑔_𝜎𝑧𝑟. (15)

So (1) implies the following evolution equation for 𝑟: ̇1𝑟=𝑛det^1/𝑛𝐷²_𝑥𝑢𝑢^𝑖𝑗𝑟_𝑖𝑗−2𝑛𝑒^𝜆𝑡det^1/𝑛𝐷²_𝑥𝑢𝑢^𝑖𝑗𝑢̇_𝑖𝑢̇_𝑗+𝜆−2𝑔_𝜎𝑧𝑟. (16)

Theorem 6. As long as a strictly convex solution of (1) exists, one obtains the estimates |𝑢̇|_0,𝑄𝑇≤𝑀, (17) where 𝑀 is a controllable constant.

Proof. If (𝑢̇)² admits a positive local maximum in 𝑥∈𝜕Ω for a positive time, then we differentiate the Neumann boundary condition and obtain from (2) that (𝑢̇)²_𝜈=2𝑢̇(𝑢̇)_𝜈=2𝑢̇𝑢̇_𝜈=2(𝑢̇)²𝜑_𝑧>0 (18) which contradicts the maximality of (𝑢̇)² at 𝑥.
Now we choose 𝜆=0 in (16) and get 𝑑(𝑢̇)²=1𝑑𝑡𝑛𝑢^𝑖𝑗det^1/𝑛𝐷²_𝑥𝑢(𝑢̇)²_𝑖𝑗−2𝑛𝑢^𝑖𝑗det^1/𝑛𝐷²_𝑥𝑢𝑢̇_𝑖𝑢̇_𝑖−2𝑔_𝜎𝑧(𝑢̇)²≤1𝑛det^1/𝑛𝐷²_𝑥𝑢𝑢^𝑖𝑗(𝑢̇)²_𝑖𝑗−2𝑔_𝜎𝑧(𝑢̇)². (19) Since det𝐷²_𝑥𝑢>0, (𝑔_𝜎)_𝑧≥0, we obtain from the parabolic maximum principle that max_𝑄𝑇(𝑢̇)²≤max_{𝜕𝑃𝑄𝑇}(𝑢̇)²⁺. (20) From the aforementioned a positive local maximum of (𝑢̇)² cannot occur at a point of 𝜕Ω for a positive time, so (𝑢̇)²≤max_𝑡=0(𝑢̇)²⟹|𝑢̇|≤max_𝑡=0|𝑢̇|. (21) From the fact that the solution is smooth up to the initial time 𝑡=0, we get 𝑢̇=det^1/𝑛𝐷²_𝑥𝑢₀−𝑔₁𝑥,𝑢₀onΩ×{𝑡=0}. (22) By (21) and (22), there exists a controllable constant 𝑀 such that |𝑢̇|_0,𝑄𝑇≤𝑀. Here we have used the fact that 𝑢₀∈𝐶⁴(Ω).

Lemma 7. If 0≤𝑢̇(𝑥,0)≢0 for 𝑡=0, then a solution of (1) satisfies 𝑢̇>0 or equivalently det^1/𝑛(𝐷²_𝑥𝑢)−𝑔_𝜎(𝑥,𝑢)>0 for 𝜎=1,2 and 𝑡>0.

Proof. We use the methods known from [6]. Differentiating the equation 𝑢̇=det^1/𝑛𝐷²_𝑥𝑢−𝑔_𝜎(𝑥,𝑢) (23) yields ̈1𝑢=𝑛det^1/𝑛𝐷²_𝑥𝑢𝑢^𝑖𝑗𝑢̇_𝑖𝑗−𝑔_𝜎𝑧𝑢̇. (24)
From (24) and parabolic maximum principle, we see that inf_𝑄𝑇(𝑢̇)≥inf_{𝜕𝑃𝑄𝑇}(𝑢̇)₋, (25) where (𝑢̇)₋=min{𝑢̇,0}.
If 𝑢̇ admits a negative local minimum in 𝑥∈𝜕Ω for a positive time, then we differentiate the Neumann boundary condition and get from (2) that (𝑢̇)_𝜈=𝑢̇_𝜈=𝜑_𝑧𝑢̇<0 (26) which contradicts the minimum of (𝑢̇) at 𝑥. Since 0≤𝑢̇(𝑥,0), it follows that inf_{𝜕𝑃𝑄𝑇}(𝑢̇)₋=0. That is, inf_𝑄𝑇(𝑢̇)≥inf_{𝜕𝑃𝑄𝑇}(𝑢̇)₋=0. (27) So 𝑢̇≥0 or equivalently det^1/𝑛(𝐷²_𝑥𝑢)−𝑔_𝜎(𝑥,𝑢)≥0 for 𝜎=1,2 and 𝑡>0.
From (24) and the strong parabolic maximum principle [13], we obtain that 𝑢̇ has to vanish identically if it vanishes in (𝑥₀,𝑡₀)∈Ω×(0,𝑇], contradicting 𝑢̇≢0 for 𝑡=0. If 𝑢̇=0 for 𝑥₀∈𝜕Ω, the Neumann boundary condition implies that (𝑢̇)_𝜈=(𝑢̇)𝜑_𝑧=0, (28) but this is impossible in view of the Hopf lemma applied to (24).
Consequently, if 0≤𝑢̇(𝑥,0)≢0 for 𝑡=0, then a solution of (1) satisfies 𝑢̇>0 or equivalently det^1/𝑛(𝐷²_𝑥𝑢)−𝑔_𝜎(𝑥,𝑢)>0 for 𝜎=1,2 and 𝑡>0.

5. 𝐶⁰- and 𝐶¹-Estimates

In this section we derive the 𝐶⁰- and 𝐶¹-estimates of the solution to problem (1).

Theorem 8. Let Ω be a bounded, uniformly convex domain in 𝑅^𝑛. Also, 𝑢∈𝐶^2,1(𝑄_𝑇)∩𝐶^1,0(𝑄_𝑇) is a strictly convex solution of (1). Then there exists a controllable constant 𝑀₀, such that |𝑢|_0,𝑄𝑇≤𝑀₀.

Proof. Since (4) is satisfied, we obtained from Lemma 7 that 𝑢̇≥0 in 𝑄_𝑇. So 𝑢(𝑥,𝑡)≥𝑢(𝑥,0)=𝑢₀(𝑥). As 𝑢₀(𝑥)∈𝐶^4+𝛼(Ω), then there exists a controllable constant 𝑁₁ such that 𝑢(𝑥,𝑡)≥𝑁₁in𝑄_𝑇. (29)
Next we will prove that 𝑢 is uniformly a priori bounded from above.
At a maximum of 𝑢, which necessarily occurs on 𝜕Ω since 𝑢 is convex, we have 𝑢_𝜈≤0. Since 𝑢_𝜈=𝜑(𝑥,𝑢) on 𝜕Ω×[0,𝑇], then 𝜑(𝑥,𝑢)≤0(𝑥,𝑡)∈𝜕Ω×[0,𝑇]. (30) From (2) we get that 𝜑(⋅,𝑧)→∞ uniformly as 𝑧→∞. Then we can deduce that 𝜑(𝑥,𝑧)>0 for all 𝑥∈𝜕Ω and 𝑧>𝑁₂, where 𝑁₂ is controllable constant. Combining (30) yields 𝑢≤𝑁₂. (31) This completes the proof of the theorem.

Theorem 9. Let Ω be a bounded, uniformly convex domain in 𝑅^𝑛, and 𝑢∈𝐶^4,2(𝑄_𝑇)∩𝐶^1,0(𝑄_𝑇) is a strictly convex solution of (1). Then one has sup_𝑄𝑇|𝐷𝑢|≤𝑀^∗, (32) where 𝑀^∗ is a controllable constant.

Proof. For any 𝑡₀∈[0,𝑇], 𝑢(𝑥,𝑡₀) is a continuous differentiable, convex function. From 𝑢_𝜈=𝜑(𝑥,𝑢) and the 𝐶⁰-estimates, we get 𝑢_𝜈≥−𝑀, (33) where 𝑀 is a controllable constant. Then using Theorem 2.2 in [7], we have 𝐷𝑢𝑥,𝑡₀≤𝑀₁onΩ. (34)
Since 𝑡₀ is arbitrary, we obtain that |𝐷𝑢|≤𝑀^∗, (35) where 𝑀^∗ is a controllable constant. This completes the proof of the theorem.

6. 𝐶²- and 𝐶^{2+𝛽,1+𝛽/2}-Estimates

This section is concerned with the 𝐶²-estimates and the 𝐶^{2+𝛽,1+𝛽/2}-estimates of the solution to problem (1).

Theorem 10. Assume that Ω is a 𝐶⁴ bounded, uniformly convex domain in 𝑅^𝑛 and 𝑢∈𝐶^4,2(𝑄_𝑇) is a strictly convex solution of (1). Let 𝑔_𝜎∈𝐶^2,2(Ω×𝑅), 𝜎=1,2, 𝜑∈𝐶^3,3(Ω×𝑅). Then one has sup_𝑄𝑇𝐷²_𝑥𝑢≤𝑀^′′, (36) where 𝑀^′′ is a controllable constant.

Proof. Let 𝜉∈𝑆^𝑛−1. First we observe that 𝐷_𝜉𝜉𝑢>0, since 𝑢 is strictly convex. So we only need to prove the fact that 𝐷_𝜉𝜉𝑢 is a priori bounded from above.
We define for (𝑥,𝑡,𝜉)∈Ω×[0,𝑇]×𝑆^𝑛−1 that 𝜔(𝑥,𝑡,𝜉)=𝐷_𝜉𝜉𝑢−𝑉(𝑥,𝑡,𝜉)+𝐾|𝑥|², (37) where 𝑉(𝑥,𝜉,𝑡) is given by 𝑉(𝑥,𝜉,𝑡)=2𝜉,𝜈𝜉^′_𝑖𝐷_𝑖𝜑−𝐷_𝑘𝑢𝐷_𝑖𝜈_𝑘. (38) Here 𝜈 is a smooth extension of the inner unit normal on 𝜕Ω that is independent of 𝑡. 𝜉^′ is given by 𝜉^′=𝜉−𝜉,𝜈𝜈, (39)𝐾 is a constant to be chosen, and 𝐷 indicates that the chain rule has not yet been applied to the respective terms.
Let 𝑎_𝑘=2𝜉,𝜈𝜑_𝑧𝜉^′_𝑘−𝜉^′_𝑖𝐷_𝑖𝜈_𝑘,𝑏=2𝜉,𝜈𝜉^′_𝑖𝜑_𝑖, (40) then 𝑉(𝑥,𝜉,𝑡)=𝑎_𝑘𝑢_𝑘𝜔()+𝑏,𝑥,𝑡,𝜉=𝐷_𝜉𝜉𝑢−𝑎_𝑘𝑢_𝑘−𝑏+𝐾|𝑥|². (41) We compute that −𝜕𝜔𝜕𝑡+𝐹^𝑖𝑗𝐷_𝑖𝑗𝜔=−𝐷_𝜉𝜉𝑡𝑢+𝑎_𝑘𝐷_𝑡𝑘𝑢+𝐷_𝑡𝑎_𝑘⋅𝑢_𝑘+𝐷_𝑡𝑏+𝐹^𝑖𝑗𝐷_{𝑖𝑗𝜉𝜉}𝑢−𝑎_𝑘𝐹^𝑖𝑗⋅𝑢_𝑖𝑗𝑘−2𝐹^𝑖𝑗𝐷_𝑖𝑎_𝑘⋅𝑢_𝑘𝑗−𝐹^𝑖𝑗⋅𝑢_𝑘𝐷_𝑖𝑗𝑎_𝑘−𝐹^𝑖𝑗𝐷_𝑖𝑗𝑏+2𝐾𝐹^𝑖𝑗𝛿_𝑖𝑗. (42)
Next, we estimate the right-hand side of (42), respectively.
Let 𝐹(𝐷²_𝑥𝑢)=det^1/𝑛(𝐷²_𝑥𝑢). From Lemma 2, we have that 𝐹(𝐷²_𝑥𝑢) is a concave function, (𝐹^𝑖𝑗(𝐷²_𝑥𝑢)) is a positive matrix, and tr(𝐹^𝑖𝑗(𝐷²_𝑥𝑢))=∑^𝑛_𝑖=1𝐹^𝑖𝑖(𝐷²_𝑥𝑢)≥1.
Differentiating the equation 𝑢̇=𝐹𝐷²_𝑥𝑢−𝑔_𝜎(𝑥,𝑢) (43) twice in the direction 𝜉, 𝜉∈𝑆^𝑛−1, we therefore obtain −𝐷_𝑡𝜉𝜉𝑢+𝐹^𝑖𝑗𝐷_{𝑖𝑗𝜉𝜉}𝑢+𝐹^{𝑖𝑗,𝑘𝑙}𝐷_𝑖𝑗𝜉𝑢𝐷_𝑘𝑙𝜉𝑢=𝐷_𝜉𝜉𝑔_𝜎(𝑥,𝑢). (44) Using the concavity of 𝐹, we have 𝐹^{𝑖𝑗,𝑘𝑙}𝐷_𝑖𝑗𝜉𝑢𝐷_𝑘𝑙𝜉𝑢≤0; (45) then −𝐷_𝑡𝜉𝜉𝑢+𝐹^𝑖𝑗𝐷_{𝑖𝑗𝜉𝜉}𝑢≥𝐷_𝜉𝜉𝑔_𝜎(𝑥,𝑢). (46)
Differentiating the equation 𝑢̇=𝐹𝐷²_𝑥𝑢−𝑔_𝜎(𝑥,𝑢) (47) in the 𝑘th coordinate direction, we obtain −𝐷_𝑡𝑘𝑢+𝐹^𝑖𝑗𝐷_𝑖𝑗𝑘𝑢=𝐷_𝑘𝑔_𝜎. (48) From (𝐹^𝑖𝑗)=𝜕𝐹(𝐷²_𝑥𝑢)/𝜕𝑢_𝑖𝑗=(𝐹(𝐷²_𝑥𝑢)/𝑛)𝑢^𝑖𝑗, where (𝑢^𝑖𝑗) is the inverse of (𝑢_𝑖𝑗), we have 𝐹^𝑖𝑗𝐷_𝑖𝑎_𝑘⋅𝑢_𝑘𝑗=𝐹𝐷²_𝑥𝑢𝑛𝑢^𝑖𝑗𝑢_𝑘𝑗⋅𝐷_𝑖𝑎_𝑘=𝐹𝐷²_𝑥𝑢𝑛𝐷_𝑖𝑎_𝑖. (49) Using the estimates of 𝑢̇ and 𝑢, we obtain that 𝐹(𝐷²_𝑥𝑢)=𝑢̇+𝑔_𝜎 is bounded. From 𝐷_𝑗𝑎_𝑗=2𝜉_𝑙𝐷_𝑗𝜈_𝑙𝜑_𝑧𝜉^′_𝑗−𝜉^′_𝑖𝐷_𝑖𝜈_𝑗+2𝜉,𝜈𝜑_𝑧𝑗+𝜑_𝑧𝑧𝐷_𝑗𝑢𝜉^′_𝑗−𝜉^′_𝑖𝐷_𝑗𝑖𝜈_𝑗, (50) as well as 𝐶⁰- and 𝐶¹-estimates, it follows that |𝐷_𝑗𝑎_𝑗| is bounded. Thus there exists a controllable constant 𝐶 such that 𝐹^𝑖𝑗𝐷_𝑖𝑎_𝑘⋅𝑢_𝑘𝑗≤𝐶. (51)
Since (𝐹^𝑖𝑗) is positive definite, we can get that 𝐹^𝑖𝑗≤12𝐹^𝑖𝑖+𝐹^𝑗𝑗. (52) Applying (52), 𝐶¹-estimates, and the following equality: 𝐷_𝑗ℎ𝑎_𝑘=2𝜉_𝑙𝐷_𝑗ℎ𝜈_𝑙𝜑_𝑧𝜉^′_𝑘−𝜉^′_𝑖𝐷_𝑖𝜈_𝑘+2𝜉_𝑙𝐷_𝑗𝜈_𝑙𝜑_𝑧ℎ+𝜑_𝑧𝑧𝑢_ℎ𝜉^′_𝑘−𝜉^′_𝑖𝐷_𝑖ℎ𝜈_𝑘+2𝜉_𝑠𝐷_ℎ𝜈_𝑠𝜑_𝑧𝑗+𝜑_𝑧𝑧𝑢_𝑗𝜉^′_𝑘−𝜉^′_𝑖𝐷_𝑖𝑗𝜈_𝑘+2𝜉,𝜈𝜑_𝑧𝑗ℎ+𝜑_𝑧𝑗𝑧𝑢_ℎ+𝜑_𝑧𝑧ℎ+𝜑_𝑧𝑧𝑧𝑢_ℎ𝑢_𝑗+𝜑_𝑧𝑧𝑢_𝑗ℎ𝜉^′_𝑘−𝜉^′_𝑖𝐷_𝑖𝑗ℎ𝜈_𝑘, (53) we obtain that 𝐹^ℎ𝑗⋅𝑢_𝑘𝐷_ℎ𝑗𝑎_𝑘≤𝑢_𝑘𝐹^ℎ𝑗2𝜉,𝜈𝜑_𝑧𝑧𝑢_𝑗ℎ𝜉^′_𝑘+𝐶𝐹^ℎ𝑗=2𝑢_𝑘𝜉,𝜈𝜑_𝑧𝑧𝜉^′_𝑘𝐹𝐷²_𝑥𝑢𝑛𝑢^ℎ𝑗𝑢_𝑗ℎ+𝐶𝐹^ℎ𝑗≤𝐶₁+𝐶₂tr𝐹^𝑖𝑗, (54) where 𝐶₁ and 𝐶₂ are positive controllable constants.
From (51), (54), and the estimates like these, it follows that 𝐷_𝑡𝑎_𝑘⋅𝑢_𝑘+𝐷_𝑡𝑏−2𝐹^𝑖𝑗𝐷_𝑖𝑎_𝑘⋅𝑢_𝑘𝑗−𝐹^𝑖𝑗⋅𝑢_𝑘𝐷_𝑖𝑗𝑎_𝑘−𝐹^𝑖𝑗𝐷_𝑖𝑗𝑏≤𝑐₁tr𝐹^𝑖𝑗+𝑐₂, (55) where 𝑐₁ and 𝑐₂ are positive controllable constants. Then using (46) and (48), we can obtain −𝜕𝜔𝜕𝑡+𝐹^𝑖𝑗𝐷_𝑖𝑗𝜔≥𝐷_𝜉𝜉𝑔_𝜎−𝑎_𝑘𝐷_𝑘𝑔_𝜎−𝑐₁tr𝐹^𝑖𝑗+𝑐₂+2𝐾𝐹^𝑖𝑗𝛿_𝑖𝑗=𝑔_𝜎𝜉𝜉+2𝑔_𝜎𝜉𝑧𝑢_𝜉+𝑔_𝜎𝑧𝑧𝑢_𝜉𝑢_𝜉+𝑔_𝜎𝑧𝑢_𝜉𝜉−𝑎_𝑘𝑔_𝜎𝑘+𝑔_𝜎𝑧𝑢_𝑘−𝑐₁tr𝐹^𝑖𝑗+𝑐₂+2𝐾𝐹^𝑖𝑗𝛿_𝑖𝑗≥𝑔_𝜎𝜉𝜉+2𝑔_𝜎𝜉𝑧𝑢_𝜉+𝑔_𝜎𝑧𝑧𝑢_𝜉𝑢_𝜉−𝑎_𝑘𝑔_𝜎𝑘+𝑔_𝜎𝑧𝑢_𝑘−𝑐₁tr𝐹^𝑖𝑗+𝑐₂+2𝐾𝐹^𝑖𝑗𝛿_𝑖𝑗, (56) where we have used the structure condition (3) and the convexity of 𝑢. Using 𝐶⁰- and 𝐶¹-estimates, there exist positive controllable constants 𝑐₃ and 𝑐₄ such that −𝜕𝜔𝜕𝑡+𝐹^𝑖𝑗𝐷_𝑖𝑗𝜔≥2𝐾−𝑐₃tr𝐹^𝑖𝑗−𝑐₄. (57)
Since tr(𝐹^𝑖𝑗)≥1, we fix 𝐾≥(1/2)(𝑐₃+𝑐₄+1) and deduce that −𝜕𝜔𝜕𝑡+𝐹^𝑖𝑗𝐷_𝑖𝑗𝜔≥1. (58) Thus by the parabolic maximum principle, we have 𝜔≤sup_{𝜕𝑃𝑄𝑇}𝜔. (59) As 𝜔 is known on Ω×{𝑡=0}×𝑆^𝑛−1, we need only to estimate 𝜔 on 𝜕Ω×[0,𝑇]×𝑆^𝑛−1.
The estimation of 𝜔 on 𝜕Ω×[0,𝑇]×𝑆^𝑛−1 splits into four stages according to the direction 𝜉. The first three stages: (i) the mixed tangential normal second derivatives of 𝑢 on 𝜕Ω×[0,𝑇]×𝑆^𝑛−1, (ii) 𝜉 tangential, and (iii) 𝜉 nontangential, can be carried out as in [7]. The details of this procedure could be seen in [7]. Stage (i) is readily estimated. Stages (ii) and (iii) are reduced to the purely normal case. So we only give the proof of the fourth stage: (iv) 𝜉 normal. We extend the argument given in [2] and modified for the parabolic case.
Set ℎ(𝑥,𝑡)=𝜈_𝑘𝐷_𝑘𝑢−𝜑(𝑥,𝑢)=𝐷_𝜈𝑢−𝜑(𝑥,𝑢). By (48), a direct calculation yields 𝐿ℎ=−𝐷_𝑡ℎ+𝐹^𝑖𝑗𝐷_𝑖𝑗ℎ=−𝜈_𝑘𝐷_𝑘𝑡𝑢+𝜑_𝑧𝐷_𝑡𝑢+𝜈_𝑘𝐹^𝑖𝑗𝐷_𝑖𝑗𝑘𝑢+2𝐹^𝑖𝑗𝐷_𝑖𝜈_𝑘𝐷_𝑗𝑘𝑢+𝐹^𝑖𝑗𝐷_𝑖𝑗𝜈_𝑘𝐷_𝑘𝑢−𝐹^𝑖𝑗𝐷_𝑖𝑗𝜑=𝜈_𝑘𝑔_𝜎𝑘+𝑔_𝜎𝑧𝑢_𝑘+𝜑_𝑧𝐷_𝑡𝑢+2𝐹^𝑖𝑗𝐷_𝑖𝜈_𝑘𝐷_𝑗𝑘𝑢+𝐹^𝑖𝑗𝐷_𝑖𝑗𝜈_𝑘𝐷_𝑘𝑢−𝐹^𝑖𝑗𝜑_𝑖𝑗+2𝜑_𝑖𝑧𝑢_𝑗+𝜑_𝑧𝑧𝑢_𝑖𝑢_𝑗+𝜑_𝑧𝑢_𝑖𝑗. (60) Thus, using (𝐹^𝑖𝑗)=(𝐹(𝐷²_𝑥𝑢)/𝑛)𝑢^𝑖𝑗, (52), and our a priori estimates, we have |𝐿ℎ|=−𝐷_𝑡ℎ+𝐹^𝑖𝑗𝐷_𝑖𝑗ℎ≤𝐶₀1+tr𝐹^𝑖𝑗≤𝐶tr𝐹^𝑖𝑗, (61) where 𝐶 is a controllable constant.
Let (𝑥₀,𝑡₀)∈𝜕Ω×[0,𝑇], and (𝑥₀,𝑡₀) is arbitrary. We observe that Ω is a bounded, uniformly convex domain in 𝑅^𝑛, so there exists a uniformly closed ball 𝐵_𝑅(𝑥^∗) such that Ω⊂𝐵_𝑅𝑥^∗⊂𝑅^𝑛,𝜕𝐵_𝑅𝑥^∗∩𝜕Ω=𝑥₀. (62) Meanwhile, we assume that |𝑥−𝑥^∗|>1 for all 𝑥∈Ω.
We consider the auxiliary function in 𝐵_𝑅(𝑥^∗)×[0,𝑇]𝑞(𝑥,𝑡)=𝑒^𝐾1𝑅2−𝑒^{𝐾1|𝑥−𝑥∗|2}, (63) where 𝐾₁ is a positive constant to be determined.
If we choose 𝐾₁ sufficiently large, it is easy to see that 𝑞(𝑥,𝑡)≥ℎ(𝑥,𝑡) on 𝜕_𝑃𝑄_𝑇. For sufficiently large 𝐾₁, we have −𝜕𝑞𝜕𝑡+𝐹^𝑖𝑗𝐷_𝑖𝑗𝑞=−2𝐾₁tr𝐹^𝑖𝑗𝑒^{𝐾1|𝑥−𝑥∗|2}−4𝐾²₁𝐹^𝑖𝑗𝑥−𝑥^∗_𝑖𝑥−𝑥^∗_𝑗𝑒^{𝐾1|𝑥−𝑥∗|2}≤−2𝐾₁tr𝐹^𝑖𝑗𝑒^{𝐾1|𝑥−𝑥∗|2}≤−𝐶tr𝐹^𝑖𝑗, (64) where we have used the fact that (𝐹^𝑖𝑗) is positive definite.
By (61) and (64), we get −𝜕𝑞−ℎ𝜕𝑡+𝐹^𝑖𝑗𝐷_𝑖𝑗𝑞−ℎ≤0; (65) thus we obtain 𝑞−ℎ≥inf_{𝜕𝑃𝑄𝑇}(𝑞−ℎ)≥0 on 𝑄_𝑇 in view of the parabolic maximum principle. Since 𝑞(𝑥₀,𝑡₀)=ℎ(𝑥₀,𝑡₀)=0, it follows that 𝑞−ℎ𝑥₀+𝜌𝜈−𝑞−ℎ𝑥₀,𝑡₀𝜌≥0 (66) thus lim_𝜌→0+𝑞−ℎ𝑥₀+𝜌𝜈−𝑞−ℎ𝑥₀,𝑡₀𝜌≥0. (67) Therefore, 𝐷_𝜈𝑞−ℎ𝑥₀,𝑡₀≥0. (68) Hence, ℎ_𝜈𝑥₀,𝑡₀≤𝑞_𝜈𝑥₀,𝑡₀≤𝑐₁, (69) where 𝑐₁ is a controllable constant.
For −𝑞, in a similar fashion we can obtain ℎ_𝜈𝑥₀,𝑡₀≥−𝑐₂, (70) where 𝑐₂ is a controllable constant.
Since (𝑥₀,𝑡₀)∈𝜕Ω×[0,𝑇] is arbitrary, we obtain sup_{[]𝜕Ω×0,𝑇}𝐷_𝜈𝜈𝑢≤𝐶, (71) where 𝐶 is a controllable constant.
Combining the estimates of the four stages, we obtain that there exists a controllable constant 𝐶 such that 𝐷_𝜉𝜉𝑢≤𝐶 on 𝑄_𝑇.
Since 𝜉 is an arbitrary direction in 𝑆^𝑛−1, now let 𝜉=𝑒_𝑖±𝑒_𝑗/2^1/2, where 𝑒_𝑖=(0,0…,1,…0)=𝑖th standard coordinate vector. Thus we can get the required bounded for 𝐷²_𝑥𝑢 immediately. This completes the proof of the theorem.

From the uniform 𝐶⁰-estimates, 𝑢̇-estimates, and the assumptions on 𝑔_𝜎, 𝜎=0,1, we can conclude that 𝐹(𝐷²𝑢) has a priori positive bound from below. And using the uniform 𝐶²-estimates for 𝑢, we obtain that (1) is uniformly parabolic. So we can apply the method of [14] to obtain the 𝐶^{2+𝛽,1+𝛽/2} interior estimates and the estimates near the bottom. Using the estimates near the side in [15], we can get the Hölder seminorm estimates for 𝑢̇ and 𝐷²_𝑥𝑢. Thus we have the 𝐶^{2+𝛽,1+𝛽/2}-estimates.

7. The Proof of Theorem 1

In Section 3 we proved the uniqueness of the strictly convex solution for (1). The existence of the strictly convex solution for (1) is obtained by using the continuity method. Applying Theorem 5.3 in [16], the implicit function theorem, and the Arzela-Ascoli theorem, we can get the desired result. Then the standard regularity of parabolic equation implies that 𝑢∈𝐶^{4+𝛽,2+𝛽/2}(𝑄_𝑇). Since there are sufficient a priori estimates, we can extend a solution of (1) on a time interval [0,𝑇] to [0,𝑇+𝜖) for a small 𝜖>0. In this way we obtain existence for all 𝑡≥0 from the a priori estimates. We then need the following lemma to prove the asymptotic behavior of a classical solution of (1).

Lemma 11. If a solution of (1) exists for all 𝑡≥0 and (4) is satisfied, then as 𝑡→∞, the functions 𝑢|_𝑡 converge to a limit function 𝑢^∞(𝑥) such that 𝑢^∞(𝑥) satisfies the Neumann boundary value problem det^1/𝑛𝐷²_𝑥𝑢^∞=𝑔₂𝑥,𝑢^∞𝑢𝑥∈Ω,_∞𝜈=𝜑𝑥,𝑢^∞𝑥∈𝜕Ω, (72) where 𝜈 is the unit inner normal on 𝜕Ω. Moreover, 𝑢(𝑥,𝑡)→𝑢^∞(𝑥) in 𝐶³-norm.

Proof. We may assume that 𝑢̇(⋅,0)≢0 and proceed as in [17]. Integrating the equation 𝑢̇=det^1/𝑛𝐷²_𝑥𝑢−𝑔₂(𝑥,𝑢) (73) with respect to 𝑡 yields 𝑢(𝑥,𝑡)−𝑢𝑥,𝑇₀=^𝑡_𝑇0det^1/𝑛𝐷²_𝑥𝑢(𝑥,𝜏)−𝑔₂(𝑥,𝑢(𝑥,𝜏))𝑑𝜏. (74) The left-hand side is uniformly bounded in view of the 𝐶⁰-estimates. By applying Lemma 7, det^1/𝑛(𝐷²_𝑥𝑢)−𝑔₂ is nonnegative, and we can find that 𝑡_𝑘=𝑡_𝑘(𝑥)→∞ such that det^1/𝑛𝐷²_𝑥𝑢−𝑔_{2𝑡=𝑡𝑘}⟶0. (75)
On the other hand, 𝑢(𝑥,⋅) is monotone, and therefore lim_𝑡→∞𝑢(𝑥,𝑡)=∶𝑢^∞(𝑥) (76) exists and is of class 𝐶⁴(Ω) in view of the a priori estimates.
From differential interpolation inequality in Lemma 3, we can obtain the interpolation inequality of the form ‖𝐷𝑢‖≤𝐶‖𝑢‖⋅𝐷²_𝑥𝑢+‖𝑢‖ (77) for 𝑢=𝑢−𝑢^∞, where ‖⋅‖ denotes the sup-norm.
Dini’s theorem and interpolation inequalities of the form (77) yield 𝑢(𝑥,𝑡)→𝑢^∞(𝑥) in 𝐶³-norm. We finally, obtain in view of (75) that 𝑢^∞ is a solution of the problem (72). This complete, the proof of the lemma.
Now we completed the proof of Theorem 1.

Acknowledgment

This project is supported by Inner Mongolia Natural Science Foundation of China under Grant 2011MS0107.