If the adhesive has simply failed, you can re-glue it with the type of adhesive used on car headliners. 3M Upholstery & Headliner Adhesive is probably the best known brand, but most auto parts stores carry at least one brand. It is a spray.
But if the problem is that the backing of the material itself is crumbling (it is sometimes a thin layer of foam or other "polyxxx" material), then it probably needs to be replaced.
What does the material look like? Ozite (a short nap, carpet-like fiber)? Vinyl/leatherette? Some sort of cloth?