KYIV, Ukraine (AP) — Ukrainian forces said they retook a strategically important suburb of Kyiv early Tuesday, as Russian forces squeezed other areas near the capital and their attack on the embattled southern port of Mariupol raged unabated.

Explosions and bursts of gunfire shook Kyiv, and black smoke rose from a spot in the north. Intensified artillery fire could be heard from the northwest, where Russia has sought to encircle and capture several suburban areas of the capital, a crucial target.

Residents sheltered at home or underground under a 35-hour curfew imposed by city authorities that runs to Wednesday morning.

Russian forces also carried on with their siege of Mariupol after the southern port city's defenders refused demands to surrender, with fleeing civilians describing relentless bombardments and corpses lying in the streets. But the Kremlin’s ground offensive in other parts of the country advanced slowly or not at all, knocked back by lethal hit-and-run attacks by the Ukrainians.

Early Tuesday, Ukrainian troops forced Russian forces out of the Kyiv suburb of Makariv after a fierce battle, Ukraine's Defense Ministry said. The regained territory allowed Ukrainian forces to retake control of a key highway and block Russian troops from surrounding Kyiv from the northwest.

Still, the Defense Ministry said Russian forces battling toward Kyiv were able to partially take other northwest suburbs, Bucha, Hostomel and Irpin, some of which had been under attack almost since Russia's military invaded almost a month ago.

Russian President Vladimir Putin’s forces are increasingly concentrating their air power and artillery on Ukraine’s cities and the civilians living there. Moscow's invasion has driven nearly 3.5 million people from Ukraine, according to the United Nations, with another 6.5 million displaced inside the country. The U.N. has confirmed over 900 civilian deaths while saying the real toll is probably much higher. Estimates of Russian deaths vary, but even conservative figures are in the low thousands.

U.S. and British officials say Kyiv remains Russia's primary objective. The bulk of Moscow’s forces remain miles from the center, but missiles and artillery have destroyed apartment buildings and a large shopping mall, which was left a smoking ruin after being hit late Sunday by strikes that killed eight people, according to emergency officials.

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.

U.S. President Joe Biden, who is heading to Europe later in the week to meet with allies, suggested Monday evening that worse may be still to come.

“Putin’s back is against the wall,” Biden said. “He wasn’t anticipating the extent or the strength of our unity. And the more his back is against the wall, the greater the severity of the tactics he may employ."

Biden reiterated accusations that Putin is considering resorting to using chemical weapons.

As Russian forces try to squeeze Kyiv, talks to end the fighting have continued by video but failed to bridge the chasm between the two sides. Ukrainian President Volodymyr Zelenskyy told Ukrainian television late Monday that he would be prepared to consider waiving any NATO bid by Ukraine — a key Russian demand — 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. Zelenskyy plans to speak to Italy's lawmakers Tuesday and Japanese lawmakers on Wednesday, part of a series of addresses to foreign legislatures as he seeks to drum up support.

In Mariupol, 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 is crucial port for Ukraine and lies along a stretch of territory between Russia and Crimea. As such, it is a key target that has been besieged for more than three weeks and has seen some of the worst suffering of the war.

It is not clear how close its capture might be. Ukraine's Defense Ministry said Tuesday that their forces were still defending the city and had destroyed a Russian patrol boat and electronic warfare complex.

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 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 the humanitarian corridors. Other attempts have been thwarted by the fighting.

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.

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 stood on a road in Bezimenne, east of Mariupol, as residents of the besieged city 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.”

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, Zelenskyy said.

Matthew Saltmarsh, a spokesman for the U.N. refugee agency called the speed and scale of people fleeing danger in Ukraine "unprecedented in recent memory.”

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Anna reported from Lviv, Ukraine. Associated Press writer Yuras Karmanau in Lviv, and other AP journalists around the world contributed to this report.