The rallies were sparked by President Viktor Yanukovych's decision last month to choose ties with Russia over integration with the 28-nation European Union.
That deeply angered many Ukrainians, who favor the democratic structures of the West over Russia's autocratic government.
After a violent police crackdown on a peaceful rally, the demonstrators turned against Yanukovych himself and have transformed Kiev into a giant protest encampment.
"We will create such a hell for the authorities that the ground will burn under their feet," said Oleh Tyahnybok, head of the opposition nationalist party Svoboda.
Yanukovych's concessions of releasing some jailed opposition activists and suspending several top officials over the crackdown have failed to end the protests.
After several attempts to clear the protesters by force drew strong condemnation from the West, he now appears set on waiting them out.
But Yanukovych's stance was strengthened this week by a major bailout package from Russia to help Ukraine fend off a possible default.
The aid includes a $15 billion pledge to buy Ukrainian government bonds and a sharp decrease in the price Russia charges Ukraine for natural gas.
The opposition, however, has dismissed the agreements with Russia as a sell-out and insisted that Ukraine's future lies with the European Union.
Opposition leader Vitali Klitschko urged the crowd to spend New Year's and the following weeks on Kiev's Independence Square to force Yanukovych into calling early presidential and parliamentary elections.
"They think that we will get tired, they think that we will go home," Klitschko said Sunday.
"This will never dissipate, because we understand that we need to implement reforms and the only way to implement reforms is by changing the leaders."
The movement is also trying to widen opposition support in the east of the country, which remains largely loyal to Yanukovych.
While Sunday's demonstration was smaller than on previous weekends, it was still a strong showing for a protest movement that has entered its second month in the frigid cold.
Kiev's main square, the Maidan, and an adjacent street were filled with protest tents, field kitchens and giant barricades made of sacks of snow, car tires and wooden planks.
"The Maidan has become a symbol of Ukraine's change," said Mykola Razdel, 35.
"Simple people want change and it is in our power to change everything."
Meanwhile, Maidan received another important message Sunday.
Mikhail Khodorkovsky, once Russia's richest men and the country's most famous prisoner until two days ago, expressed hope that Ukraine's most prominent prisoner, the former premier and key opposition leader Yulia Tymoshenko, would also soon be released from jail.
Tymoshenko, Yanukovych's top foe, is serving a seven-year sentence on charges of abuse of office which the West condemns as political.
"I hope that President Yanukovych, who frequently communicates with the president of my country will follow his example in this matter — the release of a political prisoner," Khodorkovsky told a news conference in Berlin.
Source: FOX News