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 strong 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 ? _t u=? _xx u+f(u). If f(u)=u ^p , 1<p??3, we also obtain a law of large numbers for the explosion time.