A particle system with explosions: law of large numbers for the density of particles and the blow-up time

Consider a system of independent random walks in the discrete torus with
creation-annihilation of particles and possible explosion of the total number of particles
in finite time. Rescaling space and rates for diffusion/creation/annihilation of particles, we
obtain a stong law of large numbers for the density of particles in the supremum norm. The
limiting object is a classical solution to the semilinear heat equation ∂tu = ∂xxu + f(u). If
f(u) = u
p
, 1 < p ≤ 3, we also obtain a law of large numbers for the explosion time.