Cracked & Heaving Garage Doorway
At this stage you can only try to keep it from getting worse.  heaving is caused by the freeze thaw cycle.  You can either keep the area warm or keep it dry.  Often neither is practical.

Keeping it dry requires figuring out how water gets to the area below the concrete and stopping that flow.  Water is very clever, and may be moving in after getting in elsewhere, but my first guess is the crack between the curb and the garage slab, and the one between the asphalt and the kerb.  Backer rod and silicon grout come to mind.

At the core you have a bad installation.  The curb has heaved, but not the slab.  Fixing it is going to be expensive.  Unless there are signs of problems elsewhere, I'd be tempted to live with it.  Check the entire garage foundation to see if there are gaps opening anywhere else.

You can do a cosmetic repair.  Rent a jack hammer to take off the protruding bits and about 1/2 to 3/4" more, do surface prep, and use a concrete patching compound to make it flush again.  Finish by sealing the cracks with a combination of backer rod.  A grout seal will accommodate something like half its width in moving.  E.g., if you have 1/4" wide bead of grout, the two surfaces can shift something like 1/8" before the seal breaks.  So a wide seal will help you here.  Take this into account when you do the patch.  You may want so preserve space for this.  

Also, save a bunch of chips you removed from the prep.  You can use that on top of the ground to partially colour match the floor.