I'm not sure how 'nice' the repair job would look. Yes, there are epoxy mixes for wood but it's not really meant for chewed and frayed edges. In this case, there really isn't anthing for the epoxy to 'grab' onto unless you do some 'intentional' manual hole drilling in the damaged area, or drive in some nails for the epoxy to grab on to.
YOu might want to consider a brass 'sleeve' that they make for kinda like 'kickplates'. It should cover MORE than you want but other than replacing the door (which I'd consider doing), I don't know what else to suggest. You could visit your local Home Center and talk to the folks there about your problem and the fix. As for my brass sleeve idea, you may have to go to a Door Store, or something.
Also, you didn't say what kind of door you had. WHen you're at the home center, check around for prices to replace the door. Oh, and do take measurements of the one that's chewed up to the store w/you. Perhaps others will have more to say.