Answer: Some bromeliads are more temperamental than others, but there are several reasons why this might be happening. I suspect that the plant may be following its natural cycle of dying off after blooming. The surrounding baby plants or pups will then grow and replace it, or can be divided and replanted.
Also, make sure you are not under or overwatering it. The soil needs to drain quickly and should never be waterlogged. You might use rain water to fill the cup and water the roots as well. Depending on the ambient humidity, the type of material it is potted in, and the ambient temperature and air flow you might need to water once or twice a week.
The humidity level around it should be high -- both winter heating and air conditioning can dry the air in our homes so you might put it on a pebble tray. This is a shallow tray filled with pebbles and water. Set the pot on the pebbles and keep the water just below the bottom of the pot (it should not sit in the water).
Check the plant carefully for signs of mealy bugs or scale insects. Mealy bugs tend to be cottony looking, scale will look like flat raised disks stuck on the foliage. These can be removed with a cotton swab dipped in rubbing alcohol. Check for these on all sides of the foliage and as far into the crevices as you can. Also check for spider mites which leave fine webbing on the undersides of the foliage. These can be treated by washing the plant (especially the undersides of the leaves) with plain tepid water once a week and increasing the humidity level.
If none of these seem plausible, you might want to consult with your local county extension to see if they can give you a more specific diagnosis. I hope this helps.
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