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

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*} τ\partial_t^3ψ+\partial_t^2ψ+\mathcal{A}ψ+(δ+τ)\mathcal{A}\partial_tψ=(1+\tfrac{B}{2A})\partial_t[(\partial_tψ)^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.