This paper deals with the solution u to the parabolic Anderson equation ¿u/¿t=¿¿u+¿u on the lattice . We consider the case where the potential ¿ is time-dependent and has the form ¿(t,x)=d0(x-Yt) with Yt being a simple random walk with jump rate 2d. The solution u may be interpreted as the concentration of a reactant under the influence of a single catalyst particle Yt. In the first part of the paper we show that the moment Lyapunov exponents coincide with the upper boundary of the spectrum of certain Hamiltonians. In the second part we study intermittency in terms of the moment Lyapunov exponents as a function of the model parameters ¿ and .