Kagoshima Governor Satoshi Mitazono requested the help of military personnel in the rescue effort.
One woman has so far been killed in the rains that began at the weekend, after her house was buried in a landslide.
Rainfall since Monday totalled more than 24 inches (620mm) in Miyakonojo, and more than 20 inches (520 mm) in Kagoshima.
Kyushu island, where Kagoshima, Aira and Kirishima are located, has received almost 3 feet of rain since last week and is expected to continue to be rained upon at a rate of 3 inches per hour until Thursday, weather officials said. That's nearly the island's average rainfall of 319 millimeters (12.5 inches) for the entire month of July.
A flooded road is seen during heavy rain on July 3, 2019, in Soo City, Kagoshima Prefecture, southwest Japan.
The heavy rain is expected to hang over the entire Japanese archipelago until Saturday. Officials are warning of the risk of flooding and mudslides.
A final round of torrential rain will threaten eastern Japan on Saturday, with lingering rainfall expected Saturday night across eastern Honshu.
From early June to late July, Japan traditionally experiences a rainy "baiu" season, as humid air from the south converges with cooler, drier air from the north.
The rain has been responsible for at least one fatality, with an elderly woman found dead in a mudslide in Kagoshima.
Trump congratulates U.S. Navy SEAL cleared of battlefield murder
As a result of Gallagher's reduced rank and forfeiture, his pension will be reduced, though it's unclear at this time by how much. The jury comprised five Marines and two sailors, including a SEAL, most of whom are seasoned combat veterans who served in Iraq .