Heavy fighting has broken out between Somali government forces and al-Shabab rebels in the capital of Mogadishu, as Kenyan forces continue to advance across southern Somalia in an effort to hunt down members of the Somali insurgent group.

Officials and witnesses said African Union-backed Somali forces launched an assault Thursday in Mogadishu's Daynile district, one of the few remaining pockets of the Somali capital still controlled by the al-Qaida-linked militants. Somali officials say they are now in control of most parts of the district.

Al-Shabab abandoned most of the city in August, but has continued to carry out guerilla-style attacks against Somali citizens and government forces.

Meanwhile, a Kenyan army spokesperson said Wednesday that Kenyan forces have killed 73 al-Shabab fighters since launching an incursion into neighboring Somalia several days ago.

Kenyan forces are reported to be in the Gedo and Lower Jubba regions of southern Somalia, as part of what the government calls a “pursuit” operation against al-Shabab. Kenya accuses the militants of kidnapping several foreigners on Kenyan territory — an allegation al-Shabab has denied.

France's foreign ministry said Wednesday that one of the kidnapping victims, French national Marie Dedieu, has died.

Officials say Dedieu probably died because she was not given her required medication.

Dedieu was in her mid-60's and used a wheelchair, which the kidnappers did not take with them.

Unidentified gunmen kidnapped Dedieu from her private beach home in northern Kenya on October 1.

Al-Shabab has denied responsibility for the abductions, and has called on Somalis to defend against what it says are Kenyan “aggressors” and “occupiers.”

Al-Shabab is fighting to topple the Somali government and set up an strictly Islamic state. The militants have threatened to attack targets inside Kenya unless the Kenyan government withdraws its troops from Somalia.