The reasons a carved pumpkin decays or rots sooner than a uncarved pumpkin are:

Other organisms (fungi, bacteria, protozoans, insects, etc.) can now enter easily and steal the stored up high energy molecules of the pumpkin. This process is called rotting.

Oxygen can easily enter and cause oxidation.

Dehydration will begin when the pumpkin is carved. Low humidity.

Light can now enter as well as dust and other particles.

Tips to help prevent early decaying:

Rub vegetable oil or petroleum jelly onto freshly cut areas to delay aging. After carving your pumpkin, wipe the inside with bleach to deter the growth of mould. After 30 minutes, rub the cut areas and the inside with a dry cloth, then coat with petroleum jelly or vegetable oil. To keep the jack-o’-lantern from drying out, cover it with a damp towel when it’s not on display.

If you have a spare refrigerator, you can empty it out and keep them in it over night to slow down decomposition.

Another way to keep them fresh is to use a pumpkin dip that is available at most grocery stores or stands that sell pumpkins during the Halloween season. Pumpkin dip is mixed with water in a large container and the pumpkins are submerged in it, sometimes over night, in order to preserve them. This is a tedious and messy way to keep your pumpkins fresh as you have to do it every night after they have been carved.

If all else fails and you still find your carving starting to shrivel, submerse the pumpkin in cold water to rehydrate the carving. It should soak anywhere from about 1/2 an hour to 2 hours, depending the amount of shriveling that has occurred and the intricacy of the carving.

Realistically speaking, you should be able to get anywhere from 4 to 7 days out of your carved Jack O'Lantern.