Blur with better control over the curve?

[Asterra]

Not sure what terminology to use.  The Gaussian blur by default is applying a gentle bell curve to the blur, so that it tapers off more and more gradually as the value approaches 0% (or 100%).  In the end, the blur winds up extending extremely far—much further than the actual limit suggested by the specified radius, for example.

I'm hoping there's a way to tighten that curve.  Make it less of a bell.  Ideally, the ability to directly program the curve would be nice.

Edit: Since it's on-topic, some convenient means of forcing the blur to affect all four channels (rather than literally only alpha) would also be nice.

[Ofnuts]

You can completely shape the alpha by adding a layer mask (transfer alpha to mask) and then using Curves on the mask.