President Vladimir Putin ordered his army on Monday to "dislodge" Ukrainian troops who have entered Russian territory as authorities said over 120,000 people had been evacuated away from the fighting.
Kyiv launched a surprise offensive into Russia's western Kursk region last Tuesday, capturing over two dozen settlements in the most significant cross-border attack on Russian soil since World War II.
"One of the obvious goals of the enemy is to sow discord, strife, intimidate people, destroy the unity and cohesion of Russian society," Putin told a televised meeting with government officials.
"The main task is, of course, for the defence ministry to dislodge the enemy from our territories," he said.
Some 121,000 people have fled the Kursk region since the start of the fighting, which has killed at least 12 civilians and injured 121 more, regional governor Alexei Smirnov told the meeting with Putin.
Authorities in Kursk announced on Monday they were widening their evacuation area to include Belovsky district, home to some 14,000 residents. The neighbouring Belgorod region also said it was evacuating its border district of Krasnoyaruzhsky.
Ukraine has pierced into the region by at least 12 kilometres (seven miles) and has captured 28 towns and villages, with the new front 40 kilometres long, Smirnov said.
A top Ukrainian official told AFP over the weekend that the operation was aimed at stretching Russian troops and destabilising the country after months of slow Russian advances across the frontline.
Putin said Russia would respond by showing "unanimous support for all those in distress" and claimed there had been an increase in men signing up to fight.
"The enemy will receive a worthy riposte," he said.
'Maximum losses'
The assault appeared to catch the Kremlin off guard. Russia's army rushed in reserve troops, tanks, aviation, artillery and drones in a bid to quash it.
But it conceded on Sunday that Ukraine had penetrated up to 30 kilometres (20 miles) into Russian territory in places.
It said some forces were near the villages of Tolpino and Obshchy Kolodez, around 25 kilometres and 30 kilometres from the Russia-Ukraine border.
A Ukrainian security official told AFP, on condition of anonymity, that "the aim is to stretch the positions of the enemy, to inflict maximum losses and to destabilise the situation in Russia as they are unable to protect their own border".
The Ukrainian official said thousands of Ukrainian troops were involved in the operation.
Russia's defence ministry said on Monday that its air defence systems had destroyed 18 Ukrainian drones -- including 11 over the Kursk region.
'It's scary'
Russia's emergency situations ministry said on Sunday that over 44,000 residents in the Kursk region have applied for financial assistance, TASS news agency reported.
Russia's rail operator has meanwhile organised emergency trains from Kursk to Moscow, around 450 kilometres away, for those fleeing.
"It's scary to have helicopters flying over your head all the time," said Marina, refusing to give her surname, who arrived by train in Moscow on Sunday. "When it was possible to leave, I left."
Across the border in Ukraine's Sumy region, AFP journalists on Sunday saw dozens of armoured vehicles daubed with a white triangle -- the insignia apparently being used to identify Ukrainian military hardware deployed in the attack.
At an evacuation centre in the regional capital of Sumy, 70-year-old retired metal worker Mykola, who fled his village of Khotyn some 10 kilometres from the Russian border, welcomed Ukraine's push into Russia.
"Let's let them find out what it's like," he told AFP. "They don't understand what war is. Let them have a taste of it."
Analysts think Kyiv may have launched the assault to relieve pressure on its troops in other parts of the frontline.
But the Ukrainian official said: "Their pressure in the east continues, they are not pulling back troops from the area," even if "the intensity of Russian attacks has gone down a little bit".
The Ukrainian official said he expected Russia would "in the end" stop the incursion.
Ukraine was bracing for a large-scale retaliatory missile attack, including "on decision-making centres" in Ukraine, the official said.