Global well-posedness of the Cauchy problem for the Jordan--Moore--Gibson--Thompson equation with arbitrarily large higher-order Sobolev norms

TL;DR

This paper establishes global existence and decay properties for solutions to the 3D Jordan--Moore--Gibson--Thompson equation in nonlinear acoustics, allowing large higher-order norms in initial data.

Contribution

It proves global well-posedness with small lower-order Sobolev norms and large higher-order norms, and introduces a new decay estimate without the $L^1$-assumption.

Findings

Global existence for solutions with small lower-order norms.

Decay estimates for the linearized model.

Removal of the $L^1$-assumption on initial data.

Abstract

In this paper, we consider the 3D Jordan--Moore--Gibson--Thompson equation arising in nonlinear acoustics. First, we prove that the solution exists globally in time provided that the lower order Sobolev norms of the initial data are considered to be small, while the higher-order norms can be arbitrarily large. This improves some available results in the literature. Second, we prove a new decay estimate for the linearized model and removing the $L^1$-assumption on the initial data. The proof of this decay estimate is based on the high-frequency and low-frequency decomposition of the solution together with an interpolation inequality related to Sobolev spaces with negative order.