All wood is prone to splitting. I had this occur in a Victorian door recently. The split worked its way from the jamb on one side all the way to the plate behind the doorknob. To get this fixed before somebody stabbed a hand on it, I used a semi-stiff splint.
Strips cut from the hard plastic of a blister pack can be the right combination of thin and stiff to work glue into the crack. Or if you need something with a bit of a radius in it, you can try cutting a strip out of a soda bottle. The price is right, too. You can shape a piece to fit a specific need, then toss it afterward if you don’t feel like washing it off.
Bill Houghton, Sebastopol, CA