Reaction-diffusion with a time-dependent reaction rate: the single-species diffusion-annihilation process

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We study the single-species diffusion-annihilation process with a time-dependent reaction rate, lambda(t)=lambda₀ t-omega. Scaling arguments show that there is a critical value of the decay exponent omegac(d) separating a reaction-limited regime for omega > omegac from a diffusion-limited regime for omega < omegac. The particle density displays a mean-field, omega-dependent, decay when the process is reaction limited whereas it behaves as for a constant reaction rate when the process is diffusion limited. These results are confirmed by Monte Carlo simulations. They allow us to discuss the scaling behaviour of coupled diffusion-annihilation processes in terms of effective time-dependent reaction rates.