We consider the isothermal Euler system with damping. We explicitly compute the propagation and the behavior of Gaussian initial data, then we show the weak $ L^1 $ convergence of the density as well as the asymptotic behavior of its first and second moments. We also rigorously show the convergence of Barenblatt solutions towards a limit Gaussian profile in the isothermal limit $ \gamma \rightarrow 1 $.
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