If you replace it with concrete, it'd have to be done right otherwise the freeze and thaws cracking it will be an even bigger headache than cracked asphalt.
If you want to rip it out for fairly cheap, assuming you do the work yourself, turn the end section into an apron of pavers. It'd be annoying but straight-forward work.
I wonder how a gravel apron would look there for that 10 or so feet. Put brick edging on the sides, remove some of the asphalt so water drains better in the middle, then put nice colored gravel on top. That's sort of how they make tar-and-chip driveways anyway. The height difference between the two sections might be a concern.
If you're just trying to maintain it, and if water pooling is a problem in that area, maybe install a trench drain right through the middle of the low area.
Just throwing some ideas at you.