MUKACHEVO, Ukraine — Russia has sent nearly all its assembled combat power into Ukraine and on Thursday unleashed some of the most intense fighting since the invasion began, with local officials pleading for help as ground troops seized or encircled strategically important southern cities.

Russian President Vladimir Putin said Moscow’s mission was “going according to plan and in full compliance with the timetable,” despite widespread agreement among Western military analysts that the invasion had been slowed by unexpectedly fierce Ukrainian resistance.

Amid an ongoing mass exodus of people across the country, Ukraine and Russia said they had agreed to temporary local cease-fires to create “humanitarian corridors” so civilians can be evacuated and food and medicine can be delivered. But the cease-fires would not apply everywhere, Ukrainian presidential adviser Mykhailo Podolyak said – and logistical details remained unclear.

Kherson, among the first Ukrainian cities to be encroached upon by Russian forces, was running out of medicines and is facing disaster within days if a humanitarian corridor isn’t established, according to the secretary of the city council, Galina Luhova.

“People are in a panic, people are tense, people are frightened just to the core of their souls,” she said.

Reports from other cities in Ukraine’s south told a story of increasing desperation as communications and transport routes were cut off and supplies dwindled.

The mayor of Mariupol said a Russian siege and hours of shelling that battered rail links and bridges had cut off water, power and food supplies. The mayor of Odessa, a major port on the Black Sea, said the population was preparing to mount a defense amid unverified reports that a large fleet of Russian warships was heading toward the waters off the coast.

“Who are you saving us from,” Mayor Gennady Trukhanov said in a video posted online, using an expletive as he addressed Russian leaders who have said the purpose of the invasion is to save ethnic Russians and Russian-language communities in Ukraine.

“Our job today is to act so that every meter of our Odessa land is under control of us, Odessians,” Trukhanov told reporters, according to a separate video posted to his Facebook page.

Ukrainian President Volodymyr Zelensky said in an emotional video posted on Thursday said that 16,000 foreigners had volunteered to fight for Ukraine against the Russian invasion in what he referred to as an “international legion” of foreign volunteers.

Earlier this week, Ukraine temporarily lifted visa requirements for foreign volunteers who wish to enter the country and join the fight against Russian forces. In Washington, two men arrested with firearms near the Ukrainian embassy told police they had driven from Indiana to volunteer for battle, law enforcement officials said.

A senior U.S. defense official told reporters that 90 percent of the combat power Russia had assembled outside Ukraine was now within the country – up from 80 percent on Tuesday. More than 150,000 troops, along with tanks, aircraft, personnel carriers and missile launchers had been massed on the border in recent weeks under the pretext of training exercises.

The capital city, Kyiv, and Ukraine’s second-largest city, Kharkiv – as well as the northern city of Chernihiv – have come under fierce air, missile and artillery attack but remain in Ukrainian hands, according to local officials and assessments from Britain, which defense analysts say has substantial intelligence capabilities in the region.

A large explosion lit up the night sky in Kyiv in the early hours of Thursday, local time, according to video footage verified by The Washington Post. The blast, which was captured by a camera in the city’s southeastern neighborhoods, apparently took place in a region far west of the city center. Between late Wednesday night and Thursday morning, air raid sirens sounded at least six times, spurring residents to take shelter, according to messages in the official municipal government Telegram channel.

The Pentagon said Russia had fired nearly 500 missiles of all sizes since last week, including in population centers. A majority were launched from inside Ukraine, often with mobile launchers, but about 160 have come from inside Russia and 70 from Belarus, a U.S. defense official said, speaking on the condition of anonymity to discuss the Pentagon’s current assessments. Another 10 were launched from the Black Sea.

“They’ve showed a willingness to hit civilian infrastructure on purpose,” the official said, adding that the strikes also appeared to deliberately target government and media operations.

The official said the Pentagon could not verify reports of the use of cluster munitions, which often fail to explode initially but can detonate if disturbed on the ground, extending their harm to civilians long after the missile delivering them is fired. Amnesty International said it has assembled evidence that cluster munitions were fired indiscriminately in Kharkiv, killing and injuring civilians.

In a sign of how the conflict has heightened regional tensions, the U.S. and Russian militaries opened a special line to communicate with one another, the Pentagon official told The Post on Thursday. The United States does not have troops in Ukraine, but the line will be used to ensure no miscalculations or accidents occur outside Ukraine’s borders, where U.S. troops are positioned. Such lines have been used in Syria to avoid instances such as fighter jets using the same airspace. The opening of the communication line was first reported by NBC News.

The White House on Thursday afternoon announced new sanctions on dozens of new Russian oligarchs and elites in the latest round of U.S. sanctions designed to cripple the Russian economy in retaliation for the invasion. President Biden said the sanctions effort has had “a profound impact already.” The sanctioned individuals include Putin’s press secretary Dmitry Peskov, as well as 19 oligarchs and 47 associates and family members.

The Department of Homeland Security announced on Thursday that it will grant entry to tens of thousands of Ukrainians under protected status, allowing them to live and work in the United States for 8 months without fear of deportation, it said on Thursday.

Britain’s Ministry of Defense said Thursday morning that the massive Russian military convoy that has been poised north of the Ukrainian capital remains more than 18 miles away from Kyiv’s center. The 40-mile line of armored vehicles, tanks and towed artillery had drawn within 20 miles of the city center on Monday, satellite images captured by U.S. firm Maxar Technologies showed. But since then, it has made little progress, the British said, with delays stemming in part from “Ukrainian resistance, mechanical breakdown and congestions.”

A Western intelligence official, speaking on the condition of anonymity to discuss sensitive assessments, said the slowdown may also be strategic. “It could also be that they’re waiting to mask the proper numbers and types of forces around the city itself. This is the logistics train that will support them, so maybe it’s more of a pause, so that the siege can be laid properly, more so than it is a gross reflection of some mismanagement,” the official said.

“There’s plenty of mismanagement,” the official added. “But I think we shouldn’t over or underestimate many things at this point.”

Putin and French President Emmanuel Macron held a call on Thursday that did not deliver any major diplomatic breakthroughs and left Macron convinced that “the worst is yet to come,” according to a senior French official. In the terse exchange initiated by Moscow, Macron “called on Vladimir Putin not to lie to himself,” the official said.

The Kremlin’s state-controlled news service cited Putin as having told Macron that Russia’s mission in Ukraine “will be fulfilled in any case.”

The latest casualty estimates provided by the Russian government were from Wednesday, when officials said 498 service members have died in the Ukraine war and 1,597 have been wounded. Defense Ministry spokesman Igor Konashenkov said 572 service members have been captured.

There was no way to verify the toll, and the Western intelligence official said, “It’s fair to say that’s almost certainly a very large undercount.”

Kherson, a city of roughly 290,000 people, is currently inundated with Russian equipment and soldiers, but Moscow’s forces have yet to take control of the city, said Luhova, the city council secretary. Large portions of the city have been cut off from water and electricity, and residents have begun running out of food, she said.

According to eyewitness accounts, Kherson was one of the first cities to fall to Moscow’s forces, though Ukraine’s Defense Ministry also has not confirmed its capture.

Officials in Mariupol said Thursday that the southern coastal city was still under Ukrainian control but was completely surrounded by Russian troops.

“They impede the supply of food, create a blockade for us,” Mayor Vadym Boychenko, said in a Telegram post from the city council. He said people could not leave the city because of the damage to trains and bridges, but his team was trying to secure a safe corridor to bring in supplies or help evacuate residents.

The Pentagon has pointed to Russian operations in southern Ukraine as their most successful, in part because of logistical support Russia has established in Crimea, a former Ukrainian territory Russia occupied and annexed in 2014.

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