  We study a class of reaction-diffusion model extrapolating continuously
between the pure coagulation-diffusion case ($A+A\to A$) and the pure
annihilation-diffusion one ($A+A\to\emptyset$) with particles input
($\emptyset\to A$) at a rate $J$. For dimension $d\leq 2$, the dynamics
strongly depends on the fluctuations while, for $d >2$, the behaviour is
mean-field like. The models are mapped onto a field theory which properties are
studied in a renormalization group approach. Simple relations are found between
the time-dependent correlation functions of the different models of the class.
For the pure coagulation-diffusion model the time-dependent density is found to
be of the form $c(t,J,D) = (J/D)^{1/\delta}{\cal F}[(J/D)^{\Delta} Dt]$, where
$D$ is the diffusion constant. The critical exponent $\delta$ and $\Delta$ are
computed to all orders in $\epsilon=2-d$, where $d$ is the dimension of the
system, while the scaling function $\cal F$ is computed to second order in
$\epsilon$. For the one-dimensional case an exact analytical solution is
provided which predictions are compared with the results of the renormalization
group approach for $\epsilon=1$.
