We characterize the well-posedness of a third order in time equation with infinite delay in Hölder spaces, solely in terms of spectral properties concerning the data of the problem. Our analysis includes the case of the linearized Kuznetzov and Westerwelt equations. We show in case of the Laplacian operator the new and surprising fact that for the standard memory kernel $g(t)=\frac{t^{ν-1}}{Γ(ν)}e^{-at}$ the third order problem is ill-posed whenever $0<ν ≤q 1$ and $a$ is inversely proportional to one of the terms of the given model.