FALLUJAH, Iraq -- Iraqi commanders are preparing to dislodge Islamic State group fighters from pockets of territory in Fallujah's northern and western neighborhoods where the militants have dug in after largely fleeing their positions in the city center last week.

Before Iraqi forces rolled into central Fallujah under cover of U.S.-led coalition airstrikes, they were bogged down for weeks, trying to push through deep defensive trenches, tunnels and houses converted into bunkers by IS militants on the city's southern edge. Now looking to the city's north, Iraqi commanders expect to encounter a similarly fierce fight.

"It's not going to be easy," Iraqi special forces Brig. Gen. Ali Jameel said of the upcoming battle for the last pockets of IS resistance, where an estimated 100 militants largely are surrounded.

"They are going to fight to the death because they have nowhere to run," he said.

Prime Minister Haider al-Abadi launched the offensive to retake Fallujah from the Islamic State group May 22.

The Sunni-led extremist group has held the city west of Baghdad for more than two years.

Last week, Iraqi forces raised their flag above a government complex in central Fallujah and declared victory, saying 80 percent of Fallujah was under their control.

It was some much-needed positive news for al-Abadi amid growing anti-government protests and civil unrest in Baghdad.

But nearly a week later, Iraq appears to have only a fragile hold on the territory its forces claim to have liberated.

The U.S.-led coalition, which has been conducting airstrikes in the offensive, said Tuesday only a third of Fallujah can be described as "cleared," while other territory remains contested.

One of the militants' remaining strongholds is Fallujah's Jolan neighborhood, the northwestern corner of the city that was the scene of some of the most persistent skirmishes between U.S. forces and insurgents in 2004 and 2007.

Its jumble of narrow streets and dense concentration of residential buildings is expected to make it harder to use airstrikes and Iraqi armor.

Additionally, thousands of civilians are believed to be trapped in the IS-held territory, according to the United Nations.