Singular convergence to nonlinear diffusion waves for solutions to the Cauchy problem for the compressible Euler equations with damping

We investigate the singular limit for the solutions to the compressible gas dynamics equations with damping term, after a parabolic scaling, in the one-dimensional isentropic case. In particular, we study. the convergence in Sobolev norms towards diffusive prophiles, in case of well-prepared initial data and small perturbations of them. The results are obtained by means of symmetrization and energy estimates.