Global in-time existence of solutions for the complex-valued Jordan-Moore-Gibson-Thompson equations of Westervelt-type under different conditions on initial data

TL;DR

This paper proves the global existence of solutions for complex-valued Jordan-Moore-Gibson-Thompson equations of Westervelt-type under various initial data conditions, including rough data and data with different regularities, without requiring smallness assumptions.

Contribution

It establishes new global existence results for the JMGT equations with less restrictive initial data conditions, including rough data and data with different Sobolev regularities.

Findings

Global existence without smallness of initial data.

Results for rough initial data with Fourier support restrictions.

Existence for data with different Sobolev and Lebesgue regularities.

Abstract

We are interested in the global in-time existence of solutions for the complex-valued Jordan-Moore-Gibson-Thompson (JMGT) equations of Westervelt-type, namely, \begin{align*} \tau\partial_t^3\psi+\partial_t^2\psi+\mathcal{A}\psi+(\delta+\tau)\mathcal{A}\partial_t\psi=(1+\tfrac{B}{2A})\partial_t[(\partial_t\psi)^2] \end{align*} in the whole space $\mathbb{R}^n$, with $\tau,\delta,\frac{B}{A}\in\mathbb{R}_+$ and the fractional Laplacian $\mathcal{A}:=(-\Delta)^{\sigma}$ equipping $\sigma\in\mathbb{R}_+$. Our aims are twofold. For one thing, by considering the rough initial data with their Fourier support restrictions in a suitable subset of first octant, we demonstrate a global in-time existence result without requiring the smallness of initial data. For another, by removing these Fourier support restrictions, we prove another global in-time existence result for the equivalent strongly coupled JMGT systems, where the real and imaginary parts of initial data, respectively, belong to regular Sobolev spaces with different additional Lebesgue integrabilities.