Ukraine’s presidential office reported heavy fighting Tuesday in the Kherson region in southern Ukraine, an area occupied by Russian forces where Ukraine says it has launched a counteroffensive to try to retake territory.

Ukrainian President Volodymyr Zelenskyy vowed in his nightly address Monday that Ukrainian forces would take back their territory. He said Ukraine would chase Russia’s forces “to the border.”

“If they want to survive — it’s time for the Russian military to run away. Go home,” he said.

Britain’s defense ministry said Tuesday that as of early Monday, “several brigades of the Ukrainian Armed Forces increased the weight of artillery fires in front line sectors across southern Ukraine.”

It added that since the start of August, Russia has worked to reinforce its presence on the western bank of the Dnipro River in the Kherson area.

“Most of the units around Kherson are likely undermanned and are reliant upon fragile supply lines by ferry and pontoon bridges across the Dnipro,” the British defense ministry said.

Russia’s defense ministry said Monday that Russian forces had stopped Ukrainian attacks in the Kherson and Mykolaiv regions and inflicted “heavy losses” on Ukrainian forces.

A senior U.S. defense official told reporters Monday that the United States would know more about Ukraine’s offensive near Kherson “in the next 24-36 hours.” The official said Ukrainian force numbers are gaining parity with Russian forces in the south.

“Are they on the offensive? I think they are,” the official said.

Russia failed to capture the capital, Kyiv, in northern Ukraine in its initial attack that began in late February, but later took control of wide swathes of land in the south along the Black Sea coast.

Fighting for months has centered on eastern Ukraine in the Donbas region, where Russia-supported separatists and Kyiv’s forces have fought since 2014, the same year Moscow seized Ukraine’s Crimean Peninsula in a move not recognized by the international community.

The conflict in eastern Ukraine has been at somewhat of a standstill for weeks, with Russia and Ukraine gaining or losing territory incrementally.

But Western allies, led by the United States, have continued to ship armaments to the Kyiv government, possibly giving Ukraine new confidence to attack farther in the southern reaches of the country.

Some information for this story came from The Associated Press, Agence France-Presse and Reuters. VOA’s Carla Babb contributed to this report.