In this paper, we consider the Jordan–Moore–Gibson–Thompson with a time-fractional damping term of the type \(\delta \text {D}_t^{1-\alpha } \Delta \psi _t\) where we allow the challenging so-called critical case ( \(\delta =0\) ). This equation arises in the context of acoustic propagation through thermally relaxed media. We tackle the question of long-time existence of the solution. More precisely, the goal of the paper is twofold: First, we establish local well-posedness of the initial boundary value problem, where we also provide a lower bound on the final time of existence as a function of initial data. Second, we prove a regularity result which guarantees, under the hypothesis of boundedness of certain quantities, that the local solution can be extended to be global-in-time.
