E. Generalizations: I. Winds
The simplest generalization of the previous models is to allow a variable circuburst density with n(R) R-k. The hydrodynamic evolution of a relativistic blast wave in such a medium was considered already in the original paper of BLmc1. The synchrotron light curve was considered first by Meszaros et al.  and by Dai and Lu .
Chevalier and Li [49, 50] stressed the importance of the n(R) R-2 case which arises whenever there is a stellar wind ejected by the GRB's progenitor prior to the burst. This arises naturally in the Collapsar model that is based on the collapse of a massive star. The calculations follow those outlines in the previous sections, with the only difference that the relations determining R(t) and (t) for homogeneous circumburst medium, Eqs. 86, should be replaced by Eqs. 78 with k = 2
The high initial densities in a wind density profile implies a low initial cooling frequency. Unlike the constant density case the cooling frequency here increase with time . This leads to a different temporal relations between the different frequencies and cooling regimes. For example it is possible that the cooling frequency will be initially below the synchrotron self absorption frequency. Chevalier and Li  consider five different evolution of the light curves for different conditions and observed frequencies. We list below the two most relevant cases, the first fits the X-ray and optical afterglows while the second is typical for the lower radio frequencies.
|c < < m||-1/4||-1/2|
|m, c <||-(3p-2)/4||- p / 2 = (2 - 1) / 3|
|m < < c||-(3p-1)/4||- (p - 1) / 2 = (2 + 1) / 3|
Note that for m, c < both the spectral slop and the temporal evolution are similar for a wind and for a constant density profile. This poses, of course, a problem in the interpretation of afterglow light curves.
|c < < a < m||7/4||5/2|
|< c < a < m||2||2|
|< a < m < c||1||2|
|a < < m < c||0||1/3|
|a < m < < c||-(3p-1)/4||- (p - 1) / 2 = (2 + 1) / 3|