Russian forces pressed on with their offensive across several regions in Ukraine on Monday, while Ukrainian President Volodymyr Zelenskyy warned of the potential for more serious attacks ahead of Ukraine's 31st anniversary of independence from Soviet rule.

Artillery and rocket fire close to the Zaporizhzhia nuclear reactor complex, on the Russian-occupied south bank of the Dnipro River, has stirred fears of a nuclear disaster and calls for the surrounding area to be demilitarized.

Ukraine and Russia have traded blame for the repeated shelling, some of which has grazed the plant. It was seized by Russian forces shortly after they invaded Ukraine in February but is still run mainly by Ukrainian technicians.

Overnight Russian rockets targeting Nikopol, across the Dnipro from Russian-occupied Enerhodar where the Zaporizhzhia plant is situated, and nearby Krivyi Rih and Synelnykovsky districts injured at least four people, regional Governor Valentyn Reznichenko wrote on Telegram on Monday.

Ukraine also reported a Russian missile strike on Voznesensk, to the southwest and not far from the country's second-largest atomic power station.

After the warnings, the capital Kyiv banned rallies this week to commemorate independence from Soviet rule for fear of Russian attacks.

Zelenskyy has called for vigilance, saying Moscow could try "something particularly ugly" ahead of Wednesday, which marks Ukraine's Independence Day and also half a year since Russia invaded Ukraine on Feb. 24.

Zelenskyy said he had discussed "all the threats" with French President Emmanuel Macron and word had also been sent to other leaders including Turkish President Tayyip Erdoğan and U.N. Secretary-General Antonio Guterres.

"All of Ukraine's partners have been informed about what the terrorist state can prepare for this week," Zelenskyy said in his nightly video address, referring to Russia.

He also said if Russia went ahead with plans to try captured Ukrainian defenders in Mariupol, then it would have violated international rules and cut itself off from negotiations.

"If this despicable show trial were to go ahead ... this would be the line beyond which negotiations are no longer possible," he said. "There will be no more conversations. Our state has said everything."

The Financial Times, in an article published Sunday, quoted Gennady Gatilov, Moscow's ambassador to the United Nations in Geneva, as saying Erdoğan had tried to facilitate dialogue.

But he dismissed speculation about talks between Zelenskyy and Russian President Vladimir Putin, saying there "was not any practical platform for having this meeting," the report said.

In Russia, authorities are investigating a suspected car bomb attack outside Moscow that killed the daughter of Alexander Dugin, an ultra-nationalist Russian ideologue who advocates Russia absorbing Ukraine.

While investigators said they were considering "all versions" when it came to establishing who was responsible, the Russian Foreign Ministry speculated there could be a link to Ukraine, something a Zelenskyy adviser dismissed.

"Ukraine, of course, had nothing to do with this because we are not a criminal state, like the Russian Federation, and moreover we are not a terrorist state," Mykhailo Podolyak said on Ukrainian TV.

More Russian strikes

As Ukraine prepared to mark its Independence Day embroiled in a war that has flattened towns and cities, killed thousands and forced millions to flee, officials reported more Russian strikes on targets in the east and south of the country.

In the eastern Bakhmut region, Russian forces inflicted damage from artillery and multiple rocket launcher systems in the areas of Soledar, Zaytseve and Bilogorivka settlements, Ukraine's General Staff said in its daily update on Monday.

They continued to focus their efforts on establishing full control over the territories of the Luhansk and Donetsk regions, maintaining the captured areas of Kherson and parts of Kharkiv, Zaporizhzhia and Mykolaiv regions, the General Staff added.

Of particular concern is the shelling of Nikopol.

As a result of overnight rocket attacks on Nikopol, Krivyi Rih and Synelnykovsky districts of the Dnipropetrovsk region, four people were injured, regional Governor Valentyn Reznichenko wrote on Telegram on Monday.

Over the past 24 hours, two civilians were killed in the Donetsk region in Ukraine's east, the regional administration said.

Russia denies targeting civilians.

The fighting near Zaporizhzhia and a missile strike on the southern town of Voznesensk, not far from Ukraine's second-largest atomic plant, have spurred fears of a nuclear accident.

On Sunday, U.S. President Joe Biden, British Prime Minister Boris Johnson, German Chancellor Olaf Scholz and Macron held a phone call stressing the importance of ensuring the safety of nuclear installations, while underlining their "steadfast commitment" to Ukraine.

Moscow says it is engaged in a "special military operation" to disarm and "denazify" its neighbor. Ukraine and its allies say Moscow has launched an unprovoked war of aggression.

Russia said on Sunday that its Kalibr missiles had destroyed an ammunition depot containing missiles for U.S.-made HIMARS rockets in Ukraine's southeastern Odesa region, home to ports critical to a U.N.-brokered plan to help Ukrainian agricultural exports reach world markets again.

Kyiv said a granary had been hit.