A substitute for the maximum principle for singularly perturbed time-dependent semilinear reaction-diffusion problems --- Part I

As the maximum principle does not hold true for nonlinear problems unless global, restrictive and often nonphysical assumptions are imposed over the whole domain, we introduce less restrictive, viable assumptions and show that the theory of upper and lower solutions is an appropriate substitute for the maximum principle in the case of singularly perturbed time-dependent semilinear reaction-diffusion problems with Dirichlet boundary conditions. The upper and lower solutions capture the boundary, interior and corner layers and the boundary conditions