KYIV, Ukraine (AP) — Ukrainian fighters who were holed up in a massive steel plant in the last known pocket of resistance inside the shattered city of Mariupol ignored a surrender-or-die ultimatum from Russia on Sunday and held out against the capture of the strategically vital port.

The fall of Mariupol, the site of a merciless 7-week-old siege that has reduced much of the city to a smoking ruin, would be Moscow's biggest victory of the war and free up troops to take part in a potentially climactic battle for control of Ukraine’s industrial east.

Capturing the southern city would also allow Russia to fully secure a land corridor to the Crimean Peninsula, which it seized from Ukraine in 2014, and deprive Ukraine of a major port and its prized industrial assets.

As its missiles and rockets slammed into other parts of the country, Russia estimated that 2,500 Ukrainian troops and about 400 foreign mercenaries were dug in at the sprawling Azovstal steel mill, which covers more than 11 square kilometers (4 square miles) and is laced with tunnels.

Many Mariupol civilians, including children, are also sheltering at the Azovstal plant, Mikhail Vershinin, head of the city’s patrol police, told Mariupol television on Sunday. He said they are hiding from Russian shelling, and from any occupying Russian soldiers.

Moscow had given the defenders a midday deadline to surrender and "keep their lives,” but the Ukrainians rejected it, as they've done with previous ultimatums.

“We will fight absolutely to the end, to the win, in this war,” Ukrainian Prime Minister Denys Shmyhal vowed on ABC's “This Week.” He said Ukraine is prepared to end the war through diplomacy if possible, “but we do not have intention to surrender.”

As for besieged Mariupol, there appeared to be little hope Sunday of military rescue by Ukrainian forces anytime soon. Foreign Minister Dmytro Kuleba told CBS’ “Face the Nation” that the remaining Ukrainian troops and civilians in Mariupol are basically encircled. He said they “continue their struggle,” but that the city effectively doesn’t exist anymore because of massive destruction.

Ukrainian President Volodymyr Zelenskyy sent Easter greetings via Twitter, saying: "The Lord’s Resurrection is a testimony to the victory of life over death, good over evil.”

If Mariupol falls, Russian forces there are expected to join an all-out offensive in the coming days for control of the Donbas, the eastern industrial region that the Kremlin is bent on capturing after failing in its bid to take Kyiv, Ukraine's capital.

The relentless bombardment and street fighting in Mariupol have killed at least 21,000 people, by the Ukrainians' estimate. A maternity hospital was hit by a lethal Russian airstrike in the opening weeks of the war, and about 300 people were reported killed in the bombing of a theater where civilians were taking shelter.

An estimated 100,000 remained in the city out of a prewar population of 450,000, trapped without food, water, heat or electricity in a siege that has made Mariupol the scene of some of the worst suffering of the war.

"All those who will continue resistance will be destroyed,” Maj. Gen. Igor Konashenkov, the Russian Defense Ministry's spokesman, said in announcing the latest ultimatum.

Drone footage carried by the Russian news agency RIA-Novosti showed towering plumes of smoke over the steel complex, which sits on the outskirts of the bombed-out city, on the Sea of Azov.

Ukrainian Deputy Defense Minister Hanna Malyar described Mariupol as a “shield defending Ukraine” as Russian troops prepare for battle in the mostly Russian-speaking Donbas, where Moscow-backed separatists already control some territory.

Russian forces, meanwhile, carried out aerial attacks near Kyiv and elsewhere in an apparent effort to weaken Ukraine’s military capacity ahead of the anticipated assault.

After the humiliating sinking of the flagship of Russia's Black Sea Fleet last week in what the Ukrainians boasted was a missile attack, the Kremlin had vowed to step up strikes on the capital.

Russia said Sunday that it had attacked an ammunition plant near Kyiv overnight with precision-guided missiles, the third such strike in as many days.

Explosions were also reported overnight in Kramatorsk, the eastern city where rockets earlier this month killed at least 57 people at a train station crowded with civilians trying to evacuate ahead of the Russian offensive.

At least five people were killed by Russian shelling in Kharkiv, Ukraine’s second-largest city, on Sunday, regional officials said. The barrage slammed into apartment buildings and left the streets scattered with broken glass and other debris, including part of at least one rocket.

Kharkiv Mayor Igor Terekhov, in an impassioned address marking Orthodox Palm Sunday, lashed out at Russian forces for not letting up the bombing campaign on such a sacred day.

And Zelenskyy, in his nightly address to the nation, called the bombing in Kharkiv “nothing but deliberate terror.”

A regional official in eastern Ukraine said at least two people were killed when Russian forces fired at residential buildings in the town of Zolote, near the front line in the Donbas.

Zelenskyy said Russian troops in parts of southern Ukraine have been carrying out torture and kidnappings, and he called on the world to respond with more weapons and tougher sanctions.

“Torture chambers are built there,” he said in his address. “They abduct representatives of local governments and anyone deemed visible to local communities.”

Malyar, the Ukrainian deputy defense minister, said the Russians continued to hit Mariupol with airstrikes and could be getting ready for an amphibious landing to reinforce their ground troops.

The looming offensive in the east, if successful, would give Russian President Vladimir Putin a vital piece of the country and a badly needed victory that he could sell to the Russian people amid the war's mounting casualties and the economic hardship caused by the West's sanctions.

Austrian Chancellor Karl Nehammer, who met with Putin in Moscow this week — the first European leader to do so since the invasion Feb. 24 — said the Russian president is “in his own war logic” on Ukraine.

In an interview on NBC’s “Meet the Press,” Nehammer said he thinks Putin believes he is winning the war, and “we have to look in his eyes and we have to confront him with that, what we see in Ukraine.’’

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Chernov reported from Kharkiv. Yesica Fisch in Kramatorsk, Ukraine, and Associated Press journalists around the world contributed to this report.