On the regularity of solutions to the Moore–Gibson–Thompson equation: a perspective via wave equations with memory

We undertake a study of the initial/boundary value problem for the (third order in time) Moore–Gibson–Thompson (MGT) equation. The key to the present investigation is that the MGT equation falls within a large class of systems with memory, with affine term depending on a parameter. For this model equation, a regularity theory is provided, which is also of independent interest; it is shown in particular that the effect of boundary data that are square integrable (in time and space) is the same displayed by the wave equation. Then, a general picture of the (interior) regularity of solutions corresponding to homogeneous boundary conditions is specifically derived for the MGT equation in various functional settings. This confirms the gain of one unity in space regularity for the time derivative of the unknown, a feature that sets the MGT equation apart from other partial differential equation models for wave propagation. The adopted perspective and method of proof enable us to attain as well boundary regularity results for both the integro-differential equation and the MGT equation.