The first initial-boundary value problem of parabolic Monge–Ampère equations outside a bowl-shaped domain

Abstract

Abstract In this paper, we study the parabolic Monge–Ampère equations − u t det ( D 2 u ) = g $-u_{t}\det (D^{2}u)=g$ outside a bowl-shaped domain with g being the perturbation of g 0 ( | x | ) $g_{0}(|x|)$ at infinity. Under the weaker conditions compared with the problem outside a cylinder, we obtain the existence and uniqueness of viscosity solutions with asymptotic behavior for the first initial-boundary value problem by using the Perron method.

Keywords

• Parabolic Monge–Ampère equations
• Initial-boundary value problem
• Bowl-shaped domain
• Perron method
• Asymptotic behavior