Ukrainians said on Wednesday (local time) they were battling on in the port of Kherson, the first sizeable city Russia claimed to have seized, as air strikes and bombardment caused devastation in cities that Moscow's bogged down forces have failed to capture.

After nearly a week, Russia has yet to achieve its aim of overthrowing Ukraine's government, but has, according to the Ukrainian emergency service, killed more than 2,000 civilians and destroyed hospitals, kindergartens and homes.

The invasion has sent more than 870,000 people fleeing over Ukraine's borders and retaliatory sanctions have shaken the world economy, with surging oil prices exacerbating fears of inflation. 

Bombing of Kharkiv, an eastern city of 1.5 million people, has left its centre a wasteland of ruined buildings and debris.

"The Russian 'liberators' have come," one Ukrainian volunteer lamented sarcastically, as he and three others strained to carry the dead body of a man wrapped in a bedsheet out of the ruins on a main square.

After an air strike on Wednesday morning, the roof of a police building in central Kharkiv collapsed as it was engulfed in flames. Authorities said 21 people were killed by shelling and air strikes in the city in the past 24 hours, and four more on Wednesday morning.

Moscow denies targeting civilians and says it aims to disarm Ukraine, a country of 44 million people, in a "special military operation".

Apple, Exxon, Boeing and other firms joined an exodus of international companies from Russian markets that has left Moscow financially and diplomatically isolated since President Vladimir Putin ordered the Feb. 24 invasion.

"He thought he could roll into Ukraine and the world would roll over. Instead, he met a wall of strength he could never have anticipated or imagined: he met Ukrainian people," U.S. President Joe Biden said on Tuesday in his annual State of the Union address.

U.S. lawmakers stood, applauded and roared, many waving Ukrainian flags and wearing its blue and yellow colours. Russia said it had sent delegates for a second round of peace talks in Belarus. Ukrainian President Volodymyr Zelenskiy said Russia must stop bombing if it wanted to negotiate.

Moscow said on Wednesday it had captured Kherson, a southern provincial capital of around 250,000 people strategically placed where the Dnipro River flows into the Black Sea.

Zelenskiy advisor Oleksiy Arestovych denied Kherson was fully under Russian control, saying: "The city has not fallen, our side continues to defend."

Also in the south, Russia was bombarding the port of Mariupol, which it says it has surrounded in a ring around the entire Sea of Azov. The besieged city's mayor said Mariupol had suffered mass casualties after a night of intense strikes. He gave no full casualty figure, but said it was impossible to evacuate the wounded and that water supplies were cut.

"The enemy occupying forces of the Russian Federation have done everything to block the exit of civilians from the city of half a million people," mayor Vadym Boichenko said in a live broadcast on Ukrainian TV.

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On the other two main fronts in the east and north, Russia so far has little to show for its advance, with Ukraine's two biggest cities, Kyiv and Kharkiv, holding out in the face of increasingly intense bombardment.

"We are going to see ... his brutality increase," British Defence Secretary Ben Wallace said of Putin in a radio interview. "He doesn't get his way, he surrounds cities, he ruthlessly bombards them at night ... and he will then eventually try and break them and move into the cities."

'ORDERS TO ERASE US'

In Kyiv, the capital of 3 million people where residents have been sheltering at night in the underground metro, Russia blasted the main television tower near a Holocaust memorial on Tuesday, killing bystanders.

Zelenskiy, in his latest update to his nation, said that attack proved the Russians "don't know a thing about Kyiv, about our history. But they all have orders to erase our history, erase our country, erase us all."

Earlier, a tired and unshaven Zelenskiy, wearing green battle fatigues in a heavily guarded government compound, told Reuters and CNN in an interview that the bombing must stop for talks to end the war.

"It's necessary to at least stop bombing people, just stop the bombing and then sit down at the negotiating table."

Russia's main advance on the capital - a huge armoured column stretched for miles along the road to Kyiv - has been largely frozen in place for days, Western governments say. A senior U.S. defense official on Tuesday cited problems including shortages of food and fuel, and signs of flagging morale among Russian troops. 

"While Russian forces have reportedly moved into the centre of Kherson in the south, overall gains across axes have been limited in the past 24 hours," Britain's ministry of defence said in an intelligence update on Wednesday morning.

"This is probably due to a combination of ongoing logistical difficulties and strong Ukrainian resistance."

It said Russia was carrying out intensive air and artillery strikes, especially on Kharkiv, Kyiv, Mariupol and the eastern city of Chernihiv.

The Kremlin's decision to launch war - after months of denying such plans - has shocked Russians accustomed to viewing Putin, their ruler of 22 years, as a methodical strategist. have been forced to queue at banks to salvage their savings, an echo of the post-Soviet economic collapse of the 1990s.

Ukraine said more than 1,000 volunteers from 16 countries were on their way to fight alongside Ukrainian forces, and that it would free any Russian prisoners whose mothers come to collect them.

Moscow has given no full account of its losses so far, but Ukraine says it has killed nearly 6,000 Russian troops and captured hundreds more. Pictures online have shown burnt-out columns of Russian tanks surrounded by corpses.

Russia has largely eliminated domestic opposition, with Putin's main critics jailed or forced into exile. Leading opposition figure Alexey Navalny said from prison that Russians should protest daily against the war, a spokesperson tweeted.