KRAMATORSK, Ukraine —

       Missile strikes on Monday hit western Ukraine’s biggest city, Lviv, killing at least six people, according to local officials, as Russian forces pressed ahead with their drive to seize the strategic southern port of Mariupol, where Ukraine has vowed that desperate defenders would fight to the end.

       In the war’s eighth week, the country’s east braced for a full-scale Russian assault that President Volodymyr Zelensky said was intended to “literally finish off and destroy” Ukraine’s industrial heartland.

       Civilians have for weeks been urged to escape the eastern battle zone, but on Monday, for a second day in a row, Ukrainian officials said the two sides had failed to agree on setting up humanitarian corridors for civilians to use to flee.

       In a bleak benchmark, the United Nations says more than 5 million people have now fled the country since Feb. 24, when Russian President Vladimir Putin launched an unprovoked invasion to control Ukraine and topple its government.

       While seizing chunks of territory in the east and south, Russian forces have also suffered a series of stinging military setbacks. Those include a failed effort to seize the capital, Kyiv, and the sinking last week of Russia’s Black Sea flagship in what Western and Ukrainian military officials say was a Ukrainian missile strike.

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       The renewed long-distance attacks on Kyiv and the country’s western hub, Lviv, are viewed as part of a concerted effort by Moscow to target infrastructure and strike at armaments manufacturers in preparation for its eastern offensive.

       Russia claimed Monday to have staged hundreds of strikes overnight aimed at what it said were military targets across Ukraine. A statement from the Defense Ministry said 16 Ukrainian military facilities had been destroyed.

       That account could not be independently verified, but the intensifying tempo of strikes is in line with Western military assessments that Russian preparations for a devastating assault in the east are rapidly moving forward.

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       Russia’s war in Ukraine is causing a European refugee crisis on a scale not seen since World War II and raising fears of a return to Cold War-era dynamics.

       Pivotal to that offensive is Mariupol, on the Sea of Azov, whose capture would allow Russian forces to link up along a land corridor between Russian-controlled areas and the Crimean Peninsula, which Moscow seized in 2014.

       The city’s defenders ignored a surrender demand on Sunday by Russia, but Ukrainian officials have acknowledged that the holed-up Ukrainian forces have only one real redoubt, a sprawling steelworks complex.

       The city itself is an all-but-annihilated prize. Under siege and bombardment since the war’s first days, Mariupol lies largely in ruins, with unburied bodies lying in the streets. More than 450,000 people lived in what was a busy, bustling metropolis before the invasion; about 100,000 are thought to remain, under desperate humanitarian conditions.

       “Just as the Russian troops are destroying Mariupol, they want to wipe out other cities and communities” in two eastern regions, in Donetsk and Luhansk, Zelensky said in his overnight address.

       Russia controls part of both areas now, in the form of self-declared independent statelets, but Putin has signaled he intends to try to take the entire Donbas, as the larger region is called.

       Zelensky, appealing again for international shipments of heavy weaponry to try to beat back the expected offensive, declared that every delay in providing armaments is “permission for Russia to take the lives of Ukrainians – this is how Russia interprets it.”

       In Lviv, a city less than 50 miles from the Polish border where many of those fleeing violence elsewhere in Ukraine have taken shelter, the city’s mayor and regional governor said at least six people died in a volley of missile strikes early Monday.

       Mayor Andriy Sadovyi said in addition to those killed, 11 people were injured, including a child. A strike apparently aimed at a vehicle-repair facility blew out windows in a nearby hotel where people displaced by fighting elsewhere were staying, he said on social media.

       The number of people fleeing the war has reached epic proportions. The U.N.’s refugee agency says more than 4.8 Ukrainians have sought haven outside the country, in addition to the departures of more than 200,000 students and migrant workers who were residing in Ukraine before the war.

       The U.N. High Commissioner for Refugees, Filippo Grandi, over the weekend cited those figures on what he called a “somber Easter Sunday,” saying the holiday should serve as a reminder of the hope that the “methods and language of war” would not prevail over those of peace.

       In Ukraine’s east, amid the drumbeat of shelling and the sustained notes of air-raid sirens, evacuations have slowed, particularly after a horrific strike earlier this month on a rail station in the eastern city of Kramatorsk that killed about 60 people.

       Oxana Federova, from the city of Toretsk, some 23 miles southeast of Kramatorsk, had driven to the city with her grandfather in their Soviet-era car to try to find an open bank. All the ones in their town were closed, the 32-year-old shop manager said.

       But although Kramatorsk is a jumping-off point for departures by bus, Federova had no intention of going.

       “Why do I need to leave here? Would you leave your house? If everything you own is here, your family is here, would you go?” she said. “We’re not rich people, but this is our Ukraine, not Russia’s Ukraine.”

       Bulos reported from Kramatorsk and King from Berlin.