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We discuss the front propagation in the $A+B\rightarrow 2A$ reaction undersubdiffusion which is described by continuous time random walks with aheavy-tailed power law waiting time probability density function. Using acrossover argument, we discuss the two scaling regimes of the frontpropagation: an intermediate asymptotic regime given by the front solution ofthe corresponding continuous equation, and the final asymptotics, which isfluctuation-dominated and therefore lays out of reach of the continuous scheme.We moreover show that the continuous reaction subdiffusion equation indeedpossesses a front solution that decelerates and becomes narrow in the course oftime. This continuous description breaks down for larger times when the frontgets atomically sharp. We show that the velocity of such fronts decays in timefaster than in the continuous regime.

Asymptotic front behavior in an $A+B\rightarrow 2A$ reaction under subdiffusion