Is a husband Islamically obligated to provide for his wife's mother, who is financially capable, given that she has financially capable sons who refuse to provide for her?

Legally, one is not obligated to spend on the wife's mother, but it is an act of kindness for which one will be rewarded. Her able-bodied children are obligated to spend on her, and neglecting this obligatory spending is a great sin, especially concerning parents. Allah has enjoined kindness to them after worshipping Him. So advise her children to treat her kindly and spend on her to the best of their ability.