This article studies the diffusion problems with a concentrated source which is provided at a sequential time steps in 1 dimensional space. The problems are considered for both Gaussian and fractional diffusion operators. For the fractional diffusion case, Riemann-Liouville operator with fractional order is used to describe the model with diffusion rate slower than normal time scale, which is known as sub diffusive problems. Due to this sub diffusive property, the existence and nonexistence behavior of the solution will be studied. Since the forcing term will experience a concentrated source at a sequence of time steps, the frequency, the time difference and strength of the source may affect the growth rate of the solution. Criteria for these effects which may cause for the quenching behavior of the solution will be given. The existence of the solution is investigated. The monotone behavior in spatial will be given. The quenching behavior of the solution will be studied. The location of the quenching point will be discussed.