The vanishing relaxation time behavior of multi-term nonlocal Jordan-Moore-Gibson-Thompson equations

TL;DR

This paper studies the behavior of generalized multi-term nonlocal Jordan-Moore-Gibson-Thompson equations in nonlinear acoustics, focusing on their vanishing relaxation time limit and the conditions for uniform bounds on solutions.

Contribution

It introduces a generalized class of JMGT equations with singular memory kernels and analyzes their vanishing relaxation time behavior, highlighting the role of kernel properties and nonlinearities.

Findings

Established uniform bounds for solutions depending on kernel regularity and coercivity.

Identified conditions relating kernel behavior to the nonlinear terms for the singular limit.

Connected the generalized equations to classical wave equations through the vanishing relaxation limit.

Abstract

The family of Jordan-Moore-Gibson-Thompson (JMGT) equations arises in nonlinear acoustics when a relaxed version of the heat flux law is employed within the system of governing equations of sound motion. Motivated by the propagation of sound waves in complex media with anomalous diffusion, we consider here a generalized class of such equations involving two (weakly) singular memory kernels in the principal and non-leading terms. To relate them to the second-order wave equations, we investigate their vanishing relaxation time behavior. The key component of this singular limit analysis are the uniform bounds for the solutions of these nonlinear equations of fractional type with respect to the relaxation time. Their availability turns out to depend not only on the regularity and coercivity properties of the two kernels, but also on their behavior relative to each other and the type of nonlinearity present in the equations.