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We investigate the damping of Alfv\'en waves generated by the cosmic rayresonant streaming instability in the context of the cosmic ray escape andpropagation in the proximity of supernova remnants. We consider ion-neutraldamping, turbulent damping and non linear Landau damping in the warm ionizedand warm neutral phases of the interstellar medium. For the ion-neutraldamping, up-to-date damping coefficients are used. We investigate in particularwhether the self-confinement of cosmic rays nearby sources can appreciablyaffect the grammage. We show that the ion-neutral damping and the turbulentdamping effectively limit the residence time of cosmic rays in the sourceproximity, so that the grammage accumulated near sources is found to benegligible. Contrary to previous results, this also happens in the most extremescenario where ion-neutral damping is less effective, namely in a medium withonly neutral helium and fully ionized hydrogen. Therefore, the standardpicture, in which CR secondaries are produced during the whole time spent bycosmic rays throughout the Galactic disk, need not to be deeply revisited.

Grammage of cosmic rays in the proximity of supernova remnants embedded in a partially ionized medium