Marina, who did not want her last name revealed out of fears for her safety, is one of them.
She decided to stay in her home city of Kramatorsk, about 10 miles from the front lines, despite having the means to escape to Kyiv or even western Ukraine, far from active fighting.
She said the prospect of having to collect rainwater to drink or use wood for heat did not scare her or others who are guided by hope.
Shes also wary of signing the form. First of all I will sit and read it carefully, Marina, 60, said on the phone from Kramatorsk, where she has lived all her life.
I believe in God and I believe in the Ukrainian army, she added. This is my destiny.
Ukraine is eager to get people out of the area to minimize civilian casualties and give it more operational freedom, analysts said.
Kyiv intends to mount a fierce defense of the key cities of Slavyansk and Kramatorsk, said Michael Clarke, a professor of war studies at Kings College London.
Ukraines military is hoping for a strategic victory in the Russian-occupied south, Clarke said, where its expected to mount a counteroffensive. For any success there to have political impact, Clarke said, it must not lose the rest of the Donbas.
Logistically, the evacuations give Ukrainians more leeway to defend or leave a city without having to take civilians into account, said the head of intelligence at the consultancy Le Beck, Michael A. Horowitz, a geopolitical and security analyst.
Ukrainians have also been asking people to evacuate since the very beginning, and the presence of civilians has made the job of the Ukrainian military far more difficult, Horowitz said.
So Zelenskys order may just be an escalation in efforts to remove civilians from whats become the main area of fighting, Horowitz said.
Human rights group Amnesty International said in a much-criticized report published Thursday that Ukrainian forces have exposed civilians to Russian attacks at times by basing themselves in schools, residential buildings and other places in populated areas.
Zelenskyy denounced the report, as did other officials in Kyiv and allies across social media.
Amnesty said in a statement that it had made it clear the Ukrainian military practices we described in no way justify Russias routine violations of international humanitarian law.
When we find violations of international humanitarian law, as we did in this case, we will report them fairly and accurately. Ignoring violations committed by a favoured side would not be meaningful human rights reporting, it said.
Oleksandr Ivanov, a volunteer who works for a local aid organization helping with the evacuations, told NBC News that people want to stay for a variety of reasons.
Many people cant leave their elderly parents, Ivanov said. For older people, its psychologically difficult to leave. And of course, there are people who believe that until a shell hits their house, they will not leave.