Well-posedness of the time-periodic Jordan-Moore-Gibson-Thompson equation

TL;DR

This paper establishes well-posedness and stability of the time-periodic Jordan-Moore-Gibson-Thompson equation, including the singular limit as relaxation time approaches zero, with implications for nonlinear acoustics and inverse problems.

Contribution

It provides the first comprehensive well-posedness analysis of the periodic JMGT equation, including gradient nonlinearities and boundary conditions, and demonstrates Lipschitz differentiability of the source-to-state map.

Findings

Proved well-posedness of periodic JMGT equations.

Established energy bounds and stability results.

Validated the singular limit as relaxation time vanishes.

Abstract

Motivated by applications of nonlinear ultrasonics under continuous wave excitation, we study the Jordan-Moore-Gibson-Thompson (JMGT) equation -- a third order in time quasilinear PDE -- under time periodicity conditions. Here the coefficient of the third order time derivative is the so-called relaxation time and a thorough understanding of the limiting behaviour for vanishing relaxation time is essential to link these JMGT equations to classical second order models in nonlinear acoustics, As compared to the meanwhile well understood initial value problem for JMGT, the periodic setting poses substantial challenges due to a loss of temporal regularity, while the analysis still requires an $L^\infty$ control of solutions in space and time in order to maintain stability or equivalently, to avoid degeneracy of the second time derivative coefficient. We provide a full well-posedness analysis with and without gradient nonlinearity, as relevant for modelling non-cumulative nonlinear effects, under practically relevant mixed boundary conditions. The source-to-state map is thus well-defined and we additionally show it to be Lipschitz continuously differentiable, a result that is useful for inverse problems applications such as acoustic nonlinearity tomography. The energy bounds derived for the well-posedness analysis of periodic JMGT equations also allow to fully justify the singular limit for vanishing relaxation time.