Electronic Theses and Dissertations



Document Type


Degree Name

Doctor of Philosophy


Mathematical Sciences

Committee Chair

Irena Lasiecka

Committee Member

Roberto Triggiani

Committee Member

Hongqiu Chen

Committee Member

Tsz Ho Chan


The (third-order in time) JMGT equation is a nonlinear (quasilinear) Partial Differential Equation (PDE) model introduced to describe the acoustic velocity potential in ultrasound wave propagation. One begins with the parabolic Westervelt equation governing the dynamics of the pressure in nonlinear acoustic waves. In its derivation from constitutive laws, one then replaces the Fourier law with the Maxwell-Cattaneo law, to avoid the paradox of the infinite speed of propagation. This process then gives rise to a new third time derivative term, with a small constant coefficient $\tau$, referred to as relaxation time. As a consequence, the mathematical structure of the underlying model changes drastically from the parabolic character of the Westervelt model (whose linear part generates a s.c, analytic semigroup) to the hyperbolic-like character of the JMGT model (whose linear part generates a s.c, group on a suitable function space). This is a particularly delicate issue since the $\tau-$ dynamics is governed by a generator which is singular as $\tau\rightarrow 0.$ It is therefore of both mathematical and physical interest to analyze the asymptotic behavior of hyperbolic solutions of the JMGT model as the relaxation parameter $\tau \geq 0 $ tends to zero. In particular, it will be shown that for suitably calibrated initial data one obtains at the limit exponentially time-decaying solutions. The rate of convergence allows one then to estimate the relaxation time needed for the signal to reach the target. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences. These include applications to welding, lithotripsy, ultrasound technology, noninvasive treatment of kidney stones.


Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to ProQuest