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

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 behavior 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 a pointwise (in space and time) control on the magnitude of solutions in order to maintain stability or equivalently, to avoid degeneracy of the second time derivative coefficient.

We provide a full well-posedness analysis both in the presence and absence of gradient nonlinearity, as relevant for modeling non-cumulative nonlinear effects, under practically relevant mixed boundary conditions. The source-to-state map is thus well-defined and in addition we 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.