Edit: I wrote a Gimp script that does the steps below, and another one for Ilmaris answer. Both scripts are available for download on GitHub. The suggested way to go is this one.

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I’m answering this right here because I have been searching for a solution for quite some time and found a simple and working one. Let’s get to the result right away:

1. Create a background layer by copying a vertical stripe of the background only and stretching it to the size of the original image.

   In Gimp: Select r a rectangle from top to bottom containing background, paste Shift+Ctrl+V to a new image, and expand Filter > Map > Tile it to image size, run a gaussian blur over it, then copy it back to the image as new layer.

2. Set the layer mode to Division.

3. Reduce layer opacity as desired, e.g. to 90 %. Keeping it on 100 % is usually a bit too bright.

What this does, in essence, is that the background layer defines the RGB value w of “white” on each pixel. The division layer mode then stretches the value on the image from [0,w] to [0,255] (see GIMP layer modes) per pixel.

(Bad) Example using curves and linear gradient

First attempts were always about duplicating the layer, changing the brightness such that the darkest part on the bottom were lit properly, and adding a layer mask with a linear gradient to lighten up the darker regions only. The stronger the gradient, the worse it looked.

(Better) Example using division

Division yields much better results; lighting is fixed evenly, and there are no burnt areas as especially in the left part of the drawing.