Exponents appearing in heterogeneous reaction-diffusion models in one dimension

29 pages. The three figures are not included, but are available upon requestWe study the following 1D two-species reaction diffusion model : there is a small concentration of B-particles with diffusion constant $D_B$ in an homogenous background of W-particles with diffusion constant $D_W$; two W-particles of the majority species either coagulate ($W+W \\longrightarrow W$) or annihilate ($W+W \\longrightarrow \\emptyset$) with the respective probabilities $ p_c=(q-2)/(q-1) $ and $p_a=1/(q-1)$; a B-particle and a W-particle annihilate ($W+B \\longrightarrow \\emptyset$) with probability 1. The exponent $\\theta(q,\\lambda=D_B/D_W)$ describing the asymptotic time decay of the minority B-species concentration can be viewed as a generalization of the exponent of persistent spins in the zero-temperature Glauber dynamics of the 1D $q$-state Potts model starting from a random initial condition : the W-particles represent domain walls, and the exponent $\\theta(q,\\lambda)$ characterizes the time decay of the probability that a diffusive \"spectator\" does not meet a domain wall up to time $t$. We extend the methods introduced by Derrida, Hakim and Pasquier ({\\em Phys. Rev. Lett.} {\\bf 75} 751 (1995); Saclay preprint T96/013, to appear in {\\em J. Stat. Phys.} (1996)) for the problem of persistent spins, to compute the exponent $\\theta(q,\\lambda)$ in perturbation at first order in $(q-1)$ for arbitrary $\\lambda$ and at first order in $\\lambda$ for arbitrary $q$