We study the blow-up behaviors of two parabolic problems on a bounded domain. One is the heat equation with nonlinear memory and the other is a parabolic system with power nonlinearity in which the coefficients of the reaction terms (potentials) are nonnegative and spatially inhomogeneous. Our aim is to show that any zero of the potential, where there is no reaction, is not a blow-up point, if the solution is monotone in time. We also give sufficient conditions for the time monotonicity of solutions.