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We present observations of the X-ray afterglow of GRB 050408, a gamma-rayburst discovered by HETE-II. Swift began observing the field 42 min after theburst, performing follow-up over a period of 38 d (thus spanning three decadesin time).The X-ray light curve showed a steepening with time, similar to manyother afterglows. However, the steepening was unusually smooth, over theduration of the XRT observation, with no clear break time. The early decay wastoo flat to be described in terms of standard models. We therefore explorealternative explanations, such as the presence of a structured afterglow or oflong-lasting energy injection into the fireball from the central GRB engine.The lack of a sharp break puts constraints on these two models. In the formercase, it may indicate that the angular energy profile of the jet was not asimple power law, while in the second model it implies that injection did notstop abruptly. The late decay may be due either to a standard afterglow (thatis, with no energy injection), or to a jetted outflow still being refreshed. Asignificant amount of absorption was present in the X-ray spectrum,corresponding to a rest-frame Hydrogen column density NH = 1.2(-0.3,+0.4)*10^22 cm^-2, indicative of a dense environment.Comment: Accepted for publication in A&A; 7 pages, 3 figure

Long-term monitoring of the X-ray afterglow of GRB 050408 with Swift/XRT