With the invasion well into its second week, Russian troops were making significant advances in southern Ukraine but stalled in some other regions.
Soldiers and volunteers fortified the capital, Kyiv, with hundreds of checkpoints and barricades designed to thwart a takeover.
A steady rain of shells and rockets fell on other population centers, including the Kyiv suburb of Bucha, where the mayor reported heavy artillery fire.
``We can't even gather up the bodies because the shelling from heavy weapons doesn't stop day or night,'' Mayor Anatol Fedoruk said. ``Dogs are pulling apart the bodies on the city streets. It's a nightmare.''
In one of the most desperate cities, the encircled southern port of Mariupol, an estimated 200,000 people _ nearly half the population of 430,000 _ were hoping to flee, and Red Cross officials waited to hear when a corridor would be established.
Russia's coordination center for humanitarian efforts in Ukraine said Russia will begin a cease-fire at 10 a.m. Moscow time (0700 GMT) to allow civilians to flee through special corridors agreed upon with Ukrainian authorities, according to Russian media.
Most of those corridors would lead to Russia, either directly or through Belarus, though people in Kharkiv would be allowed to travel to western Ukraine, the center said.
But doubts abounded, fueled by the failure of previous attempts to lead civilians to safety amid the biggest ground war in Europe since World War II.
The office of Ukrainian President Volodymyr Zelenskyy would not comment on the latest Russian proposal, saying only that Moscow's plans can be believed only if a safe evacuation begins.
Demands for effective passageways have surged amid intensifying shelling by Russian forces. The steady bombardments, including in some of Ukraine's most populated regions, have yielded a humanitarian crisis of diminishing food, water and medical supplies.
Through it all, Zelenskky said Ukrainian forces were showing unprecedented courage.
``The problem is that for one soldier of Ukraine, we have 10 Russian soldiers, and for one Ukrainian tank, we have 50 Russian tanks,'' Zelenskyy told ABC News in an interview that aired Monday night.
But he noted that the gap in strength was closing and that even if Russian forces ``come into all our cities,'' they will be met with an insurgency.
A top US official said multiple countries were discussing whether to provide the warplanes that Zelenskyy has been pleading for.
Mariupol was short on water, food and power, and cellphone networks are down. Stores have been looted as residents search for essential goods.
Police moved through the city, advising people to remain in shelters until they heard official messages broadcast over loudspeakers to evacuate.
Hospitals in Mariupol are facing severe shortages of antibiotics and painkillers, and doctors performed some emergency procedures without them.
The lack of phone service left anxious citizens approaching strangers to ask if they knew relatives living in other parts of the city and whether they were safe.
Several hundred kilometers (miles) west of Mariupol, Russian forces continued their offensive in Mykolaiv, opening fire on the Black Sea shipbuilding center of a half-million people, according to Ukraine's military.
Rescuers said they were putting out fires caused by rocket attacks in residential areas.
Ukraine's general staff of the armed forces said in a statement Tuesday that Ukrainian forces are continuing defense operations in the suburbs of the city.
The general staff said ``demoralized'' Russian forces are engaging in looting in places they have occupied, commandeering civilian buildings like farm hangars for military equipment, and are setting up firing positions in populated areas.
The claims could not be independently verified.
Ukrainian defense forces were also involved in operations in the northern city of Chernihiv and the outskirts of Kyiv, the general staff said.
In Kyiv, soldiers and volunteers have built hundreds of checkpoints to protect the city of nearly 4 million, often using sandbags, stacked tires and spiked cables.
Some barricades looked significant, with heavy concrete slabs and sandbags piled more than two stories high, while others appeared more haphazard, with hundreds of books used to weigh down stacks of tires.
``Every house, every street, every checkpoint, we will fight to the death if necessary,'' said Mayor Vitali Klitschko.
In Kharkiv, Ukraine's second-largest city, with 1.4 million people, heavy shelling slammed into apartment buildings.
``I think it struck the fourth floor under us,'' Dmitry Sedorenko said from his Kharkiv hospital bed. ``Immediately, everything started burning and falling apart.'' When the floor collapsed beneath him, he crawled out through the third story, past the bodies of some of his neighbors.
In the small town of Horenka, where shelling reduced one area to ashes and shards of glass, rescuers and residents picked through the ruins as chickens pecked around them.
``What are they doing?'' rescue worker Vasyl Oksak asked of the Russian attackers. ``There were two little kids and two elderly people living here. Come in and see what they have done.''
At The Hague, Ukraine pleaded with the International Court of Justice to order a halt to Russia's invasion, saying Moscow is committing widespread war crimes.
Russia ``is resorting to tactics reminiscent of medieval siege warfare, encircling cities, cutting off escape routes and pounding the civilian population with heavy ordnance,'' said Jonathan Gimblett, a member of Ukraine's legal team.
Russia snubbed the court proceedings, leaving its seats in the Great Hall of Justice empty.
Efforts to set up safe passage for civilians over the weekend fell apart amid continued Russian shelling.
Russia's plan announced Monday allowing civilians to leave Kyiv, Mariupol, Kharkiv and Sumy would have used routes toward Russia or its ally Belarus, which was a launch pad for the invasion.
Ukraine instead proposed eight routes allowing civilians to travel to western regions of the country where there is no shelling.
Later, Russia's UN Ambassador, Vassily Nebenzia, told the UN Security Council that Russia would carry out a cease-fire Tuesday morning and made the suggestion that humanitarian corridors leading away from Kyiv, Mariupol, Sumy and Chernigov could let people choose where they want to seek safety.
The UN humanitarian chief, Undersecretary-General Martin Griffiths, addressed the Security Council and urged safe passage for people to go ``in the direction they choose.''
The battle for Mariupol is crucial because its capture could allow Moscow to establish a land corridor to Crimea, which Russia seized from Ukraine in 2014.
The fighting has sent energy prices surging worldwide and stocks plummeting, and threatens the food supply and livelihoods of people around the globe who rely on crops farmed in the fertile Black Sea region.
The UN human rights office reported 406 confirmed civilian deaths but said the real number is much higher. The invasion has also sent 1.7 million people fleeing Ukraine.
On Monday, Moscow again announced a series of demands to stop the invasion, including that Ukraine recognize Crimea as part of Russia and recognize the eastern regions controlled by Moscow-supported separatist fighters as independent.
It also insisted that Ukraine change its constitution to guarantee it won't join international bodies like NATO and the EU. Ukraine has already rejected those demands.
Zelenskyy has called for more punitive measures against Russia, including a global boycott of its oil exports, which are key to its economy.
``If (Russia) doesn't want to abide by civilized rules, then they shouldn't receive goods and services from civilization,'' he said in a video address.