The North Korean leader, 69, died of a heart attack while on a train outside capital Pyongyang on Saturday.
President Barack Obama and UN Secretary General Ban Ki-Moon have assured South Korea's president Lee Myung-bak that they have stepped up monitoring of the country. South Korea was put on high military alert.
And Japanese PM Yoshihiko Nodo told his ministers to prepare for "any unexpected surfaces".
China, North Korea's closest international ally, offered deep condolences on the death of the dictator.
Concerns over what direction the regime will now take are stoked by predictions that Kim's death deals a shattering blow to the cult of personality with which he and his dad Kim Il-sung have ruled since 1948.
There are also fears that the North Korean military could stage a coup — nearly ten million of its 24million inhabitants are in the services — and that the death could set back efforts by the worldwide community to get Pyongyang to abandon its nuclear weapons.
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