# On the Jordan-Moore-Gibson-Thompson equation: well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time

## TL;DR

This paper analyzes the Jordan-Moore-Gibson-Thompson equation, establishing well-posedness with quadratic gradient nonlinearity and examining the singular limit as relaxation time vanishes, connecting it to classical nonlinear acoustics models.

## Contribution

It proves well-posedness for the equation with gradient nonlinearity and studies the singular limit to classical models, using energy estimates and fixed-point methods.

## Key findings

- Well-posedness in velocity potential formulation.
- Convergence to Kuznetsov and Westervelt models as relaxation time vanishes.
- Numerical experiments support theoretical results.

## Abstract

In this paper, we consider the Jordan-Moore-Gibson-Thompson equation, a third order in time wave equation describing the nonlinear propagation of sound that avoids the infinite signal speed paradox of classical second order in time strongly damped models of nonlinear acoustics, such as the Westervelt and the Kuznetsov equation. We show well-posedness in an acoustic velocity potential formulation with and without gradient nonlinearity, corresponding to the Kuznetsov and the Westervelt nonlinearities, respectively. Moreover, we consider the limit as the parameter of the third order time derivative that plays the role of a relaxation time tends to zero, which again leads to the classical Kuznetsov and Westervelt models. To this end, we establish appropriate energy estimates for the linearized equations and employ fixed-point arguments for well-posedness of the nonlinear equations. The theoretical results are illustrated by numerical experiments.