Large time asymptotics of compressible Euler equations for a polytropic gaswith and without the porous media equation are constructed in which theBarenblatt solution is embedded. Invariance analysis for these governingequations are carried out using the classical and the direct methods. A newsecond order nonlinear partial differential equation is derived and is shownto reduce to an Euler-Painlev´e equation. A regular perturbation solution ofa reduced ordinary differential equation is determined. And an exact closedform solution of a system of ordinary differential equations is derived usingthe invariance analysis.