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 admite as examples the Laplacian and the fractional Laplacian
operators, among others. We also provide a practical and general criteria
that allows Lp–Lq-well posedness.