Hölder regularity for the Moore-Gibson-Thompson equation with infinite delay

Abstract

We characterize the well-posedness of a third order in time equation with infinite delay in H\"older 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 $g(t)=\frac{t^{\nu-1}}{\Gamma(\nu)}e^{-at}$ the third order problem is ill-posed whenever $0<\nu \leq 1$ and $a$ is inversely proportional to the damping term of the given model.