Russia has been pounding Ukraine's Kharkiv region with dozens of strikes since Saturday.
Several residential buildings were destroyed or damaged, and the body of one person was found under the rubble.
Heavy fighting has been mounting since Saturday following Russia's sudden incursion in the Vovchansk area.
Over 5,000 residents have already been evacuated to safer areas.
Vovchansk city military administration head Tamaz Gambarashvili said local authorities would aim to "completely evacuate the population within one or two days".
"There were small (Russian) groups that tried to enter the city streets, but the armed forces cut off these attempts", he added.
In his Sunday address, Ukraine's President Volodymyr Zelenskyy said the city is "under constant Russian fire", describing the situation as "extremely difficult".
On the same day, Ukraine hit back with a massive shelling on Belgorod, partially destroying a residential building and killing at least 15 people.
The Belgorod region, on Russia's western border, has also been the target of a large number of strikes.
