Publisher Summary This chapter discusses the existence and stability for some systems of diffusion–reaction equations. Initial boundary value problems for semilinear systems of parabolic equations are considered. These equations describe a single, irreversible, nonisothermic, pth order chemical reaction in a permeable catalyst whose shape is described by Ω. The system of equations described in the chapter is of considerable importance in the theory of chemical reactions. However, so far there seem to be no general existence, uniqueness, and stability results, although many particular cases and approximations have been considered. There are studied isothermic reactions or the case where diffusion can be neglected, in which the system is reduced to the much simpler case of ordinary differential equations. The chapter presents a theorem to show that none of the assumptions considered is necessary to prove that the discussed system has a unique global solution. The proof of this theorem is based on the theory of semilinear evolution equations in Banach spaces as developed by Sobolevskii, Friedman, and others.