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We investigate the nonequilibrium phase transition of the disordered contactprocess in five space dimensions by means of optimal fluctuation theory andMonte Carlo simulations. We find that the critical behavior is of mean-fieldtype, i.e., identical to that of the clean five-dimensional contact process. Itis accompanied by off-critical power-law Griffiths singularities whosedynamical exponent $z'$ saturates at a finite value as the transition isapproached. These findings resolve the apparent contradiction between theHarris criterion which implies that weak disorder is renormalization-groupirrelevant and the rare-region classification which predicts unconventionalbehavior. We confirm and illustrate our theory by large-scale Monte-Carlosimulations of systems with up to $70^5$ sites. We also relate our results to arecently established general relation between the Harris criterion andGriffiths singularities [Phys. Rev. Lett. {\bf 112}, 075702 (2014)], and wediscuss implications for other phase transitions.

Rare regions and Griffiths singularities at a clean critical point: The five-dimensional disordered contact process