Without knowing the type of plastic or the type of adhesive that was used on the sticker you bought, I could only provide some general information, which will hold true for your question as well.
The problem you are having, unfortunately, is not uncommon and should have been answered before you made your purchase. I always ask "what are you going to be using this sticker for?" This avoids a lot of unhappy clients who would have ordered a sticker that won't work for their desired application.
Most adhesives used on decals are known collectively as "acrylic" adhesives, and are generally used successfully on many various surfaces, including painted surfaces, metal, glass, and even wood. However, there are a few select materials that will reject standard acrylic adhesives. Polypropylene and Polyethylene, at least in some cases such as containers or barrels made from these plastics, often need a more aggressive adhesive in order to stay put once placed on these items.
Certain types of paint will also reject standard acrylic adhesives, as well as some oil-based stains. This may be what happened with your door. If it has a certain type of finish you're not sure of, you'd need to test out a small, similar sticker ahead of time on your targeted door. That way, if, after 3 days, it was sticking well to the door, you would know that the decal you were preparing to order would work for this application.
Testing ahead of time is always a good idea, unless you're pretty sure a sticker will stick using a basic acrylic adhesive. Places where this would be more consistent? Glass, glossy painted surfaces (excepting possibly certain latex paints or wood stains), and smooth, clean metal. Still, if there's a shadow of a doubt, it's always better to test your sticker's glue ahead of time.
How to Wash your Clothes with Stickers on
You remove the decals, unless they are patches that are specifically glued using special adhesives that are used specifically for clothing such as military or scout uniforms.
If your child received a paper or vinyl sticker at some event or sale, and stuck it to their shirt or jacket or such, and you wash it in place, the best case scenario will be the sticker will come off in the wash, and the worst case scenario is that the glue will stay stuck to the clothing, in which case it is difficult to remove. Lacquer thinner will remove the glue, but may also remove the color from the clothing as well, so be careful there. It is more likely that an orange peel glue remover, available at most hardware stores, would be able to remove just the adhesive, but as we always advise, try it on an area of the shirt that won't show just in case the glue remover wants to remove more than just glue.
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ABOUT THE AUTHOR
Barry K. Brown has been in the Sign, Banner, Decal and Display Business for over 20 years. It isn't what he thought he'd do with his life, but he says he knows too much now to do anything else!
He has been marketing these products online since 1998, and the company he was general manager of in 1998 was the first sign company to be listed on Yahoo!