Split a polygon layer with a line layer?

First you have to union both shapes

Then activate the Polygonizerplugin to reshape the lines to polygones:

I solved this problem with the Processing Toolbox, using the SAGA Polygon-line intersection algorithm.

The process might fail (i.e. no split) if the line end is snapped to a vertex point of the polygon. Extending the line a bit into the neighbouring polygon or outside area sanitizes those pitfalls.