Praise be to Allah.
The wife is not obliged to spend on her husband’s children, either whilst the husband is still alive or after his death. That is because one of the conditions of spending being obligatory is that the one who spends should be an heir of the one on whom he (or she) spends.
Ibn Qudaamah (may Allah have mercy on him) said:
For it to be obligatory to spend, three conditions are stipulated. And he mentioned the third of these conditions: that the one who spends should be an heir [of the one on whom he (or she) spends], because Allah, may He be exalted, says (interpretation of the meaning): “And on the (father’s) heir is incumbent the like of that (which was incumbent on the father)” [al-Baqarah 2:233]. Moreover, between the two who may inherit from one another there are ties of kinship which dictate that the one who would inherit is more deserving than anyone else of the wealth of the one from whom he would inherit, so it is appropriate that it should be obligatory to uphold ties with him by spending, to the exclusion of others. If he is not an heir because there are no ties of kinship between them, then it is not obligatory for him to spend on that person.
End quote from al-Mughni (8/171)
It is well-known that the father’s wife does not inherit from her husband’s children, therefore she is not obliged to spend on them.
If the father is poor and is not able to spend on his wife and his children, but he has children from another wife who are well off, then in this case the children from the other wife who are well off are obliged to spend on their father and on those for whom their father is responsible, such as his wife and children.
Ibn Qudaamah (may Allah have mercy on him) said: Everyone who is obliged to help him maintain his chastity is also obliged to spend on his wife, because his chastity cannot be maintained otherwise.
End quote from al-Mughni (8/174)
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) was asked about a man who was unable to earn a living and he had no wealth, and he had a wife and children; is it permissible for his son who is well off to spend on him and his wife and on his young siblings?
He (may Allah have mercy on him) replied: Praise be to Allah, the Lord of the Worlds. Yes, the son who is well off is obliged to spend on his father and his father’s wife, and on his young siblings, and if he does not do that he is regarded as being disobedient towards his father and severing family ties, and he is deserving of the punishment of Allah, may He be exalted, in this world and the hereafter. And Allah knows best.
End quote from Majmoo‘ al-Fataawa (34/101)
He (may Allah have mercy on him) was also asked about a man who has a son and has some wealth, and he has a family, but his wife is not the mother of the oldest son; is his son obliged to spend on his father and on his siblings and his father’s wife, or not?
If the father is unable to spend (on his family’s maintenance) and the son is able to spend on them, then he must spend on them.
End quote from Majmoo‘ al-Fataawa (34/102)
And Allah knows best.