Global existence for the Jordan--Moore--Gibson--Thompson equation in Besov spaces

TL;DR

This paper establishes the global existence and optimal decay rates of solutions to the Jordan--Moore--Gibson--Thompson equation in Besov spaces, relaxing initial data regularity assumptions compared to previous studies.

Contribution

It proves global existence and decay of solutions in Besov spaces with minimal regularity, extending previous results by removing the L^1 initial data assumption.

Findings

Global existence of solutions in Besov spaces.

Optimal decay rates for solutions.

Reduced regularity assumptions on initial data.

Abstract

In this paper, we consider the Cauchy problem of a model in nonlinear acoustic, named the Jordan--Moore--Gibson--Thompson equation. This equation arises as an alternative model to the well-known Kuznetsov equation in acoustics. We prove global existence and optimal time decay of solutions in Besov spaces with a minimal regularity assumption on the initial data, lowering the regularity assumption required in \cite{Racke_Said_2019} for the proof of the global existence. Using a time-weighted energy method with the help of appropriate Lyapunov-type estimates, we also extend the decay rate in \cite{Racke_Said_2019} and show an optimal decay rate of the solution for initial data in the Besov space $\dot{{B}}_{2,\infty}^{-3/2}(\mathbb{R}^3)$, which is larger than the Lebesgue space $L^1(\R^3)$ due to the embedding $L^1(\mathbb{R}% ^3)\hookrightarrow \dot{{B}}_{2,\infty}^{-3/2}(\mathbb{R}^3)$. Hence we removed the $L^1$-assumption on the initial data required in \cite{Racke_Said_2019} in order to prove the decay estimates of the solution.