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

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 Racke and Said-Houari (Commun Contemp Math. 1–39, 2019. https://doi.org/10.1142/S0219199720500698) 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 Racke and Said-Houari (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 \(L^1(\mathbb {R}^3)\) 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 Racke and Said-Houari (2019) in order to prove the decay estimates of the solution.