Question

Why did the eastern part of the roman empire last longer than the western part

Answer 1

There are many reasons but here's two

1. The 'war' and invasion etc was fought in the West so the province that you are going to have to raise troops and revenue from to fight the war is the same one where all the fighting takes place.

Meanwhile the 'East' had Anatolia as a heartland for revenue and recruits that was untouched .

2. There is a lot of evidence that it wasn't simply 'barbarian' invasion but also a large element of rebellion against central authority by local forces. These are the people who are supposed to supply your troops and revenue.