Weather stripping comes in two forms, with adhesive and without. Most is with adhesive already applied type and the adhesive is actually transfer tape (double sided fabric tape, one side stuck to the weather stripping, the other side to be stuck to whatever it needs to be applied to). There are 100s of types of adhesives and transfer tape fabrics used by 3M alone and they are not the only manufacture. Most of the adhesive are a pressure sensitive acrylic base. You can reactivate the tackiness with heat (a heat gun, but be careful not to melt) or a wipe with xylene or MEK. If none of those work, quality contact cement and or spray adhesive (the key with both of these is you want to coat both surfaces and let it 'dry' a little, than apply pressure to secure).
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