Russian forces are pressing an offensive in the direction of Sloviansk, an important town in the Donetsk region, according to the Ukrainian military.
The offensive involves heavy shelling of Ukrainian defenses, the General Staff said in its daily update.
The enemy fired on the units of our troops on the Lyman-Siversk border in order to oust them from their positions and create conditions for the attack on Sloviansk," it said.
Some analysts say Russian forces have made modest territorial gains in this region over the past week, but Lyman remains in Ukrainian hands.
The General Staff said Russian forces were attacking a large number of towns in the Luhansk region, and had tried to improve their positions around the town of Popasna by moving one battalion tactical group from Mariupol.
Altogether 10 attacks in Donetsk and Luhansk had been repulsed and Ukrainian forces had destroyed a wide variety of Russian hardware, it claimed.
"The enemy deployed additional surface-to-air missile systems in the temporarily occupied territories of the Luhansk and Zaporizhzhia regions," it said.
Three people in Luhansk were killed as heavy shelling persisted in Lysychansk, Zolote and Popasna, according to Serhiy Hayday, head of Luhansk's military administration.
Heavy prolonged shelling prevented a full-fledged evacuation, he said.
In areas of southern Ukraine where fighting continues, Russian forces were looking for weaknesses in Ukrainian defenses to the south of Mykolaiv as they try to extend their control to the whole of the Kherson region, which borders Crimea, the General Staff said.
Parts of southern Zaporizhzhia have also seen heavy fighting. The regional command said Monday that "the enemy tried to break through in small groups with the support of armored vehicles, tanks and artillery, but failed."
The towns of Polohy and Orikhiv were among those targeted with shelling, it said.
It also claimed that Russian forces were forcing farmers "under the barrels of machine guns" to sell grain at a steep discount.
There has also been an uptick in attacks on grain stores and elevators.
Valentyn Reznichenko, head of the Dnipropetrovsk regional state administration, said Monday that a grain warehouse had been destroyed in the Synelnykove district.