Russian forces have been focused for weeks now on seizing Sievierodonetsk.
Ukrainian troops holding out in the ruins of Sievierodonetsk came under renewed heavy assault on Wednesday from Russian forces who see the capture of the industrial city as key to control of the surrounding Luhansk region.
In southern Ukraine, another major battleground in the war, authorities warned that Russian attacks on agricultural sites including warehouses were compounding a global food crisis that has stirred concerns of famine in some developing countries.
Russian forces have been focused for weeks now on seizing Sievierodonetsk, which was home to some 106,000 people before Moscow's full-scale invasion of Ukraine on 24 February.
But the Luhansk region's governor said Ukrainian forces would not surrender the city.
"Fighting is still raging and no one is going to give up the city, even if our military has to step back to stronger positions. This will not mean someone is giving up the city - no one will give up anything. But (they) may be forced to pull back," Serhiy Gaidai told Ukrainian television.
Russian forces will further increase their shelling and bombardment of both Sievierodonetsk and its smaller twin city of Lysychansk on the west bank of the Siverskyi Donets River, he said.
Luhansk and the adjacent province of Donetsk form the Donbas, claimed by Moscow for Russian-speaking separatists who have held eastern parts of the region since 2014.
"The absolutely heroic defence of Donbas is ongoing," Ukrainian President Volodymyr Zelenskiy said in a video statement on Tuesday.
"It is absolutely felt that the occupiers didn't believe the resistance of our military would be so strong and now they are trying to bring in new resources towards the Donbas," he said.
Reuters could not independently verify the situation on the ground.
Moscow says it is engaged in a "special military operation" to disarm and "denazify" its neighbour.
Ukraine and its allies call this a baseless pretext for a war that has killed thousands, flattened cities and forced millions of people to flee.
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