On the morning of Feb 24, Ms Olga Nietsche was at home in Kyiv, Ukraine's capital, when she was startled by what she thought was the sound of fireworks.

She looked out of the window and saw a rocket zipping by, exploding a short distance away from her place.

The strike came just after Russian President Vladimir Putin had declared a "special military operation" for the "demilitarisation and denazification of Ukraine". He also claimed it was aimed at "bringing to justice those who committed numerous bloody crimes against civilians" in the Donbass region of eastern Ukraine.

Days later, Ms Nietsche, a 28-year-old translator, found herself crossing into Przemysl in Poland, lost and jaded, according to The Economist, with nothing more than some documents, a sleeping bag and a few pieces of clothing.

She was among some 2.5 million people who have fled Ukraine since the onset of the Russian invasion. Six in 10 of them are now in neighbouring Poland to the west. Others went elsewhere including to Hungary, Moldova and Romania.

The United Nations said on Friday (March 11) that another two million Ukrainians were displaced in their own country and that 564 civilians had been killed in the war so far.

The situation is dire in many areas as Russia pummels Ukrainian cities from the air, land and sea.

Encircled by Russian forces, people in the southern port city of Mariupol are trapped. They have no electricity and mobile phone networks have been disconnected.

Food, water and medical stocks are depleting after 11 days of the Russian siege. Reports say fights have broken out over food and people have resorted to melting snow for drinking water.

Ms Elena Coventry, 48, who lives in Scotland, lost touch two weeks ago with her mother and brother who live in Mariupol. She believes they are hiding in the cellar of their home.

"Although they are told to always have food supplies, these will have run out by now," Ms Coventry told the BBC, adding that it was unbearable for her to not know if they were alive or dead.

She joined an Internet forum and recognised her mother's house from one of the pictures of the neighbourhood uploaded there.

"The windows are gone, it looks like a human with its eyes gouged out. People had their lives in those flats, they ate, they drank, but now it's horror," said Ms Coventry.

In Ukraine's second-largest city of Kharkiv, which is just 42km from the Russian border, Dr Alexander Dukhovskyi, head of paediatrics at Hospital No. 4, told the BBC that patients are lined up along corridors because they will not be safe in wards with large windows due to Russian shelling.

Seven-year-old Volodymyr Baklanov lies on one bed. He was caught in crossfire and was shot in the head.

His mother Dariya, who was also shot in the head, died instantly. She was at the checkpoint with her two sons when Russian and Ukrainian troops exchanged fire. Three-year-old Viktor escaped unscathed.

"They were shot at by an automatic gun... I don't know from whose side (Russian or Ukrainian) because that day was a very dangerous day - there was a lot of shooting," Volodymyr's father Stanislav Baklanov, a 34-year-old engineer, told The Sydney Morning Herald.

The hospital said there are dozens of patients, all of them civilians, being treated for trauma following rocket attacks and shelling.

On the western side of Ukraine, adjacent to Poland, the city of Lviv has been relatively peaceful, for now, as Russia trains all of its firepower mostly to the north, east and south of Kyiv.

But Lviv, which has been the gateway for goods from Poland, is now struggling to cope with an influx of refugees as well.

"We're at the limits of our capacity," Lviv mayor Andriy Sadovyy told the Financial Times. "I'm raising the red flag."

Accommodation is fast filling up and supplies like food, mattresses and toiletries are running low. Mr Sadovyy warned of hard times ahead if substantial humanitarian aid does not come this week.

Amid the desperation and chaos, there was a heartwarming story of 11-year-old Hassan Al-Khalaf who travelled on his own by train and on foot for over 1,000km from Zaporizhzhie in south-east Ukraine to seek refuge in Slovakia, where his older brother studies.

The boy arrived in Slovakia early this month with just a plastic bag, a passport and a phone number written on his hand.

His widowed mother had sent him off alone on the long trip as she had to look after her own disabled mother who could not flee.

The Slovakian authorities, who took care of the boy, managed to contact his relatives in the capital, Bratislava. They relied on the phone number scribbled on his hand.

"I want to say a big thank you to the volunteers, because they are helping people they don't even know," the boy said in an interview on Friday.

Hailed as a hero by local police who posted his story on Facebook, Hassan said that his mother gave him hope along the arduous journey, adding that he was longing to see her again.

"I believe that there will be a happy end," he was quoted as saying by Reuters news agency.