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. We ask Allaah to exalt his mention as well as that of his family and all his companions.
It is permissible for a fasting person to get up in the morning while in a state of sexual defilement, however, it is not permissible for him to delay performing Ghusl (ritual bath) until 11:00 a.m. or until the time of Thuhr- as the questioner mentioned- because he is required to perform the Fajr prayer at its prescribed fixed time while in a state of purity.
Therefore, if one wakes up in the morning and deliberately delays performing Ghusl until the prescribed fixed time of Fajr is over, then he has committed one of the great major sins, and he is obliged to repent to Allaah and perform the Fajr prayer at its prescribed fixed time. For more benefit on the obligation of performing the prayer on time, please refer to Fatwa 88066, and refer to Fatwa 82538 on congregational prayers.
However, if one intends to fast and wants to delay Ghusl after the time of Fajr had started but without delaying the performance of the Fajr prayer from its prescribed fixed time, then this is permissible.
For more benefit on the validity of the fast of a person who wakes up in the morning while in a state of sexual defilement, please refer to Fataawa 87481, 85185 and 81302.
Finally, it should be noted that if the sexual intercourse occurred after the beginning of the time of Fajr, then the fast becomes void; for more benefit in this regard, please refer to Fataawa 84189 and 88972.
Allaah Knows best.