Can I plan to grow grass in...

Grass that is shaded by thick trees is also competing with them for light, water, and soil nutrients, so it tends to have weakened root systems. One thing to do is to mow the shaded areas higher, leaving the grass longer between clippings. Water that area more in times of drought. Go light on the nitrogen fertilizers. If the trees are just too thick, some judicious pruning may be in order so that the lawn gets enough light.