All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is obligatory for the wealthy children to provide the necessities of their poor parents but not the children who are in financial difficulty. Allaah says (what means): {Let a man of wealth spend from his wealth, and he whose provision is restricted – let him spend from what Allaah has given him. Allaah does not charge a soul except [according to] what He has given it. Allaah will bring about, after hardship, ease.} [Quran 65:7]
Therefore, if the younger brother doesn't have money to provide for his mother, he is not obliged to provide for her, as Allaah says (what means): {Allaah does not charge a soul except [according to] what He has given it.} [Quran 65:7]
When that child who was in financial difficulty becomes solvent, then the obligation to provide maintenance for his mother along with his two brothers will start from that time, with each providing according to his ability, and he does not have to be the sole provider from that time unless he wishes to do so, because her maintenance was not incumbent on him when he was not solvent. He does not owe his brothers any part of the money they provided for their mother when he was not solvent.
If the mother needs to be served, then this is an obligation on all her children each according to the ability of each. Al-Khaadimi (from the Hanafi School of jurisprudence) said in Bareeqah Mahmoodiyyah, discussing the rights of parents: "If they need service, then their children are obliged to serve them."
In Ghithaa’ Al-Albaab, As-Saffaareeni says: "It is their (parents') right that if they both – or one of them – need service, their children must serve them." [End of quote]
However, the wife of the son is not obliged to serve his mother in any case unless she does so by her own proper consent. For more benefit please refer to Fatwa 84909.
Finally, it should be noted that the children should compete to be kind and dutiful to their mother and try to please her rather than avoid doing so.