What challenge did the empire face in the 3rd century?

The Roman Empire had to face quite a lot of problems. The first 35 years of the century were relatively peaceful, but when the last of the so called "Severer" died the empire fell into a heavy crises.The Crisis of the Third Century (also "Military Anarchy" or "Imperial Crisis") (235–284 AD) was a period in which the Roman Empire nearly collapsed under the combined pressures of invasion, civil war, plague, and economic depression. The Crisis began with the assassination of Emperor Alexander Severus at the hands of his own troops, initiating a fifty-year period in which 20–25 claimants to the title of Emperor, mostly prominent Roman Army generals, assumed imperial power over all or part of the Empire. By 258–260, the Empire split into three competing states, with the Gallic Empire including the Roman provinces of Gaul, Britannia and Hispania; and the Palmyrene Empire, including the eastern provinces of Syria Palaestina and Aegyptus; becoming independent from the Italian-centered Roman Empire proper between them. 

This period ended with the accession of Diocletian, who reigned from 284 until 305, and who solved many of the acute problems experienced during this crisis. 

However, the core problems would remain and cause the eventual destruction of the western empire. Diocletian saw the vast empire as ungovernable, and therefore split the empire in half and created two equal emperors to rule under the title of Augustus. In doing so, he effectively created what would become the Western Roman Empire and the Eastern Roman Empire. In 293 authority was further divided, as each Augustus took a junior Emperor called a Caesar to provide a line of succession. This constituted what is now known as the Tetrarchy ("rule of four"). The transitions of this period mark the beginnings of Late Antiquity.

He gathered information from primary sources.