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

TL;DR

This paper analyzes the regularity of solutions to the Moore-Gibson-Thompson equation, revealing its unique regularity properties and connections to wave equations with memory, with implications for wave propagation models.

Contribution

It provides a new regularity theory for the MGT equation, highlighting its distinctive gain in spatial regularity and boundary trace regularity, using a wave equations with memory perspective.

Findings

Boundary data with square integrability affects solutions similarly to wave equations.

The MGT equation exhibits a gain of one spatial regularity in the time derivative.

Sharp regularity results for boundary traces are established.

Abstract

We undertake a regularity analysis of the solutions to initial/boundary value problems 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 of also 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 wave equations. 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 PDE models for wave propagation. The adopted perspective and method of proof enables us to attain as well the (sharp) regularity of boundary traces.