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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.