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.

The second boundary value problem for a class parabolic Monge-Ampère Equation

“…Generally speaking, there are two ways to tackle the problems: one is via continuity method which involving some appropriate a prior estimates, the other is weak solution theory. The first boundary value problem of Monge-Ampère equation has been explored by many authors, as the problem studied in [1], [2], [3]. But the result of the second boundary value problem is scarce.…”