The　Liouville　type　theorem　on　the　parabolic　Monge–Ampère　equation　−utdetD2u=1　states　that　any　entire　parabolically　convex　classical　solution　must　be　of　form　−t+|x|2/2　up　to　a　re-scaling　and　transformation,　under　additional　assumption　that　partial　derivative　with　respect　to　time　variable　ut　is　strictly　negative　and　bounded.　In　this　paper,　we　study　the　case　when　ut　is　unbounded,　prove　an　existence　result　of　entire　parabolically　convex　smooth　solution　and　investigate　the　asymptotic　behavior　near　infinity.　©　2023　Elsevier　Ltd