Rebel commander Eduard Basurin told reporters that separatist fighters would not observe the truce in the key railway town of Debaltseve, where thousands of Ukraine troops are believed encircled by rebel fighters.
In comments to Reuters, he called the contested town "our territory."
Basurin is quoted by Russia's Interfax news agency as saying the truce deal "stipulates the withdrawal of heavy weapons 48 hours after the guns fall silent. That's what we will be guided by," he said.
In a statement late Sunday, U.N. Secretary-General Ban Ki-moon welcomed the start of the cease-fire based on the terms agreed to in the Belarusian capital, Minsk.
But he voiced serious concern about reports of continued hostilities in areas, including Debaltseve, and reiterated his call for all parties to abide by the truce.
Elsewhere along the jagged frontline separating the two sides, witnesses and officials on both sides on Sunday reported relative calm.
The four leaders who brokered last week's truce -- Russian President Vladimir Putin, German Chancellor Angela Merkel, French President Francois Hollande and Ukrainian President Petro Poroshenko -- agreed in a conference call that fighting should end in Debaltseve as well.
The devastated town has seen major fighting on a daily basis since earlier truce efforts failed last month.
Under terms of the cease-fire reached in the Belarusian capital of Minsk, withdrawal of heavy armor from the frontlines is set to begin Monday.
Shelling blamed on Cossacks
The head of Ukraine's security service, Valentyn Nalyvaichenko said Russian Cossacks appeared to be responsible for Sunday's mortar shelling in the Luhansk region.
He said an investigation is under way.
"And the mortar attack were committed, this crime, by the pro-Russian cossacks, and the Russian citizens cossack, currently deployed in this particular place in Ukraine," Nalyvaichenko said.
Earlier, Poroshenko ordered his forces to comply with a cease-fire that went into effect at at midnight Saturday (12 a.m. Sunday local time, 2200 UTC Saturday) in hopes of ending intense fighting in the country's east between Ukrainian forces and pro-Russian separatists that has lasted 10 months and killed nearly 5,500 civilians and fighters.
In a live midnight broadcast, Poroshenko said that as the commander of Ukraine's military forces, he wants peace.
But he also warned Russian-backed separatist rebels against breaking the deal.
The Ukraine president warned that the rebels might use Debaltseve to "undermine the cease-fire."
The rebel leader of the city of Donetsk said his forces will not allow the Ukrainian troops to leave Debaltseve during the cease-fire.
If the cease-fire holds, both sides are to begin pulling back their heavy weaponry to form a wide buffer zone.
Obama, Poroshenko speak
Earlier Saturday, U.S. President Barack Obama spoke to Poroshenko by telephone and stressed the need for all sides to stop the violence as scheduled.
Obama also expressed concern over the intense fighting that was happening in hours before the cease-fire was to come into force.
He expressed the same concerns in a call to German Chancellor Angela Merkel.
U.S. Secretary of State John Kerry spoke by telephone with this Russian counterpart, Sergei Lavrov, on Saturday to underscore the importance of implementing the cease-fire.
The State Department said Kerry expressed concern about "efforts by Russia and the separatists to cut off Debaltseve" ahead of the cease-fire.
The U.N. Security Council is expected to meet Sunday and could vote on a Russian-drafted resolution calling on all parties to implement the deal.
Also Saturday, U.S. Ambassador to Ukraine Geoffrey Pyatt posted on Twitter what he said were satellite photos showing Russian artillery systems near the town of Lomuvatka, about 20 kilometers northeast of Debaltseve.
State Department Spokesperson Jen Psaki said the U.S. considers the images to be "one of several pieces of credible evidence that lead to the conclusion that the Russian military has deployed large amounts of artillery and multiple rocket launchers around Debaltseve."
She added the U.S. is confident the systems are Russian military, not separatist.
Russia has denied sending troops or weapons across the border to aid in the fighting.
Ukraine and a host of Western governments accuse Moscow of stoking the rebellion in Ukraine's Russian-speaking east with arms and fighters.
Moscow has repeatedly denied providing direct support to rebels, and claims that Russian troops seen fighting alongside rebels are volunteers.
Source: Voice of America