In this paper the existence and uniqueness of a weak solution for a nonlinear parabolic system modeling a combustion-reaction process are proved. The combustion-reaction process obeys Arrhenius’ Law for small values of time. The boundary conditions describe the ignition process of the reaction by means of a Dirichlet condition on a portion of the boundary kept at a constant temperature, while on the rest of the boundary a homogeneous Neumann condition is maintained after ignition. Numerical results validating the efficiency of the proposed multistep method are also presented.