Blur by color?

[T0byWan]

I have a rather simple black and white image that's going to be used as a depth map for engraving.  I need the blur between the black and white areas to create shades of grey, but I want the black areas to bleed into the white in a way where the black areas will remain completely black, and only the white areas will change to become more grey in the border areas.  Hopefully that makes sense. can anyone tell me how to do this?

[Ofnuts]

You can just color select the white and apply the blur. The  black being unselected, it won't change:

   

However, on the edges you don't get a black-to-white transition, you get a sharp black to grey step, and then a grey to white transition. This can be changed with curves, by setting the "black point" to that gray value: