Do you mean the Timurid Empire? The Mongol Empire collapsed many years before Timur was born, and most of its successor states were well into their periods of decline by the time of Timur's rise to power (the Ilkhanate disintegrated in 1335, the Yuan Dynasty lost control of China in 1368 and the Golden Horde went into permanent decline from 1395 onward. Timur was born sometime in the 1320s or 30s, and rose to power in 1370).

If you did mean Timur's Empire, the Timurid Dynasty quickly plunged into heavy disputes and civil war following Timur's death, mainly over succession. Regional governors took this as an opportunity to exert their own authority and whilst the Timurids would dominate the politics of the Iranian Plateau for multiple decades after Timur's death in 1405, it would never rise to the same level of power again.

While this doesn't exactly answer your question, I don't think it's correct to describe Tamerlane's empire as "the Mongol Empire." While Timur did claim descent from the Chinggisids, he was Turkic rather than Mongolian, and (iirc) he built his state well after the end of Mongol rule in that area of Central Asia.

Source: Jack Weatherford, Ghengis Khan and the Making of the Modern World

I guess you could say it was the same as after the death of Möngke Khan? Infighting, civil war, and a division of the empire?