Will the prayer of one who is cursed and speaks obscenities on websites be answered, if he prays against the one who cursed him?

One should not supplicate against the perpetrator of a sin; rather, one should supplicate for them with advice, remembrance, and instruction. There has been disagreement regarding the permissibility of supplicating against them, and the preponderant view is the prohibition of doing so, due to the Prophet's (peace and blessings be upon him) prohibition of supplicating against the wine-drinker. An example of supplicating for them is what happened with Abu Hurairah's mother. As for the acceptance of a person's supplication against you, there is no definitive certainty about it, but one fears the acceptance of the supplication of the oppressed, even if they are a disbeliever or a wicked person.