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The long-awaited assault on Fallujah was officially dubbed Operation Dawn, to signify the promise of a new beginning. But the name the U.S. military had originally given the operation—Phantom Fury—seems more appropriate for the kind of war U.S. forces are fighting. At times the soldiers and Marines trawling Fallujah's alleyways feel as though they are chasing ghosts. Insurgents vanish as the armored columns rumble into town, only to reappear somewhere else, firing from minarets and hiding in houses booby-trapped to blow up. U.S. and Iraqi officials say that their forces have killed as many as 1,000 enemy fighters and that most of the ravaged city is under U.S. control. If the goal, as a senior U.S. official says, is to "break up the scorpion's nest'' that Fallujah has become, the military is willing to inflict as much punishment as needed to achieve it.

But after a week that witnessed the most brutal up-close combat conducted by the U.S. military since Somalia, victory over the insurgency in Iraq isn't necessarily any closer. Many fighters and the majority of the rebel leadership—including Abu Mousab al-Zarqawi, the most wanted terrorist in Iraq—apparently slipped out of the city in the weeks leading up to the assault. A Pentagon official says that at most, 10% of the enemy in Iraq has been killed or captured in Fallujah. As the U.S. fights there, violence is rippling across the center and north of Iraq, engulfing the increasingly restive city of Mosul, the third largest in the country.

The violence has raised the prospect that the siege of Fallujah could be a prelude to a series of nasty urban street fights—precisely the sort of war the U.S. military had desperately hoped to avoid when the invasion started in the spring of 2003.

U.S. commanders acknowledge that Fallujah is only the beginning. But they hope that the show of force there is the first step toward gradually eroding the insurgents' ability to coordinate activities around the country. Senior U.S. officials say the coming months will be like playing a deadly game of "whack a mole" across the country: attacking insurgents wherever they rise up and trying to take back enough rebel-held areas to hold credible elections in January. The U.S. does not have enough soldiers in Iraq to crush a growing insurgency in multiple locations at the same time. But officials believe they won't actually face that challenge. As messy as the Sunni triangle and Mosul now appear, so long as the insurgency doesn't ignite a nationwide conflagration, the Pentagon believes it can contain the threat. "What we're trying to do in the short term, through the elections, is make sure that there are no no-go zones," says a senior Western diplomat in Baghdad. "To the extent possible, we [will] attrit their capability to launch violent attacks."

Critical to that plan is making sure Fallujah stays secure once the insurgents are routed. Toward that end, the Pentagon says money will start to flow into the city as soon as the military operation is over. The Pentagon says it has some $100 million ready to pour into a variety of civil works in Fallujah, including improvements in water, sewage and electrical systems as well as the construction of schools and health clinics. Army Lieut. General Thomas Metz, U.S. ground-forces commander in Iraq, says it will take "weeks, maybe months, to get the city to a normal operating level."

Once Fallujah is pacified, the U.S. plans to rely on the newly trained Iraqi police and national guard forces to perform the bulk of security tasks required to begin the delivery of reconstruction aid.

That transition won't be easy. Among ordinary citizens, there is almost no confidence that the Iraqis will be up to the task, and they are almost certain to face fresh attacks. "Let the Americans think they are winning," a fighter in Fallujah told TIME. "We are not going anywhere."

The whack-a-mole strategy may already be getting its first test in Mosul. The city is home to a heterogenous population of 1 million—Sunni, Kurd and Turkoman—and for months after the invasion was viewed as one of the occupation's few success stories. But locals warn that the city is slipping out of control. Foreign terrorists streaming across the border from Syria have joined forces with a Baathist resistance stocked with unemployed ex-soldiers. Insurgent attacks have grown significantly in number and lethality in recent months, and at least two or three assassination victims arrive each day at al-Salaam Hospital, the city's largest, doctors say. After insurgents staged attacks against six police stations in the city last week, a unit involved in the U.S. assault on Fallujah had to peel off and head to Mosul to help put down the unrest there. Local political leaders fear that the violence may make it impossible to organize elections in Mosul by January.

The risk for the U.S. is that, rather than make the Sunni triangle secure for democracy, the assault on Fallujah may instead inflame Sunnis and scatter insurgents across a wider area, which could scuttle hopes of broad Sunni participation in the voting. The Iraqi Islamic Party, the largest Sunni political party in Prime Minister Iyad Allawi's interim government withdrew last week, saying it could not abide the attack on Fallujah. Meanwhile, the influential Association of Muslim Scholars, a Sunni group, has called for a total boycott of the elections. The association's leader, Harith al-Dhari, told TIME he was "very close to calling for jihad" against the Americans and the Allawi government.

Yet even after the violence and inflammatory rhetoric of the past week, not all Iraqis are convinced the Sunnis will sit out the vote.

Sunni leaders are acutely aware that the majority Shi'ites—who make up 60% of Iraq's population—seem united in their desire for elections. Optimistic U.S. and Iraqi officials believe that as elections draw near, at least some Sunni leaders will recognize their interest in having a say in Iraq's first elected government. As Sarmad Mohammad, a Sunni fruit vendor in Baghdad, says, "If there are no Sunni leaders in the new government, all the jobs in the government, police and army will go to Shi'as and Kurds."

However tumultuous the January elections prove to be, it's clear that the ultimate outcome in Iraq—whether it moves toward a semblance of stability or civil war—comes down to a test of wills. The U.S. command believes that the supply of suicidal Baathists, Islamic holy warriors and Iraqi nationalists will eventually exhaust itself.

Robert Scales, a retired Army major general, says history teaches that violent attacks on insurgencies such as the campaign mounted by the U.S. in Fallujah can work. "You don't just keep growing insurgents," Scales says. "By effectively eliminating the hard-core terrorists, the fellow travelers see the handwriting on the wall.

While the insurgency doesn't disappear, it tends to collapse to something down around noise level." But if Fallujah is a sign of things to come, the volume is likely to get cranked up first.

• BILL POWELL