How should I handle gap and erosion of garage slab?

I would dig out as much of the foam etc. as possible and then fill the gap with more concrete and/or motor. Honestly, concrete slowly just erodes and anything you do here won't last forever but I think you can support it better than what is already there.

Slabs can drop if the ground wasn't compacted or there is a water erosion problem due to sprinklers or downspouts not properly directed away from the area. There are companies like Slab Jack that can raise up a slab. We looked at doing that to old sidewalks that had problems, but it was expensive.

The gap can be filled with either concrete calk or a liquid concrete filler. Found in the concrete section of stores like Home Depot. There is also foam backer rod to push down into the gap before you fill with caulking of liquid. This acts like a bottom filler so the caulk or liquid doesn't keep seeping into the gap.

It looks like you have at least a 2" drop between garage floor and the driveway. I would make a ramp at least 6" wide to get a smooth transition into garage. It can be concrete or pressure treated wood.

I would first ask myself is where all this water is coming from? Are your gutters and downspouts efficent, and do the downspouts move the water far enough away from the building/slab. You can add a channel drain along the edge between the building and the slab to move the water away too.

I agree its going to be hard to fill underneath the edge of the slab, so I would have someone professional advice you on what is the best course.

If fixed incorrectly this could lead to more problems.

When a concrete slab cracks AND sinks, this indicates a more serious structural issue. In addition to looking bad, a sinking slab can create an uneven surface that poses a safety hazard. Instead of hoping the problem will go away or worrying about the potential for injuries and even lawsuits, it’s smart to fix a sunken slab.

Call someone that is experienced dealing with this. They not only would know how to handle this but having the right tools and materials to deal with this are important also.

You can get mortar in a caulking tube. It is great but has a slow setting time. It should work for that and is similar color.

You might call a company like Ram Jack. They do this for a living and could be helpful.

Many people will attempt to fill these voids with dirt, sand or gravel and while something is better than nothing none of these solutions are able to generate any upward pressure on the concrete so not much support is possible. Even worse is when homeowners pour concrete in those areas of erosion. Concrete is a very heavy material, in addition to being unable to generate any upward pressure on the slab the added burden on the soil often speeds settling.

The drop was intentional from the garage slab to the drive. it's to prevent rain water from entering the garage from the drive. Mine is similar. To fill any hollow areas under the drive I would use foam insulation like Great Stuff. You can add extra hose length to go in deep. It will move as it expands to fill any voids. Fill the gaps with foam backer rod then cover with Concrete Crack filler in a caulk tube. You want something that will flex with movement between the drive and garage slab. If the gaps are deep you can fill with fine gravel, then the foam rod and caulk.

I would just fill it in and use waterproofing metal sheets for addictional protection

Most of those gaps you’re showing are control joints which should not be filled. The cracks are due to settlement and can be repaired but best by a pro so it doesn’t keep happening. Everything looks correct and proper to me.