We study an evolution equation associated with the power of the Gross Laplacian Δ^p_G and a potential function V on an infinite dimensional space. The initial condition is a generalized function. The main technique we use is the representation of the Gross Laplacian as a convolution operator. This representation enables us to apply the convolution calculus on a suitable distribution space to obtain the explicit solution of the perturbed evolution equation. Our results generalize those previously obtained by Hochberg in the one-dimensional case with V=0, as well as by Barhoumi-Kuo-Ouerdiante for the case p=1.