Problem is caused by getting wet, drying out and repeating, which causes the edge to curl. I was able to fix a few small spots by using a small tube of vinyl cement, bought at a marine store. It is clear in color and can also be used to patch small holes or tears.
On one larger area, whereas I didn't want any water intrusion to get below, I put a length of rubber 'door threshold' piece to seal it off, well gooped/sealed with LifeCaulk by BoatLife (comparable to 3M 5200, but less expen$ive to buy and the tube lasts 2-3X longer).