The Nikkei 225 index in Tokyo was up 0.3 percent after a few hours of trading. The broader Topix index had increased 0.1 percent.
The Kospi index in South Korea had a rise of 1.2 percent.
The Hang Seng index in Hong Kong had risen 0.8 percent, and the composite index in Shanghai and Shenzhen had increased 0.4 and 0.9 percent, respectively.
Donald Trump announced on Monday what tariff levels may apply to export goods from a number of countries - including 25 percent on goods from Japan and South Korea - from August 1.
The announcement also meant that the tariff pause that has applied since April 9 was extended.
