CAIRO — Rebels pouring in from the countryside strengthened their grip on much of the Libyan capital, setting up checkpoints and securing buildings even as longtime leader Moammar Gadhafi evaded their grasp and pockets of his loyalists continued to put up fierce resistance.

Heavy fighting rumbled around Gadhafi’s Bab Azizia compound in southern Tripoli, and throughout the capital rebels said they had sustained heavy losses. The opposition forces claimed control of about 80 percent of the city, which was at once jubilant with waving flags and precarious with the rattle of gunfire.

Attention quickly focused on how the Transitional National Council, the rebels’ governing body, would impose order on a fractious tribal nation battered by six months of revolt. The rebels are prone to divisions and Western officials worry that power struggles and the desire for revenge may threaten stability much as they did after the fall of Saddam Hussein in Iraq.

But the rebels’ sense of momentum suffered a blow early Tuesday when Gadhafi’s son and onetime heir apparent, Seif Islam, appeared at a Tripoli hotel that remained in the hands of Gadhafi’s forces. The rebels had claimed Sunday that they had captured him. Another son, Mohammed, was reported to have escaped rebel custody.

Gadhafi’s tanks and snipers took strategic positions in several Tripoli neighborhoods. Still, it was uncertain if the mercurial man who referred to himself as Brother Leader and ruled the nation for nearly 42 years could muster a potent counterattack.

NATO indicated that it would continue its airstrikes against the Libyan leader’s army. That poses a high risk of civilian casualties in Tripoli, a dense urban landscape with more than 1.6 million people. But the pressing question for rebel leaders and NATO commanders was Gadhafi’s whereabouts. Unlike on previous days, the Libyan leader released no new audio broadcasts.