In this paper, we completely characterize, only in terms of the data, the well-posedness of a fourth order abstract evolution equation arising from the Moore-Gibson-Thomson equation with memory. This characterization is obtained in the scales of vector-valued Lebesgue, Besov and Triebel-Lizorkin function spaces. Our characterization is flexible enough to admit as examples the Laplacian and the fractional Laplacian operators, among others. We also provide a practical and general criteria that allows L-p-L-q-well-posedness.