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.
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., [
To guarantee the existence of the classical solutions for (
Moreover, we will always assume the following compatibility conditions to be fulfilled on
Elliptic equations of Monge-Ampère type have been explored in [
The organization of this paper is as follows. In Section
Our main result is as follows.
Assume that
Uniqueness of the strictly convex classical solution is given by Theorem
We first review some notations and definitions as follows:
Indices
Now, we state existence results.
Let
If
This section is concerned with the uniqueness of the strictly convex classical solution for (
Assume that
where
Consider
where
From the assumptions and Lemma
Combining (
Under the assumptions of Theorem
Assume that
From
Since
The proof of the
So (
As long as a strictly convex solution of (
If
Now we choose
If
We use the methods known from [
From (
If
From (
Consequently, if
In this section we derive the
Let
Since (
Next we will prove that
At a maximum of
Let
For any
Since
This section is concerned with the
Assume that
Let
We define for
Let
Next, we estimate the right-hand side of (
Let
Differentiating the equation
Differentiating the equation
Since
From (
Since
The estimation of
Set
Let
We consider the auxiliary function in
If we choose
By (
For
Since
Combining the estimates of the four stages, we obtain that there exists a controllable constant
Since
From the uniform
In Section
If a solution of (
We may assume that
On the other hand,
From differential interpolation inequality in Lemma
Dini’s theorem and interpolation inequalities of the form (
Now we completed the proof of Theorem
This project is supported by Inner Mongolia Natural Science Foundation of China under Grant 2011MS0107.