Can an estate file bankruptcy after the death of a widow. Or do the surving children have to pay the debt?

I do not believe an estate can file bankruptcy. What is an estate? "1) all that one owns in real estate and other assets. 2) commonly, all the possessions of one who has died and are subject to probate (administration supervised by the court) and distribution to heirs and beneficiaries", and creditors. You may want to speak with the judge of the probate court where the widow lived, or an attorney familiar with probate.