As Mariupol's defenders held out Monday against Russian demands that they surrender, the number of bodies in the rubble of the bombarded and encircled Ukrainian city remained shrouded in uncertainty, the full extent of the horror not yet known.

With communications crippled, movement restricted and many residents in hiding, the fate of those inside an art school flattened on Sunday and a theater that was blown apart four days earlier was unclear.

More than 1,300 people were believed to be sheltering in the theater, and 400 were estimated to have been in the art school.

Perched on the Sea of Azov, Mariupol has been a key target that has been relentlessly pounded for more than three weeks and has seen some of the worst suffering of the war. The fall of the southern port city would help Russia establish a land bridge to Crimea, seized from Ukraine in 2014.

But no clear picture emerged of how close its capture might be.

"Nobody can tell from the outside if it really is on the verge of being taken," said Keir Giles, a Russia expert at the British think tank Chatham House.

Over the weekend, Moscow had offered safe passage out of Mariupol - one corridor leading east to Russia, another going west to other parts of Ukraine - in return for the city's surrender before daybreak Monday. Ukraine flatly rejected the offer well before the deadline.

Mariupol officials said on March 15 that at least 2,300 people had died in the siege, with some buried in mass graves. There has been no official estimate since then, but the number is feared to be far higher after six more days of bombardment.

For those who remain, conditions have become brutal. The assault has cut off Mariupol's electricity, water and food supplies and severed communication with the outside world, plunging residents into a fight for survival. Fresh commercial satellite images showed smoke rising from buildings newly hit by Russian artillery.

"What's happening in Mariupol is a massive war crime," European Union foreign policy chief Josep Borrell said.

Mariupol had a prewar population of about 430,000. Around a quarter were believed to have left in the opening days of the war, and tens of thousands escaped over the past week by way of a humanitarian corridor. Other attempts have been thwarted by the fighting.

Those who have made it out of Mariupol told of a devastated city.

"There are no buildings there anymore," said 77-year-old Maria Fiodorova, who crossed the border to Poland on Monday after five days of travel.

Olga Nikitina, who fled Mariupol for the western Ukrainian city of Lviv, where she arrived Sunday, said gunfire blew out her windows, and her apartment dropped below freezing.

"Battles took place over every street. Every house became a target," she said.

A long line of vehicles lined a road in Bezimenne, Ukraine, as Mariupol residents sought shelter at a temporary camp set up by Russian-backed separatists in the Donetsk region. An estimated 5,000 people from Mariupol have taken refuge in the camp. Many arrived in cars with signs that said "children" in Russian.

A woman who gave her name as Yulia said she and her family sought shelter in Bezimenne after a bombing destroyed six houses behind her home.

"That's why we got in the car, at our own risk, and left in 15 minutes because everything is destroyed there, dead bodies are lying around," she said. "They don't let us pass through everywhere - there are shootings."

Francesco Rocca, president of the International Federation of Red Cross and Red Crescent Societies, urged Russia to abide by the Geneva Convention and allow humanitarian aid into the city.

In all, more than 8,000 people escaped to safer areas Monday through humanitarian corridors, including about 3,000 from Mariupol, Deputy Prime Minister Iryna Vereshchuk said.

Russian shelling of a corridor wounded four children on a route leading out of Mariupol, Ukrainian President Volodymyr Zelenskyy said.

As Russia intensifies its effort to pound Mariupol into submission, its ground offensive in other parts of the country has become bogged down, slowed by lethal hit-and-run attacks by the Ukrainians. Western officials and analysts say the conflict is turning into a grinding war of attrition, with Russian President Vladimir Putin's forces using air power and artillery to pulverize cities from a distance.

A senior U.S. defense official, speaking on condition of anonymity to discuss the military's assessment, said Russia had increased air sorties over the past two days, carrying out as many as 300 in the past 24 hours, and has fired more than 1,100 missiles into Ukraine since the invasion began.

In a video address Monday night, Zelenskyy hailed those who have fought back against Russia.

"There is no need to organize resistance," he said. "Resistance for Ukrainians is part of their soul."

In the Russian-occupied southern city of Kherson on Monday, Russian forces shot into the air and fired stun grenades at protestors who were chanting "Go home!" Kherson early this month became the first major city to fall to Russia's offensive.

In the capital, Kyiv, a shopping center in the densely populated Podil district near the city center was a smoking ruin after being hit late Sunday by shelling that killed eight people, according to emergency officials. The attack shattered every window in a neighboring high-rise.

Russian military spokesman Maj. Gen. Igor Konashenkov charged that Ukrainian forces had been using the shopping mall to store rockets and reload launchers. That claim could not be independently verified.

Britain's defense ministry said Ukrainian resistance has kept the bulk of Moscow's forces more than 25 kilometers (15 miles) from the center of Kyiv, but the capital "remains Russia's primary military objective."

Amid the continuing shelling, Kyiv Mayor Vitali Klitschko announced a curfew extending from Monday evening through Wednesday morning.

Ukrainian authorities also said Russia shelled a chemical plant outside the eastern city of Sumy, sending toxic ammonia leaking from a 50-ton tank, and hit a military training base in the Rivne region of western Ukraine with cruise missiles.

In the Black Sea port city of Odesa, authorities said Russian forces damaged civilian houses in a strike Monday. The city council said no one was killed.

Russia's invasion has driven nearly 3.5 million people from Ukraine, according to the United Nations. The U.N. has confirmed over 900 civilian deaths but said the real toll is probably much higher. Estimates of Russian deaths vary, but even conservative figures are in the low thousands.

Talks between Russia and Ukraine have continued by video but failed to bridge the chasm between the two sides. The Kremlin has demanded that Ukraine disarm and declare itself neutral. Zelenskyy told Ukrainian television late Monday that he would be prepared to consider waiving any NATO bid by Ukraine in exchange for a cease-fire, the withdrawal of Russian troops and a guarantee of Ukraine's security.

Zelenskyy also suggested Kyiv would be open to future discussions on the status of Crimea, which Russia seized in 2014, and the regions of the eastern Donbas region held by Russian-backed separatists. But he said that was a topic for another time, after a cease-fire and steps toward security guarantees.

Russia's Foreign Ministry warned that relations with the U.S. are "on the verge of a breach," citing "unacceptable statements" by U.S. President Joe Biden about Putin. Biden last week branded the Russian leader a war criminal.

In another worrying development, Ukraine's nuclear regulatory agency said radiation monitors around the decommissioned Chernobyl power plant, the site in 1986 of the world's worst nuclear meltdown, have stopped working.

The agency said that problem, and a lack of firefighters to protect the area's radiation-tainted forests as the weather warms, could mean a "significant deterioration" in the ability to control the spread of radiation in Ukraine and beyond.

From The Associated Press