Decay rates for the Moore-Gibson-Thompson equation with memory
Hizia Bounadja, Belkacem Said Houari
Abstract
<p style='text-indent:20px;'>The main goal of this paper is to investigate the existence and stability of the solutions for the Moore–Gibson–Thompson equation (MGT) with a memory term in the whole spaces <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula>. The MGT equation arises from modeling high-frequency ultrasound waves as an alternative model to the well-known Kuznetsov's equation. First, following [<xref ref-type="bibr" rid="b8">8</xref>] and [<xref ref-type="bibr" rid="b26">26</xref>], we show that the problem is well-posed under an appropriate assumption on the coefficients of the system. Then, we built some Lyapunov functionals by using the energy method in Fourier space. These functionals allows us to get control estimates on the Fourier image of the solution. These estimates of the Fourier image together with some integral inequalities lead to the decay rate of the <inline-formula><tex-math id="M2">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula>-norm of the solution. We use two types of memory term here: type Ⅰ memory term and type Ⅲ memory term. Decay rates are obtained in both types. More precisely, decay rates of the solution are obtained depending on the exponential or polynomial decay of the memory kernel. More importantly, we show stability of the solution in both cases: a subcritical range of the parameters and a critical range. However for the type Ⅰ memory we show in the critical case that the solution has the regularity-loss property.