\section{Introduction} 
Nonlinear acoustics arise in numerous applications, for instance, high-intensity ultrasonic waves have been used in medical imaging and therapy, ultrasound cleaning and welding (see, for example, \cite{Abramov-1999,Dreyer-Krauss-Bauer-Ried-2000,Kaltenbacher-Landes-Hoffelner-Simkovics-2002}). There are several mathematical models to describe nonlinear acoustics phenomena, e.g. Kuznetsov's models, Blackstock's models, Jordan-Moore-Gibson-Thompson (JMGT) models, Khokhlov-Zabolotskaya-Kuznetsov (KZK) models, etc. In the present paper, we focus on some qualitative properties of solutions to the JMGT equation and its corresponding linearized equation, i.e. the Moore-Gibson-Thompson (MGT) equation.
\subsection{Background of the MGT and JMGT equations}
By combining the conservations of mass, momentum and energy associated with the Fourier's law of heat conduction
\begin{align}\label{Fourier-law}
	\mathbf{q}=-\kappa\nabla \Theta,
\end{align} 
with the thermal conductivity $\kappa>0$, where $\mathbf{q}=\mathbf{q}(t,x)$ and $\Theta=\Theta(t,x)$ denote  the heat flux vector and the absolute temperature, moreover, by considering the state equation and neglecting all third- and higher-order terms in the deviations when combining the governing equations under the irrotational flow, we can derive one of the most established models in nonlinear acoustics, i.e. Kuznetsov's equation
\begin{align*}
	\varphi_{tt}-c_0^2\Delta\varphi-\delta\Delta\varphi_t=\partial_t\left(\frac{B}{2A}(\varphi_t)^2+|\nabla\varphi|^2\right),
\end{align*}
where the scalar unknown $\varphi=\varphi(t,x)\in\mb{R}$ denotes the acoustic velocity potential under the Fourier's law of heat conduction. The constant $c_0>0$ stands for the speed of sound in the undisturbed gas, $\delta>0$ is referred to the diffusivity of sound, and the constant ratio $B/A$ indicates the nonlinearity of the equation of state for a given fluid.

It is well-known that the application of the Fourier's law \eqref{Fourier-law} in the derivation of the mathematical models for acoustic waves leads to an infinite signal speed paradox. Thus, it seems to be unnatural in wave propagations. Indeed, the Cattaneo's law of heat conduction 
\begin{align}\label{Cattaneo-law}
	\tau\mathbf{q}_t+\mathbf{q}=-\kappa\nabla \Theta,
\end{align}
where the positive constant $\tau$ stands for the small thermal relaxation such that $0<\tau\ll 1$, can be used instead to eliminate this paradox for acoustic waves. The combination of three previous conservation laws associated with the Cattaneo's law \eqref{Cattaneo-law}  and the state equation again leads to the third-order (in time) nonlinear hyperbolic partial differential equation
\begin{align}\label{JMGT-ALL}
	\tau \psi_{ttt}+\psi_{tt}-c_0^2\Delta \psi-(\delta+\tau c_0^2)\Delta \psi_t=\partial_t\left(\frac{B}{2A}(\psi_t)^2+|\nabla\psi|^2\right)
\end{align}
with the acoustic velocity potential $\psi=\psi(t,x)\in\mb{R}$ under the Cattaneo's law of heat conduction. In the literature, this model is the so-called Jordan-Moore-Gibson-Thompson equation and we refer to \cite{Jordan-2014} for its derivation.

In recent years, the corresponding linearized model to \eqref{JMGT-ALL}, namely,
\begin{align}\label{MGT-ALL}
	\tau \psi_{ttt}+\psi_{tt}-c_0^2\Delta \psi-(\delta+\tau c_0^2)\Delta \psi_t=0,
\end{align}
has caught a lot of attentions (see, \cite{Kaltenbacher-Lasiecka-Marchand-2011,Marchand-McDevitt-Triggiani-2012,Conejero-Lizama-Rodenas-2015,Dell-Pata-2017,Pellicer-Said-2017,B-L-2019,Chen-Ikehata-2021} and references therein), where $\delta=0$ is an important threshold to distinguish behaviors of solution. Particularly, when $\delta>0$, it is called \emph{the dissipative case} since the solution is exponentially stable on smooth bounded domains; when $\delta=0$, it is called \emph{the conservative case} since an energy of the equation is conserved; when $\delta<0$, it is called \emph{the chaotic case} since the solution is unstable. Actually, the Cauchy problem for \eqref{MGT-ALL} in the dissipative case was studied by \cite{Pellicer-Said-2017} firstly, where some decay estimates of energy for higher dimensions have been derived by applying energy methods associated with Lyapunov's functional in the Fourier space. Later, the authors of \cite{Chen-Ikehata-2021} improved results in the paper \cite{Pellicer-Said-2017} by considering the combined effect of dissipation and dispersion, which allows us to get optimal growth or decay estimates. Moreover, some singular limit results with respect to the small thermal relaxation were obtained by applying classical energy methods for the integral unknown $\Psi(t,x):=\int_0^t\psi(\eta,x)\mathrm{d}\eta$. However, since the application of Hardy's inequality, they restricted the singular limit for the acoustic velocity potential (the solution itself) holding only when $n\geqslant 3$ with the weighted $L^2$ assumption on initial data.

Let us turn to the JMGT equation \eqref{JMGT-ALL}, which has been studied by \cite{Kaltenbacher-Lasiecka-Pos-2012,Kaltenbacher-Nikolic-2019,Racke-Said-2020,B-L-2020,Said-2021} and references therein. Additionally, the semilinear MGT equation  in the conservative case for $\mb{R}^n$ has been investigated by \cite{Chen-Palmieri-2020,ChenPalmieri201902}. Concerning the Cauchy problem for \eqref{JMGT-ALL} in the dissipative case, \cite{Racke-Said-2020} demonstrated global (in time) well-posedness result in 3D by using higher-order energy methods together with a bootstrap argument, and derived some decay estimates of solutions with weighted $L^1$ assumption on initial data. Quite recently, the author of \cite{Said-2021} improved the global (in time) existence result for 3D, where the smallness assumption for higher-order regular data was removed, whose idea is to utilize negative Sobolev spaces. To the best of the author's knowledge, the application of energy method to prove global (in time) existence of solutions required some suitable constructions of energy. For this reason, the total energy estimates rather than the estimates for each component will be established. Moreover, some higher-order regularity for initial data are assumed (see Remarks \ref{Rem_02} and \ref{Rem_03} later).
\subsection{Main purposes of the paper}
In the present paper we will improve the known results in the previous researches for the MGT equation and the JMGT equation in the whole space $\mb{R}^n$. More precisely, we take into account the following Cauchy problem for the JMGT equation (we assume from now on $c_0^2=1$ without loss of generality):
\begin{align}\label{JMGT_Dissipative}
	\begin{cases}
		\displaystyle{\tau \psi_{ttt}+\psi_{tt}-\Delta \psi-(\delta+\tau)\Delta \psi_t=\partial_t\left(\frac{B}{2A}(\psi_t)^2+|\nabla\psi|^2\right),}&x\in\mb{R}^n,\ t>0,\\
		\psi(0,x)=\psi_0(x),\ \psi_t(0,x)=\psi_1(x),\ \psi_{tt}(0,x)=\psi_2(x),&x\in\mb{R}^n,
	\end{cases}
\end{align}
where $0<\tau\ll 1$ to be a small constant, $\delta>0$ and $A,B>0$. Moreover, we study the corresponding linearized Cauchy problem to \eqref{JMGT_Dissipative}, i.e. the MGT equation,
\begin{align}\label{Linear_MGT_Dissipative}
	\begin{cases}
		\tau \psi_{ttt}+\psi_{tt}-\Delta \psi-(\delta+\tau)\Delta \psi_t=0,&x\in\mb{R}^n,\ t>0,\\
		\psi(0,x)=\psi_0(x),\ \psi_t(0,x)=\psi_1(x),\ \psi_{tt}(0,x)=\psi_2(x),&x\in\mb{R}^n.
	\end{cases}
\end{align}
The idea to get some improvements throughout this work is motivated by recent studies of nonlinear damped wave models, e.g. semilinear wave equations with scale-invariant damping \cite{Palmieri-Reissig-2018}. To be specific, we will regard \eqref{Linear_MGT_Dissipative} and \eqref{JMGT_Dissipative} as the scalar wave-like model (strictly hyperbolic equation) equipping some damping mechanisms.

Our first aim in this paper is to understand the refined singular limit for the MGT equation \eqref{Linear_MGT_Dissipative} in Section \ref{Section_Singular_Limits}. By formally taking $\tau=0$ in \eqref{Linear_MGT_Dissipative}, it immediately turns to the linear Kuznetsov's equation
\begin{align}\label{Linear_Kuz_Dissipative}
	\begin{cases}
		\varphi_{tt}-\Delta \varphi-\delta\Delta\varphi_t=0,&x\in\mb{R}^n,\ t>0,\\
		\varphi(0,x)=\varphi_0(x),\ \varphi_t(0,x)=\varphi_1(x),&x\in\mb{R}^n.
	\end{cases}
\end{align}
In other words, the second-order (in time) $2$-evolution equation \eqref{Linear_Kuz_Dissipative} seems to be the limit equation for the third-order (in time) strictly hyperbolic equation \eqref{Linear_MGT_Dissipative}, where an initial condition is missing (that is the reason for singular limit). It would be interesting to study the strong convergence (with respect to the small thermal relaxation $\tau$) for the solution between these models. As we mentioned in the above, although the recent paper \cite{Chen-Ikehata-2021} has investigated this property already, the lower-dimensional cases $n=1,2$ for the strong convergence is still unknown since some technical difficulties from Hardy's inequality. In the forthcoming part, we will overcome this difficulty by using suitable energy method in the Fourier space. Then, we will derive global (in time) convergence result in the $L^2$ norm with linear rate $\tau$ for all dimensions. 

To understand behaviors of solution for the small parameter $\tau$, not only singular limit (first-order profile), but also higher-order profiles are interesting. For this reason, we propose some remarks in Subsection \ref{Sub_Sec_3.3} for formal asymptotic analysis with respect to $0<\tau\ll 1$. Due to the non-matching initial condition, an initial layer may appear. In Subsection \ref{Sub_Sec_3.3}, we also propose a conjecture on the condition for appearance of initial layer, i.e.
\begin{align*}
	\psi_0(x)\equiv\varphi_0(x),\ \ \psi_1(x)\equiv\varphi_1(x),\ \ \psi_2(x)\not\equiv\Delta\psi_0(x)+\delta\Delta\psi_1(x).
\end{align*}
In a formal way by perturbation theory, there is an initial layer profile when we consider the third-order asymptotic profile, precisely,
\begin{align*}
	\psi_{\mathrm{Lay}}(t,x;\tau)=\tau^2\mathrm{e}^{-\frac{t}{\tau}}\left(\psi_2(x)-(\Delta\psi_0(x)+\delta\Delta\psi_1(x))\right),
\end{align*}
which is changed rapidly for $\tau\downarrow0$ in finite time. The initial layer profile is relatively new in the field of acoustic waves from the author's viewpoint.

The second aim in this paper is to derive global (in time) existence result for the JMGT equation \eqref{JMGT_Dissipative} for any $n\geqslant 2$ in Section \ref{Section_GESDS_JMGT}, whose proof is based on the optimal growth or decay estimates for the linearized problem. Especially, our proof is easier than the one in \cite{Racke-Said-2020}, and our approach can be applied in other models in nonlinear acoustics (see Remark \ref{Rem_01}). To construct the suitable time-weighted Sobolev space, we will extend the result from the recent work \cite{Chen-Ikehata-2021} by taking nontrivial initial data for the solution and its higher-order derivatives in Section \ref{Sec_Linear} as a preparation. Furthermore, to perform the estimates for the nonlinear term, we will apply some tools in Harmonic Analysis (see Appendix \ref{Appendix}), including the fractional Gagliardo-Nirenberg inequality, the fractional Leibniz rule and the fractional Sobolev embedding. As byproduct of our approach, estimates of solution to the nonlinear problem will be shown, which are coincide with the optimal estimates for the corresponding linearized Cauchy problem.

Eventually, some remarks for open problems on the JMGT equation in $\mb{R}^n$ will be proposed in Section \ref{Sec_Final} to complete the paper.

\medskip


\noindent\textbf{Notations:} In this paper,  $\dot{H}^s_q$ with $s\geqslant0$ and $1\leqslant q<\infty$, denote Riesz potential spaces based on $L^q$. Furthermore, $|D|^s$ and $\langle D\rangle^s$ with $s\geqslant0$ stand for the pseudo-differential operators with symbol $|\xi|^s$ and $\langle \xi\rangle^s$, respectively, where the Japanese bracket means $\langle\xi\rangle^2:=1+|\xi|^2$. Moreover, let us define the following zones of the phase space: $\ml{Z}_{\intt}(\varepsilon):=\{\xi\in\mb{R}^n:|\xi|<\varepsilon\ll1\}$, $\ml{Z}_{\bdd}(\varepsilon,N):=\{\xi\in\mb{R}^n:\varepsilon\leqslant |\xi|\leqslant N\}$, $\ml{Z}_{\extt}(N):=\{\xi\in\mb{R}^n:|\xi|> N\gg1\}$. The cut-off functions $\chi_{\intt}(\xi),\chi_{\bdd}(\xi),\chi_{\extt}(\xi)\in \mathcal{C}^{\infty}$ having their supports in the zone $\ml{Z}_{\intt}(\varepsilon)$, $\ml{Z}_{\bdd}(\varepsilon/2,2N)$ and $\ml{Z}_{\extt}(N)$, respectively, so that $\chi_{\intt}(\xi)+\chi_{\bdd}(\xi)+\chi_{\extt}(\xi)=1$ for all $\xi \in \mb{R}^n$.

\section{Preliminaries: Some estimates of solutions to the MGT equation}\label{Sec_Linear}
In this part, we will prepare some estimates of solution  and its derivatives for the MGT equation \eqref{Linear_MGT_Dissipative}, which will contributes to investigations of the nonlinear problem \eqref{JMGT_Dissipative}. According to the recent paper \cite{Chen-Ikehata-2021}, the solution to the linear Cauchy problem \eqref{Linear_MGT_Dissipative} in the phase space can be expressed by using the partial Fourier transform with respect to spatial variables 
$
	\widehat{\psi}(t,\xi):=\ml{F}_{x\to\xi}(\psi(t,x))
$
 such that in the case of pairwise distinct eigenvalues, it holds
 \begin{align}\label{Representation_Fourier_u}
 	\widehat{\psi} (t,\xi)=\widehat{K}_0(t,|\xi|)\widehat{\psi}_0 (\xi)+\widehat{K}_1(t,|\xi|)\widehat{\psi}_1 (\xi)+\widehat{K}_2(t,|\xi|)\widehat{\psi}_2 (\xi),
 \end{align}
 where the kernels in the phase space have the representations
 \begin{align*}
 	\widehat{K}_0(t,|\xi|)&:=\sum\limits_{j=1,2,3}\frac{\exp(\lambda_j(|\xi|)t)\prod_{k=1,2,3,\ k\neq j}\lambda_k(|\xi|)}{\prod_{k=1,2,3,\ k\neq j}(\lambda_j(|\xi|)-\lambda_k(|\xi|))},\\
 	\widehat{K}_1(t,|\xi|)&:=-\sum\limits_{j=1,2,3}\frac{\exp(\lambda_j(|\xi|)t)\sum_{k=1,2,3,\ k\neq j}\lambda_k(|\xi|)}{\prod_{k=1,2,3,\ k\neq j}(\lambda_j(|\xi|)-\lambda_k(|\xi|))},\\
 	\widehat{K}_2(t,|\xi|)&:=\sum\limits_{j=1,2,3}\frac{\exp(\lambda_j(|\xi|)t)}{\prod_{k=1,2,3,\ k\neq j}(\lambda_j(|\xi|)-\lambda_k(|\xi|))}.
 \end{align*}
Here, the eigenvalues $\lambda_j=\lambda_j(|\xi|)$ for $j=1,2,3$ are the roots to
the $|\xi|$-dependent cubic equation
\begin{align}\label{Cubic_Equation}
	\tau\lambda^3+\lambda^2+(\delta+\tau)|\xi|^2\lambda+|\xi|^2=0.
\end{align} 
From the asymptotic expansions and stability analysis in \cite{Pellicer-Said-2017,Chen-Ikehata-2021}, it holds
\begin{itemize}
	\item $\lambda_{1,2}(|\xi|)=\pm i|\xi|-\frac{\delta}{2}|\xi|^2+\ml{O}(|\xi|^3),\ \lambda_3(|\xi|)=-\frac{1}{\tau}+\delta|\xi|^2+\ml{O}(|\xi|^3)$ if $|\xi|\to0$;
	\item $\lambda_1(|\xi|)=-\frac{1}{\delta+\tau}+\ml{O}(|\xi|^{-1}), \
	\lambda_{2,3}(|\xi|)=\pm i\sqrt{\frac{\delta+\tau}{\tau}}|\xi|-\frac{\delta}{2\tau(\delta+\tau)}+\ml{O}(|\xi|^{-1})$ if $|\xi|\to\infty$;
	\item $\Re\lambda_{j}(|\xi|)<0$ for $j=1,2,3$ if $|\xi|\not\to0$ and $|\xi|\not\to\infty$.
\end{itemize}
\begin{remark}
	The case for multiple eigenvalues, i.e. the zero-discriminant, appears only when $\xi\in\ml{Z}_{\bdd}(\varepsilon,N)$. Namely, although the solution formula \eqref{Representation_Fourier_u} associated with the corresponding kernels is valid in the case of pairwise distinct eigenvalues, we will just use the solution formula \eqref{Representation_Fourier_u} for $\xi\in\ml{Z}_{\intt}(\varepsilon)\cup\ml{Z}_{\extt}(N)$ that three roots to \eqref{Cubic_Equation} are pairwise distinct (see more detail in Section 4 of \cite{Pellicer-Said-2017} or Section 2.1 of \cite{Chen-Ikehata-2021}). Concerning the case for multiple eigenvalues, the exponential stability of solution when $\delta>0$ is favorable for derive some pointwise estimates for $\xi\in\ml{Z}_{\bdd}(\varepsilon,N)$.
\end{remark}

\begin{remark}
From the last statement of asymptotic expansions for the eigenvalues, one may easily see the instable situation for the MGT equation \eqref{Linear_MGT_Dissipative} if $\delta<0$ (the chaotic case). Indeed, the stability of solution is determined by large frequencies in the phase space. While $|\xi|\to\infty$ and $\delta<0$, we observe either $\Re\lambda_1(|\xi|)<0$ or $\Re\lambda_{2,3}(|\xi|)<0$. Namely, the positive real part of eigenvalues lead to instability.
\end{remark}

Let us analyze some asymptotic behaviors of the kernels for $|\xi|\to0$ and $|\xi|\to\infty$, respectively. We should underline that $\widehat{K}_2(t,|\xi|)$ has been estimated in an appropriate way in Section 2 of \cite{Chen-Ikehata-2021}, and the idea of this section is strongly motivated from it. For one thing, by direct computations, one derives
\begin{align*}
	\chi_{\intt}(\xi)|\partial_t^{\ell}\widehat{K}_0(t,|\xi|)|&\lesssim\chi_{\intt}(\xi)\left(|\xi|^{\ell}\mathrm{e}^{-\frac{\delta}{2}|\xi|^2t}+|\xi|^2\mathrm{e}^{-\frac{1}{\tau}t}\right)\ \ \mbox{if}\ \ \ell=0,1,2,\\
		\chi_{\intt}(\xi)|\partial_t^{\ell}\widehat{K}_1(t,|\xi|)|&\lesssim\begin{cases}
			\chi_{\intt}(\xi)\left(|\xi|^{-1}|\sin(|\xi|t)|\mathrm{e}^{-\frac{\delta}{2}|\xi|^2t}+|\xi|^2\mathrm{e}^{-\frac{1}{\tau}t}\right)&\mbox{if}\ \ \ell=0,\\
			\chi_{\intt}(\xi)\left(|\xi|^{\ell-1}\mathrm{e}^{-\frac{\delta}{2}|\xi|^2t}+|\xi|^2\mathrm{e}^{-\frac{1}{\tau}t}\right)&\mbox{if}\ \ \ell=1,2,
		\end{cases}\\
			\chi_{\intt}(\xi)|\partial_t^{\ell}\widehat{K}_2(t,|\xi|)|&\lesssim\begin{cases}
				\chi_{\intt}(\xi)\left(|\xi|^{-1}|\sin(|\xi|t)|\mathrm{e}^{-\frac{\delta}{2}|\xi|^2t}+\mathrm{e}^{-\frac{1}{\tau}t}\right)&\mbox{if}\ \ \ell=0,\\
				\chi_{\intt}(\xi)\left(|\xi|^{\ell-1}\mathrm{e}^{-\frac{\delta}{2}|\xi|^2t}+\mathrm{e}^{-\frac{1}{\tau}t}\right)&\mbox{if}\ \ \ell=1,2.
			\end{cases}
\end{align*}
For another, by the same way as before, in the case $|\xi|\to\infty$, we obtain the next pointwise estimates: 
\begin{align*}
	\chi_{\extt}(\xi)|\partial_t^{\ell}\widehat{K}_0(t,|\xi|)|&\lesssim\chi_{\extt}(\xi)\left(\mathrm{e}^{-\frac{1}{\delta+\tau}t}+|\xi|^{\ell-1}\mathrm{e}^{-\frac{\delta}{2\tau(\delta+\tau)}t}\right),\\
    \chi_{\extt}(\xi)|\partial_t^{\ell}\widehat{K}_1(t,|\xi|)|&\lesssim\chi_{\extt}(\xi)\left(|\xi|^{-2}\mathrm{e}^{-\frac{1}{\delta+\tau}t}+|\xi|^{\ell-1}\mathrm{e}^{-\frac{\delta}{2\tau(\delta+\tau)}t}\right),\\
	\chi_{\extt}(\xi)|\partial_t^{\ell}\widehat{K}_2(t,|\xi|)|&\lesssim\chi_{\extt}(\xi)\left(|\xi|^{-2}\mathrm{e}^{-\frac{1}{\delta+\tau}t}+|\xi|^{\ell-2}\mathrm{e}^{-\frac{\delta}{2\tau(\delta+\tau)}t}\right),
\end{align*}
for all $\ell=0,1,2$. Clearly, thanks to negative real parts of eigenvalues in the bounded zone, we are able to get
\begin{align*}
	\chi_{\bdd}(\xi)\left(|\partial_t^{\ell}\widehat{K}_0(t,|\xi|)|+|\partial_t^{\ell}\widehat{K}_1(t,|\xi|)|+|\partial_t^{\ell}\widehat{K}_2(t,|\xi|)|\right)\lesssim \chi_{\bdd}(\xi)\mathrm{e}^{-ct}
\end{align*}
with some positive constants $c>0$, which means exponential stabilities of kernels.

\begin{prop}\label{Prop_Estimate_Solution_itself}
	Let us consider $(\psi_0 ,\psi_1 ,\psi_2 )\in (H^{s+2} \cap L^1 )\times (H^{s+1} \cap L^1)\times (H^{s} \cap L^1)$ for $s\in[0,\infty)$. Then, the solution $\psi =\psi (t,x)$ to the Cauchy problem \eqref{Linear_MGT_Dissipative} fulfills the following estimates:
	\begin{align}
		\|\psi (t,\cdot)\|_{L^2 }&\lesssim\ml{D}_{n}(t)\|(\psi_0 ,\psi_1 ,\psi_2 )\|_{(L^2 \cap L^1 )^3},\label{Est_01}\\
		\|\,|D|^{s+2}\psi (t,\cdot)\|_{L^2 }&\lesssim(1+t)^{-\frac{1}{2}}\|(\psi_0 ,\psi_1 ,\psi_2 )\|_{\dot{H}^{s+2}\times \dot{H}^{s+1} \times \dot{H}^{s} },\label{Est_08}\\
		\|\,|D|^{s+2}\psi (t,\cdot)\|_{L^2 }&\lesssim(1+t)^{-\frac{1}{2}-\frac{s}{2}-\frac{n}{4}}\|(\psi_0 ,\psi_1 ,\psi_2 )	\|_{(\dot{H}^{s+2} \cap L^1 )\times(\dot{H}^{s+1} \cap L^1 )\times (\dot{H}^{s} \cap L^1)},\label{Est_06}
	\end{align}
where the time-dependent coefficients are defined by
\begin{align*}
	\ml{D}_n(t):=\begin{cases}
		(1+t)^{\frac{1}{2}}&\mbox{if}\ \ n=1,\\
		(\ln (\mathrm{e}+t))^{\frac{1}{2}}&\mbox{if}\ \ n=2,\\
		(1+t)^{\frac{1}{2}-\frac{n}{4}}&\mbox{if} \ \ n\geqslant 3.
	\end{cases}
\end{align*}
\end{prop}
\begin{proof}
	To prove our desired results, we will use H\"older's inequality and the Hausdorff-Young inequality such that
	\begin{align*}
		\|\chi_{\intt}(\xi)\hat{f}_1\hat{f}_2\|_{L^2}&\lesssim\|\chi_{\intt}(\xi)\hat{f}_1\|_{L^2}\|\hat{f}_2\|_{L^{\infty}}\lesssim\|\chi_{\intt}(\xi)\hat{f}_1\|_{L^2}\|f_2\|_{L^1},\\
		\|\chi_{\intt}(\xi)\hat{f}_1\hat{f}_2\|_{L^2}&\lesssim\|\chi_{\intt}(\xi)\hat{f}_1\|_{L^{\infty}}\|\hat{f}_2\|_{L^2}\lesssim  \|\chi_{\intt}(\xi)\hat{f}_1\|_{L^{\infty}}\|f_2\|_{L^2},
	\end{align*}
where $f_1$ and $f_2$ denote the kernels and the corresponding initial data, respectively. Then, associated with the sharp estimates
	\begin{align*}
		\|\chi_{\intt}(\xi)|\xi|^s\mathrm{e}^{-c|\xi|^2t}\|_{L^2}&\lesssim (1+t)^{-\frac{s}{2}-\frac{n}{4}},\\
		\|\chi_{\intt}(\xi)|\xi|^s\mathrm{e}^{-c|\xi|^2t}\|_{L^{\infty}}&\lesssim(1+t)^{-\frac{s}{2}},\\
				\|\chi_{\intt}(\xi)|\xi|^{-1}|\sin(|\xi|t)|\mathrm{e}^{-c|\xi|^2t}\|_{L^2}&\lesssim \ml{D}_n(t),
	\end{align*}
	with $s\in[0,\infty)$, and exponential decay estimates for bounded and large frequencies, we follow the same approaches as the proofs of Theorems 2.1 and 2.2 in \cite{Chen-Ikehata-2021} to easily complete the derivations of all estimates. Here, the requirements of the regularity for initial data come from the derived estimates when $|\xi|\to\infty$.
\end{proof}

\begin{prop}\label{Prop_Estimate_Time_Derivative}
Let us consider $(\psi_0 ,\psi_1 ,\psi_2 )\in(H^{s+1}\cap L^1 )^2\times (H^{s} \cap L^1) $ for $\ell=1,2$ and $s\in[0,\infty)$.	Then, the first- and second-order time derivatives of the solution $\psi =\psi (t,x)$ to the Cauchy problem \eqref{Linear_MGT_Dissipative} fulfills the following estimates:
	\begin{align}
		\|\partial_t^{\ell}\psi (t,\cdot)\|_{L^2 }&\lesssim(1+t)^{\frac{1}{2}-\frac{\ell}{2}}\|(\psi_0 ,\psi_1 ,\psi_2 )\|_{(\dot{H}^{\ell-1} )^2\times L^2} ,\label{Est_05}\\
		\|\partial_t^{\ell}\psi (t,\cdot)\|_{L^2 }&\lesssim(1+t)^{\frac{1}{2}-\frac{\ell}{2}-\frac{n}{4}}\|(\psi_0 ,\psi_1 ,\psi_2 )\|_{(\dot{H}^{\ell-1}\cap L^1 )^2\times (L^2\cap L^1) },\label{Est_04}\\
		\|\,|D|^{s+2-\ell}\partial_t^{\ell}\psi (t,\cdot)\|_{L^2 }&\lesssim(1+t)^{-\frac{1}{2}}\|(\psi_0 ,\psi_1 ,\psi_2 )\|_{(\dot{H}^{s+1} )^2\times \dot{H}^{s} },\label{Est_09}\\
		\|\,|D|^{s+2-\ell}\partial_t^{\ell}\psi (t,\cdot)\|_{L^2 }&\lesssim(1+t)^{-\frac{1}{2}-\frac{s}{2}-\frac{n}{4}}\|(\psi_0 ,\psi_1 ,\psi_2 )\|_{(\dot{H}^{s+1}\cap L^1 )^2\times (\dot{H}^{s} \cap L^1) }.\label{Est_07}
	\end{align}
\end{prop}
\begin{proof}
With the aid of the derived pointwise estimates, and the same philosophy as the proof of Proposition \ref{Prop_Estimate_Solution_itself}, one can complete the proof.
\end{proof}
\begin{remark}
	The combined effect from dissipation and dispersion only appears when we estimate the solution itself. To be specific, concerning the term
	\begin{align*}
		|\xi|^{-1}|\sin(|\xi|t)|\mathrm{e}^{-\frac{\delta}{2}|\xi|^2t}\ \ \mbox{as}\ \ |\xi|\to0,
	\end{align*}
the singularity from $|\xi|^{-1}$ will be compensated by the oscillating part $|\sin(|\xi|t)|$ from the dispersion. Eventually, in the $L^p$ norm for some $p>1$, together with $\mathrm{e}^{-\frac{\delta}{2}|\xi|^2t}$ from the dissipation, it will provide some estimates of the solution itself. More detailed explanations have been shown in \cite{Chen-Ikehata-2021}.
\end{remark}


\section{Singular limit and initial layer for the MGT equation}\label{Section_Singular_Limits}
This section contributes to asymptotic analysis for the MGT equation with the small parameter $\tau$, namely,
\begin{align}\label{Linear_MGT_Dissipative_tau}
	\begin{cases}
		\tau \psi^{\tau}_{ttt}+\psi^{\tau}_{tt}-\Delta \psi^{\tau}-(\delta+\tau)\Delta \psi^{\tau}_t=0,&x\in\mb{R}^n,\ t>0,\\
		\psi^{\tau}(0,x)=\psi_0(x),\ \psi^{\tau}_t(0,x)=\psi_1(x),\ \psi^{\tau}_{tt}(0,x)=\psi_2(x),&x\in\mb{R}^n,
	\end{cases}
\end{align}
where $\psi^{\tau}=\psi^{\tau}(t,x)$ with $\delta>0$ and sufficiently small thermal relaxation $0<\tau\ll 1$.
 This section is organized as follows. In Subsection \ref{Sub_Sec_3.1}, we prepare a pointwise estimate for the inhomogeneous MGT equation. In Subsection \ref{Sub_Sec_3.2}, with the help of the derived pointwise estimate, singular limits in the framework of $L^{\infty}([0,\infty),L^p(\mb{R}^n))$ for $p=2,\infty$ and any $n\geqslant 1$ will be investigated. Finally, in Subsection \ref{Sub_Sec_3.3}, we will propose a condition for existence of initial layer and higher-order asymptotic profiles in the initial layer.

\subsection{The inhomogeneous MGT equation in the phase space}\label{Sub_Sec_3.1}
Throughout this part, we will consider some estimates of the solution $\hat{u}=\hat{u}(t,\xi)$ to the following inhomogeneous Cauchy problem for the MGT equation in the phase space:
\begin{align}\label{Inh_Linear}
\begin{cases}
	\tau \hat{u}_{ttt}+\hat{u}_{tt}+|\xi|^2\hat{u}+(\delta+\tau)|\xi|^2\hat{u}_t=\hat{f},&\xi\in\mb{R}^n,\ t>0,\\
	\hat{u}(0,\xi)=\hat{u}_t(0,\xi)=\hat{u}_{tt}(0,\xi)=0,&\xi\in\mb{R}^n,
\end{cases}
\end{align}
with a sufficiently small $\tau$, where $\hat{f}=\hat{f}(t,\xi)$ is a suitable function as a source term. In the next subsection, the precise choices of $\hat{u}$ and $\hat{f}$ will be given.
\begin{remark}
	We should underline that the condition $0<\tau<\delta$ when $0<\tau\ll 1$ is natural. From the physical viewpoint, the diffusivity of sound can be expressed by
	\begin{align*}
		\delta=\kappa\left(\frac{1}{c_{\mathrm{V}}}-\frac{1}{c_{\mathrm{P}}}\right)+\frac{4}{3}\mu_{\mathrm{V}}+\mu_{\mathrm{B}},
	\end{align*}
	where we denoted the constant pressure and constant volume by $c_{\mathrm{P}}$ and $c_{\mathrm{V}}$, respectively. Moreover, $\mu_{\mathrm{B}}$ and $\mu_{\mathrm{V}}$ stand for the bulk and shear viscosities. Consequently, for a given fluid in the consideration of acoustic waves, we can consider $0<\tau\ll 1$ such that $0<\tau<\delta$ always holds.
\end{remark}

 Let us state a pointwise estimate in the phase space.
\begin{prop}\label{Prop_Pointwise_HAT_u}
The solution $\hat{u}$ to the inhomogeneous Cauchy problem \eqref{Inh_Linear} fulfills the next pointwise estimates:
	\begin{align}\label{Pointwise}
	&\left|\frac{1}{2}\hat{u}_t+\tau\hat{u}_{tt}\right|^2+\frac{\tau}{\delta+\tau}|\xi|^2\left|\hat{u}+(\delta+\tau)\hat{u}_t\right|^2+\frac{\delta-\tau}{2(\delta+\tau)}|\xi|^2|\hat{u}|^2+\frac{1}{4}|\hat{u}_t|^2\notag\\
	&\qquad\leqslant\frac{1+4\tau(\delta-\tau)|\xi|^2}{8(\delta-\tau)|\xi|^2}\int_0^t|\hat{f}(s,\xi)|^2\mathrm{d}s.
	\end{align}
\end{prop}
\begin{proof}
To begin with the construction of suitable energy, by the equation in \eqref{Inh_Linear}, we can immediately obtain
\begin{align}\label{Eq_01}
	\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}\hat{u}_t+\tau\hat{u}_{tt}\right)=-\frac{1}{2}\hat{u}_{tt}-(\delta+\tau)|\xi|^2\hat{u}_t-|\xi|^2\hat{u}+\hat{f}.
\end{align}
Multiplying \eqref{Eq_01} by $\frac{1}{2}\bar{\hat{u}}_t+\tau\bar{\hat{u}}_{tt}$ and taking the real part of the resultant equation to arrive at
\begin{align*}
	\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left(\left|\frac{1}{2}\hat{u}_t+\tau\hat{u}_{tt}\right|^2\right)&=-\frac{\tau}{2}|\hat{u}_{tt}|^2-\frac{\delta+\tau}{2}|\xi|^2|\hat{u}_t|^2-\frac{1}{4}\Re(\hat{u}_{tt}\bar{\hat{u}}_t)-\frac{1}{2}|\xi|^2\Re(\hat{u}\bar{\hat{u}}_t)\\
	&\quad-\tau(\delta+\tau)|\xi|^2\Re(\hat{u}_t\bar{\hat{u}}_{tt})-\tau|\xi|^2\Re(\hat{u}\bar{\hat{u}}_{tt})+\Re\left(\hat{f}\left(\frac{1}{2}\bar{\hat{u}}_t+\tau\bar{\hat{u}}_{tt}\right)\right).
\end{align*}
Moreover, we directly get
\begin{align*}
	\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\tau}{\delta+\tau}|\xi|^2|\hat{u}+(\delta+\tau)\hat{u}_t|^2\right)&=\frac{\tau}{\delta+\tau}|\xi|^2\Re(\hat{u}_t\bar{\hat{u}})+\tau|\xi|^2|\hat{u}_t|^2+\tau|\xi|^2\Re(\hat{u}_{tt}\bar{\hat{u}})\\
	&\quad+\tau(\delta+\tau)|\xi|^2\Re(\hat{u}_{tt}\bar{\hat{u}}_t),
\end{align*}
and
\begin{align*}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\delta-\tau}{2(\delta+\tau)}|\xi|^2|\hat{u}|^2+\frac{1}{4}|\hat{u}_t|^2\right)=\frac{\delta-\tau}{2(\delta+\tau)}|\xi|^2\Re(\hat{u}_t\bar{\hat{u}})+\frac{1}{4}\Re(\hat{u}_{tt}\bar{\hat{u}}_t).
\end{align*}
Summarizing the above three equalities, we find
\begin{align*}
&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left(\left|\frac{1}{2}\hat{u}_t+\tau\hat{u}_{tt}\right|^2+\frac{\tau}{\delta+\tau}|\xi|^2|\hat{u}+(\delta+\tau)\hat{u}_t|^2+\frac{\delta-\tau}{2(\delta+\tau)}|\xi|^2|\hat{u}|^2+\frac{1}{4}|\hat{u}_t|^2\right)\\
&\quad=-\frac{\tau}{2}|\hat{u}_{tt}|^2-\frac{\delta-\tau}{2}|\xi|^2|\hat{u}_t|^2+	\Re\left(\hat{f}\left(\frac{1}{2}\bar{\hat{u}}_t+\tau\bar{\hat{u}}_{tt}\right)\right)\leqslant \frac{1+4\tau(\delta-\tau)|\xi|^2}{8(\delta-\tau)|\xi|^2}|\hat{f}|^2
\end{align*}
Due to the fact that $0<\tau< \delta$, integrating the last inequality over $[0,t]$, we complete the proof of our desired estimate.
\end{proof}
\begin{remark}
Comparing the classical energy in \cite{Chen-Ikehata-2021} with those in Proposition \ref{Prop_Pointwise_HAT_u}, we include the solution itself $\hat{u}$ in the ``phase space energy'', which is the crucial idea of our improvement.
\end{remark}
\subsection{Refined singular limit}\label{Sub_Sec_3.2}
Let us define the difference of solutions between the MGT equation \eqref{Linear_MGT_Dissipative_tau} and the linear Kuznetsov's equation \eqref{Linear_Kuz_Dissipative} by $u=u(t,x)$ such that $u:=\psi^{\tau}-\varphi$, which is a solution to the following inhomogeneous MGT equation:
\begin{align}
\begin{cases}
	\tau u_{ttt}+u_{tt}-\Delta u-(\delta+\tau)\Delta u_t=f,&x\in\mb{R}^n,\ t>0,\\
	u(0,x)=\psi_0(x)-\varphi_0(x),&x\in\mb{R}^n,\\
	u_t(0,x)=\psi_1(x)-\varphi_1(x),&x\in\mb{R}^n,\\
	u_{tt}(0,x)=\psi_2(x)-(\Delta\varphi_0(x)+\delta\Delta\varphi_1(x)),&x\in\mb{R}^n,
\end{cases}
\end{align}
where the source term $f=f(t,x)$ can be expressed by
\begin{align*}
	f(t,x)=-\delta\tau\Delta\varphi_{tt}(t,x).
\end{align*}
\begin{theorem}\label{Thm_Singular_Limt}
Let us assume $\varphi_0\equiv\psi_0\in H^4\cap L^1$ and $\varphi_1\equiv\psi_1\in H^3\cap L^1$ such that
\begin{align}\label{const_condition}
\psi_2\equiv\Delta \psi_0+\delta\Delta\psi_1.
\end{align}
Then, the following estimate:
\begin{align*}
	\sup\limits_{t\in[0,\infty)}\left\|\psi^{\tau}(t,\cdot)-\varphi(t,\cdot)\right\|_{L^2}\leqslant \tau C\left(\|\psi_0\|_{H^4\cap L^1}+\|\psi_1\|_{H^3\cap L^1}\right)
\end{align*}
holds for any $0<\tau\ll 1$, where $C$ is a positive constant independent of $\tau$. Furthermore, if initial data fulfill additional regularity $\psi_0\in H^5\cap L^1$ and $\psi_1\in H^4\cap L^1$, it holds
\begin{align*}
	\sup\limits_{t\in[0,\infty)}\left(\|\psi^{\tau}(t,\cdot)-\varphi(t,\cdot)\|_{H^1}+\|\psi_t^{\tau}(t,\cdot)-\varphi_t(t,\cdot)\|_{L^2}\right)\leqslant \tau C\left(\|\psi_0\|_{H^5\cap L^1}+\|\psi_1\|_{H^4\cap L^1}\right)
\end{align*}
for any $0<\tau\ll 1$.
\end{theorem}
\begin{remark}
We improve Theorem 4.2 in \cite{Chen-Ikehata-2021} by the next points:\begin{itemize}
	\item our result is valid for any $n\geqslant 1$, particularly, the lower dimensions $n=1,2$;
	\item our result is valid for any $t>0$, namely, global (in time) convergence in the framework of $L^2$.
\end{itemize}
\end{remark}
\begin{remark}
The condition \eqref{const_condition}, actually, can be relaxed. If one removes the condition \eqref{const_condition}, the pointwise estimate \eqref{P-W-01} for the solution itself will be replaced by
\begin{align*}
	|\hat{u}|^2\leqslant \tau^2 C|\xi|^4\left(1+\tau|\xi|^2\right)\int_0^t\left(|\widehat{\varphi}_{t}(s,\xi)|^2+|\widehat{\varphi}(s,\xi)|^2\right)\mathrm{d}s+\tau^2 C|\xi|^{-2}\left|\widehat{\psi}_2+|\xi|^2\widehat{\psi}_0+\delta|\xi|^2\widehat{\psi}_1\right|^2,
\end{align*}
where we need to assume $\psi_2\in \dot{H}^{-1}$ to get the convergence result, since the singularity from  the term $\|\chi_{\intt}(\xi)|\xi|^{-1}\widehat{\psi}_2(\xi)\|_{L^2}$ appears. In other words, \eqref{const_condition} is not a necessary condition to guarantee the convergence result. But, to avoid such negative homogeneous Sobolev space, we take \eqref{const_condition} simply.
\end{remark}
\begin{proof}
From the condition \eqref{const_condition} on initial data and \eqref{Inh_Linear}, we use \eqref{Pointwise} to get
\begin{align}\label{P-W-01}
	|\hat{u}|^2&\leqslant \tau^2 C\left(1+\tau|\xi|^2\right)\int_0^t|\widehat{\varphi}_{tt}(s,\xi)|^2\mathrm{d}s\notag\\
	&\leqslant \tau^2 C|\xi|^4\left(1+\tau|\xi|^2\right)\int_0^t\left(|\widehat{\varphi}_{t}(s,\xi)|^2+|\widehat{\varphi}(s,\xi)|^2\right)\mathrm{d}s,
\end{align}
because of $\widehat{\varphi}_{tt}=-|\xi|^2(\widehat{\varphi}+\delta\widehat{\varphi}_t)$. Recalling from Lemma 2.4 of \cite{IkehataNatsume}, the solution $\widehat{\varphi}$ fulfills the pointwise estimate
\begin{align*}
	|\xi|^2|\widehat{\varphi}(s,\xi)|+|\xi|\,|\widehat{\varphi}_t(s,\xi)|\leqslant C\exp\left(-\frac{c|\xi|^2}{|\xi|^2+1}s\right)\left(|\xi|^2|\widehat{\psi}_0|+|\xi|\,|\widehat{\psi}_1|\right).
\end{align*}
It leads to
\begin{align*}
	\|\hat{u}(t,\cdot)\|_{L^2}^2&\leqslant \tau^2 C\int_{|\xi|\leqslant \varepsilon}|\xi|^4\int_0^t\left(|\widehat{\varphi}_{t}(s,\xi)|^2+|\widehat{\varphi}(s,\xi)|^2\right) \mathrm{d}s \mathrm{d}\xi\\
	&\quad+\tau^2 C\int_{|\xi|\geqslant \varepsilon}\langle\xi\rangle^6\int_0^t\left(|\widehat{\varphi}_{t}(s,\xi)|^2+|\widehat{\varphi}(s,\xi)|^2\right)\mathrm{d}s\mathrm{d}\xi\\
	&\leqslant \tau^2 C\int_0^t\left\|\chi_{\intt}(\xi)\mathrm{e}^{-c|\xi|^2s}\left(|\xi|^2|\widehat{\psi}_0(\xi)|+|\xi|\,|\widehat{\psi}_1(\xi)|\right)\right\|_{L^2}^2\mathrm{d}s\\
	&\quad+\tau^2 C\int_0^t\mathrm{e}^{-cs}\mathrm{d}s\left\|(1-\chi_{\intt}(\xi))\left(\langle\xi\rangle^4|\widehat{\psi}_0(\xi)|+\langle\xi\rangle^3|\widehat{\psi}_1(\xi)|\right)\right\|_{L^2}^2\\
	&\leqslant \tau^2 C\int_0^t(1+s)^{-1-\frac{n}{2}}\mathrm{d}s\left(\|\psi_0\|_{L^1}^2+\|\psi_1\|_{L^1}^2\right)+\tau^2 C\left(\|\psi_0\|_{H^4}^2+\|\psi_1\|_{H^3}^2\right)\\
	&\leqslant \tau^2 C\left(\|\psi_0\|_{H^4\cap L^1}^2+\|\psi_1\|_{H^3\cap L^1}^2\right),
\end{align*}
where we used H\"older's inequality and the estimate
\begin{align*}
	\left\|\chi_{\intt}(\xi)|\xi|^{\alpha}\mathrm{e}^{-c|\xi|^2s}\right\|_{L^2}^2\lesssim (1+s)^{-\alpha-\frac{n}{2}}\in L^1([0,t])\ \ \mbox{if}\ \ -\alpha-\frac{n}{2}<-1
\end{align*}
for $\alpha\geqslant0$, in the last line of the chain. Finally, with the help of Parseval's formula, we complete the proof for the convergence of the solution itself.

Concerning the derivatives of solution, we can see
\begin{align*}
	|\xi|^2|\hat{u}|^2+|\hat{u}_t|^2\leqslant\tau^2 C|\xi|^6\left(1+\tau|\xi|^2\right)\int_0^t\left(|\widehat{\varphi}_{t}(s,\xi)|^2+|\widehat{\varphi}(s,\xi)|^2\right)\mathrm{d}s.
\end{align*}
By repeating the same procedure as the previous one, we can conclude our result.
\end{proof}

In the next result, under some assumptions for initial data, we will show the convergence
\begin{align*}
	\nabla^j\partial_t^k\psi^{\tau}\to\nabla^j\partial_t^k\varphi\ \ \mbox{in}\ \ L^{\infty}([0,\infty)\times\mb{R}^n)\ \ \mbox{as}\ \ \tau\downarrow0
\end{align*}
for any $n\geqslant 1$ and $0\leqslant j+k\leqslant 1$, with the linear rate of convergence $\tau$.
\begin{theorem}\label{Thm_Linfty_singular}
	Let us assume $\varphi_0\equiv\psi_0$, $\varphi_1\equiv\psi_1$ and \eqref{const_condition} such that $\langle D\rangle^{s_0+2}\psi_0\in L^1$ and $\langle D\rangle^{s_0+1}\psi_1\in L^1$ with $s_0>n+2$. Then, the following estimate:
	\begin{align*}
		\sup\limits_{t\in[0,\infty)}\left\|\psi^{\tau}(t,\cdot)-\varphi(t,\cdot)\right\|_{L^{\infty}}\leqslant \tau C\left(\|\langle D\rangle^{s_0+2}\psi_0\|_{ L^1}+\|\langle D\rangle^{s_0+1}\psi_1\|_{ L^1}\right)
	\end{align*}
	holds for any $0<\tau\ll 1$, where $C$ is a positive constant independent of $\tau$. Furthermore, if initial data fulfill additional regularity $\langle D\rangle^{s_0+3}\psi_0 \in L^1$ and $\langle D\rangle^{s_0+2}\psi_1\in  L^1$ with $s_0>n+2$, it holds
	\begin{align*}
		\sup\limits_{t\in[0,\infty)}\left(\|\nabla(\psi^{\tau}(t,\cdot)-\varphi(t,\cdot))\|_{L^\infty}+\|\psi_t^{\tau}(t,\cdot)-\varphi_t(t,\cdot)\|_{L^{\infty}}\right)\leqslant \tau C\left(\|\langle D\rangle^{s_0+3}\psi_0\|_{ L^1}+\|\langle D\rangle^{s_0+2}\psi_1\|_{ L^1}\right)
	\end{align*}
	for any $0<\tau\ll 1$.
\end{theorem}
\begin{proof}
From the estimate $$\int_0^t\mathrm{e}^{-c|\xi|^2s}\mathrm{d}s=\frac{1}{c|\xi|^2}\left(1-\mathrm{e}^{-c|\xi|^2t}\right)\leqslant C|\xi|^{-2},$$ we are able to arrive at
\begin{align*}
	|\hat{u}|&\leqslant \tau C|\xi|^2\sqrt{1+\tau|\xi|^2}\left(\int_0^t\left(|\widehat{\varphi}_t(s,\xi)|^2+|\widehat{\varphi}(s,\xi)|^2\right)\mathrm{d}s\right)^{1/2}\\
	&\leqslant\chi_{\intt}(\xi)\tau C\left(\int_0^t\mathrm{e}^{-c|\xi|^2s}\mathrm{d}s\right)^{1/2}\left(|\xi|^2|\widehat{\psi}_0|+|\xi|\,|\widehat{\psi}_1|\right)\\
	&\quad+(1-\chi_{\intt}(\xi))\tau C\left(\int_0^t\mathrm{e}^{-cs}\mathrm{d}s\right)^{1/2}\left(\langle\xi\rangle^4|\widehat{\psi}_0|+\langle\xi\rangle^3|\widehat{\psi}_1|\right)\\
	&\leqslant \tau C(1-\chi_{\intt}(\xi))\left(\langle\xi\rangle^4|\widehat{\psi}_0|+\langle\xi\rangle^3|\widehat{\psi}_1|\right).
\end{align*}
Then, by using the Hausdorff-Young inequality $\|u(t,\cdot)\|_{L^{\infty}}\leqslant C\|\hat{u}(t,\cdot)\|_{L^1}$ and the previous estimate, we conclude
\begin{align*}
\|\hat{u}(t,\cdot)\|_{L^1}&\leqslant \tau C\int_{\mb{R}^n}(1-\chi_{\intt}(\xi))\left(\langle\xi\rangle^4|\widehat{\psi}_0|+\langle\xi\rangle^3|\widehat{\psi}_1|\right)\mathrm{d}\xi\\
	&\leqslant \tau C\int_{\mb{R}^n}(1-\chi_{\intt}(\xi))\langle\xi\rangle^{2-s_0}\mathrm{d}\xi\left(\|\langle\xi\rangle^{s_0+2}\widehat{\psi}_0\|_{L^{\infty}}+\|\langle\xi\rangle^{s_0+1}\widehat{\psi}_1\|_{L^{\infty}}\right)\\
	&\leqslant \tau C\left(\|\langle D\rangle^{s_0+2}\psi_0\|_{L^1}+\|\langle D\rangle^{s_0+1}\psi_1\|_{L^{1}}\right)
\end{align*}
for $n+1-s_0<-1$, i.e. $s_0>n+2$, which completes the proof of the first desired estimate. The estimates for energy terms can be demonstrated by the similar way as the above.
\end{proof}
\begin{remark}
	Together with Theorems \ref{Thm_Singular_Limt} and \ref{Thm_Linfty_singular}, taking sufficiently regular initial data, one can get the singular limit result in $L^{\infty}([0,\infty),L^p)$ for $2\leqslant p\leqslant\infty$ with the linear rate $\tau$ of convergence by using the Riesz-Thorin interpolation theorem.
\end{remark}

\subsection{Formal expansion and formulation of initial layer}\label{Sub_Sec_3.3}
In the view of the third-order (in time) hyperbolic equation \eqref{Linear_MGT_Dissipative_tau}, although we can formally get the limit equation \eqref{Linear_Kuz_Dissipative} by taking $\tau=0$, the third initial condition in \eqref{Linear_MGT_Dissipative_tau} will lost. For this reason, if the initial condition does not match, there may be an initial layer between \eqref{Linear_MGT_Dissipative_tau} and \eqref{Linear_Kuz_Dissipative} as $\tau\downarrow0$. In the field of acoustic waves, it seems still unknown (even in the formal way) that
\medskip

\noindent \textbf{Question I:} Does an initial layer exist in the Moore-Gibson-Thompson equation?
\medskip

\noindent\textbf{Question II:} If it exists, what is the asymptotic profile of solution in this initial layer?
\medskip

Our aim in this part is to give possible answers of last two questions. We should remark that our approaches throughout this part not only can be applied in the Cauchy problem, but also can be generalized in the Dirichlet or Neumann problems. Even in the analysis of nonlinear MGT equations, it seems an effective way.

 Motivated by the boundary layer theory and the multi-scale analysis, the solution of the MGT equation \eqref{Linear_MGT_Dissipative_tau} has the following expansions:
\begin{align}\label{Rep}
\psi^{\tau}(t,x)=\sum\limits_{j\geqslant0}\tau^j\left(\psi^{I,j}(t,x)+\psi^{L,j}(\tfrac{t}{\tau},x)\right),
\end{align}
where each term in the last representation is assumed to be smooth. Here,  $\psi^{I,j}=\psi^{I,j}(t,x)$ denotes the dominant profile for each order expansion, and $\psi^{L,j}=\psi^{L,j}(z,x)$ with $z=t/\tau$ denotes the profile including singular layer profiles as $\tau\downarrow0$.

To obtain the formulation of initial layer, we will employ perturbation theory. By 
plugging \eqref{Rep} into the Cauchy problem \eqref{Linear_MGT_Dissipative_tau}, we may obtain from the equation that
\begin{align}\label{Eq_02}
	0&=\sum\limits_{j\geqslant 0}\tau^{j+1}\left(\psi^{I,j}_{ttt}+\tau^{-3}\psi^{L,j}_{zzz}\right)+\sum\limits_{j\geqslant0}\tau^j\left(\psi^{I,j}_{tt}+\tau^{-2}\psi_{zz}^{L,j}\right)-\sum\limits_{j\geqslant0}\tau^j\left(\Delta\psi^{I,j}+\Delta\psi^{L,j}\right)\notag\\
	&\quad-\delta\sum\limits_{j\geqslant0}\tau^j\left(\Delta\psi^{I,j}_t+\tau^{-1}\Delta\psi_z^{L,j}\right)-\sum\limits_{j\geqslant0}\tau^{j+1}\left(\Delta\psi^{I,j}_{t}+\tau^{-1}\Delta\psi^{L,j}_z\right),
\end{align}
and from the initial conditions that
\begin{align*}
\psi_0(x)&=\psi^{I,0}(0,x)+\psi^{L,0}(0,x)+\sum\limits_{j\geqslant 1}\tau^j\left(\underbrace{\psi^{I,j}(0,x)+\psi^{L,j}(0,x)}_{=0}\right),\\
\psi_1(x)&=\underbrace{\tau^{-1}\psi_z^{L,0}(0,x)}_{=0}+\psi^{I,0}_t(0,x)+\psi^{L,1}_z(0,x)+\sum\limits_{j\geqslant 1}\tau^j\left(\underbrace{\psi^{I,j}_t(0,x)+\psi^{L,j+1}_z(0,x)}_{=0}\right),\\
\psi_2(x)&=\underbrace{\tau^{-2}\psi_{zz}^{L,0}(0,x)+\tau^{-1}\psi_{zz}^{L,1}(0,x)}_{=0}+\psi^{I,0}_{tt}(0,x)+\psi^{L,2}_{zz}(0,x)\\
&\quad+\sum\limits_{j\geqslant 1}\tau^j\left(\underbrace{\psi^{I,j}_{tt}(0,x)+\psi^{L,j+2}_{zz}(0,x)}_{=0}\right).
\end{align*}
In the last lines, the brackets with zero value come from the consideration of $\tau\downarrow0$.
The matching conditions for initial data hint
\begin{itemize}
	\item $\psi^{I,0}(0,x)=\psi_0(x)$, $\psi_t^{I,0}(0,x)=\psi_1(x)$, and $\psi_{tt}^{I,0}(0,x)+\psi_{zz}^{L,2}(0,x)=\psi_2(x)$;
	\item $\psi^{L,0}(0,x)=0=\psi^{L,0}_z(0,x)=0=\psi^{L,0}_{zz}(0,x)$, and $\psi_{z}^{L,1}(0,x)=0=\psi_{zz}^{L,1}(0,x)$;
	\item $\psi^{I,j}(0,x)+\psi^{L,j}(0,x)=0$, $\psi^{I,j}_t(0,x)+\psi^{L,j+1}_z(0,x)=0$, and $\psi^{I,j}_{tt}(0,x)+\psi^{L,j+2}_{zz}(0,x)=0$ for any $j\geqslant 1$.
\end{itemize}
\begin{remark}\label{Rem_04}
	To study the initial layer problem, it is natural to assume the matching condition for the first and second initial data. Nonetheless, the lack of third-order time derivative in Kuznetsov's equation implies a possible phenomenon that the third initial data may not match, namely, $\psi_{tt}^{I,0}(0,x)+\psi_{zz}^{L,2}(0,x)=\psi_2(x)$ with $\psi_{zz}^{L,2}(0,x)\not\equiv0$.
\end{remark}
For the sake of simplicity, we denote $\psi^{I,j}_k(x):=\partial_t^k\psi^{I,j}(0,x)$ for $k=0,1,2$. Then, concerning $j\in\mb{Z}$ now, we can get from \eqref{Eq_02} with the terms behaving as $\tau^{j+1}$ that
\begin{align}\label{Eq_03}
\begin{cases}
\psi^{I,j+1}_{tt}-\Delta\psi^{I,j+1}-\delta\Delta\psi^{I,j+1}_t=-\partial_t\left(\psi^{I,j}_{tt}-\Delta\psi^{I,j}\right),&x\in\mb{R}^n,\ t>0,\\
\psi^{I,j+1}(0,x)=\psi^{I,j+1}_0(x),\ \psi^{I,j+1}_t(0,x)=\psi^{I,j+1}_1(x),&x\in\mb{R}^n,
\end{cases}
\end{align}
and
\begin{align}\label{Eq_04}
	\begin{cases}
\psi^{L,j+3}_{zzz}+\psi^{L,j+3}_{zz}=\Delta\psi^{L,j+1}_z+\Delta\psi^{L,j+1}+\delta\Delta\psi_z^{L,j+2},&x\in\mb{R}^n,\ z>0,\\
\psi^{L,j+3}(0,x)=-\psi^{I,j+3}_0(x),\ \psi^{L,j+3}_z(0,x)=-\psi^{I,j+2}_1(x),&x\in\mb{R}^n,\\ \psi^{L,j+3}_{zz}(0,x)=-\psi^{I,j+1}_2(x),&x\in\mb{R}^n,
	\end{cases}
\end{align}
where we assume $\psi^{I,j}(t,x)\equiv0\equiv\psi^{L,j}(t,x)$ if $j<0$.

Let us now analyze the asymptotic profiles. The outer solution (first-order profile) is given by $j=-1$ in \eqref{Eq_03} associated with \eqref{Eq_04} carrying $j=-3$, namely,
\begin{align*}
\begin{cases}
	\psi^{I,0}_{tt}-\Delta\psi^{I,0}-\delta\Delta\psi^{I,0}_t=0,&x\in\mb{R}^n,\ t>0,\\
	\psi^{I,0}(0,x)=\psi_0(x),\ \psi^{I,0}_t(0,x)=\psi_1(x),&x\in\mb{R}^n,
\end{cases}
\end{align*}
and
\begin{align*}
		\begin{cases}
		\psi^{L,0}_{zzz}+\psi^{L,0}_{zz}=0,&x\in\mb{R}^n,\ z>0,\\
		\psi^{L,0}(0,x)=0,\ \psi^{L,0}_z(0,x)=0,\  \psi^{L,0}_{zz}(0,x)=0,&x\in\mb{R}^n.
	\end{cases}
\end{align*}
Let us recall the linear Kuznetsov's equation \eqref{Linear_Kuz_Dissipative}. Therefore, we immediately get $\psi^{I,0}(t,x)\equiv\varphi(t,x)$ and $\psi^{L,0}(z,x)\equiv0$. It formally shows
\begin{align*}
	\psi^{\tau}(t,x)=\varphi(t,x)+\sum\limits_{j\geqslant1}\tau^{j}\left(\psi^{I,j}(t,x)+\psi^{L,j}(\tfrac{t}{\tau},x)\right),
\end{align*}
which exactly corresponds to the derived result in Theorems \ref{Thm_Singular_Limt} and \ref{Thm_Linfty_singular} since $\psi^{\tau}(t,x)-\varphi(t,x)=\ml{O}(\tau)$.

Indeed, the second-order profiles consist of two parts $\psi^{I,1}(t,x)$ and $\psi^{L,1}(z,x)$, totally. We take $j=0$ in \eqref{Eq_03} and $\psi^{I,1}(0,x)=0=\psi_t^{I,1}(0,x)$ to find
\begin{align}\label{Eq_Profile_01}
	\begin{cases}
		\psi_{tt}^{I,1}-\Delta\psi^{I,1}-\delta\Delta\psi_t^{I,1}=-\delta\Delta^2\varphi-\delta^2\Delta^2\varphi_t,&x\in\mb{R}^n,\ t>0,\\
		\psi^{I,1}(0,x)=0,\ \psi^{I,1}_t(0,x)=0,&x\in\mb{R}^n,
	\end{cases}
\end{align}
where we used $-\partial_t(\psi_{tt}^{I,0}-\Delta\psi^{I,0})=-\delta\Delta\varphi _{tt}=-\delta\Delta^2\varphi-\delta^2\Delta^2\varphi_t$. By defining the kernel (see, for example, \cite{Ikehata2014,DabbiccoReissig2014})
\begin{align*}
	E_0(t,x)&:=\frac{\rho_+(|D|)\mathrm{e}^{\rho_-(|D|)t}-\rho_-(|D|)\mathrm{e}^{\rho_+(|D|)t}}{\rho_+(|D|)-\rho_-(|D|)},\\
	E_1(t,x)&:=\frac{\mathrm{e}^{\rho_+(|D|)t}-\mathrm{e}^{\rho_-(|D|)t}}{\rho_+(|D|)-\rho_-(|D|)},
\end{align*}
with the symbols
\begin{align*}
	\rho_{\pm}(|\xi|) := \begin{cases} \displaystyle{-\frac{\delta}{2}|\xi|^2 \pm \frac{|\xi|}{2} \sqrt{\delta^2|\xi|^2-4}} & \mbox{if} \ \  \displaystyle{|\xi|>\frac{2}{\delta},} \\
		\displaystyle{-\frac{\delta}{2}|\xi|^2 \pm \frac{i|\xi|}{2} \sqrt{4-\delta^2|\xi|^2} }& \mbox{if} \ \ \displaystyle{0< |\xi|<\frac{2}{\delta}},
	\end{cases}
\end{align*}
we can represent the solution to \eqref{Eq_Profile_01} by
\begin{align}\label{Solution_Layer}
\psi^{I,1}(t,x)=-\delta\int_0^tE_1(t-s,x)\ast_{(x)}\Delta^2\varphi(s,x)\mathrm{d}s-\delta^2\int_0^tE_1(t-s,x)\ast_{(x)}\Delta^2\varphi_t(s,x)\mathrm{d}s,
\end{align}
carrying
\begin{align*}
	\varphi(t,x)=E_0(t,x)\ast_{(x)}\psi_0(x)+E_1(t,x)\ast_{(x)}\psi_1(x).
\end{align*}
 Additionally, considering $j=-2$ in \eqref{Eq_04}, it yields
\begin{align}\label{Layer_Eq}
\begin{cases}
\psi_{zzz}^{L,1}+\psi_{zz}^{L,1}=0,&x\in\mb{R}^n,\ z>0,\\
\psi^{L,1}(0,x)=0,\ \psi^{L,1}_z(0,x)=0,\ \psi^{L,1}_{zz}(0,x)=0,&x\in\mb{R}^n.
\end{cases}
\end{align}
That is to say $\psi^{L,1}(z,x)\equiv0$ for any $z\geqslant 0$ and $x\in\mb{R}^n$. So far, the second-order asymptotic profile did not show the existence of layer due to Remark \ref{Rem_04} telling us that the layer may come from the third-order expansion with $\psi^{L,2}(0,x)\not\equiv0$.

Let us turn to the third-order profiles
\begin{align}\label{Eq_Profile_02}
	\begin{cases}
		\psi_{tt}^{I,2}-\Delta\psi^{I,2}-\delta\Delta\psi_t^{I,2}=-\partial_t\left(\psi_{tt}^{I,1}-\Delta\psi^{I,1}\right),&x\in\mb{R}^n,\ t>0,\\
		\psi^{I,2}(0,x)=0,\ \psi^{I,2}_t(0,x)=0,&x\in\mb{R}^n,
	\end{cases}
\end{align}
where we assumed $\psi^{I,2}(0,x)\equiv0\equiv \psi_t^{I,2}(0,x)$, and
\begin{align}\label{Layer_Eq_02}
	\begin{cases}
		\psi_{zzz}^{L,2}+\psi_{zz}^{L,2}=0,&x\in\mb{R}^n,\ z>0,\\
		\psi^{L,2}(0,x)=0,\ \psi^{L,2}_z(0,x)=0,\ \psi^{L,2}_{zz}(0,x)=\psi_c(x),&x\in\mb{R}^n.
	\end{cases}
\end{align}
In the above, we applied the relation
\begin{align*}
	\psi^{L,2}_{zz}(0,x)=\psi_2(x)-\psi_{tt}^{I,0}(0,x)=\psi_c(x):=\psi_2(x)-(\Delta\psi_0(x)+\delta\Delta\psi_1(x)).
\end{align*}
Clearly, the solution to \eqref{Layer_Eq_02} is given by
\begin{align*}
	\psi^{L,2}(z,x)=\left(z-1+\mathrm{e}^{-z}\right)\psi_c(x).
\end{align*}
Summarizing the last statements, we may conclude the next result for formal profiles of solution to the MGT equation \eqref{Linear_MGT_Dissipative_tau} with small thermal relaxation.
\begin{prop}\label{Prop_01}
	The solution $\psi^{\tau}=\psi^{\tau}(t,x)$ to the Cauchy problem for the MGT equation \eqref{Linear_MGT_Dissipative_tau} with small $\tau$ formally has the following asymptotic expansions:
\begin{align}
	\psi^{\tau}(t,x)&=\varphi(t,x)+\tau\psi^{I,1}(t,x)+\tau^2\left[\psi^{I,2}(t,x)+\left(\tfrac{t}{\tau}-1\right)\left(\psi_2(x)-(\Delta\psi_0(x)+\delta\Delta\psi_1(x))\right)\right]\notag\\
	&\quad+\tau^2\mathrm{e}^{-\frac{t}{\tau}}\left(\psi_2(x)-(\Delta\psi_0(x)+\delta\Delta\psi_1(x))\right)+\sum\limits_{j\geqslant3}\tau^{j}\left(\psi^{I,j}(t,x)+\psi^{L,j}(\tfrac{t}{\tau},x)\right),\label{Solution_01}
\end{align}
where $\varphi=\varphi(t,x)$ is the solution to the linear Kuznetsov's equation \eqref{Linear_Kuz_Dissipative}, $\psi^{I,1}=\psi^{I,1}(t,x)$ is represented by \eqref{Solution_Layer}, and $\psi^{I,2}=\psi^{I,2}(t,x)$ is the solution to \eqref{Eq_Profile_02}. Moreover, $\psi^{I,j}=\psi^{I,j}(t,x)$ is the solution to the inhomogeneous Kuznetsov's equation \eqref{Eq_03}, and $\psi^{L,j}=\psi^{L,j}(z,x)$ with $z=t/\tau$ is given by the differential equation \eqref{Eq_04}.
\end{prop}
\begin{remark}
The philosophy for the singular limit and initial layer comes form the change of Cattaneo's law of heat conduction (hyperbolic-like version) into Fourier's law of heat conduction (parabolic-like version). So, we expect that for finite time, there is a rapid change with respect to small thermal relaxation $\tau$.
\end{remark}

\noindent\textbf{Conjecture I:} The initial layer appears if and only if
\begin{align}\label{Cond}
	\psi_2(x)\neq\Delta\psi_0(x)+\delta\Delta\psi_1(x).
\end{align}
From the formal representation \eqref{Solution_01}, we notice that the term including $\mathrm{e}^{-\frac{t}{\tau}}$ will disappear providing that \eqref{Cond} does not hold.
\medskip

\noindent\textbf{Conjecture II:} In this initial layer, we can observe a
profile
\begin{align*}
	\psi_{\mathrm{Lay}}(t,x;\tau):=\tau^2\mathrm{e}^{-\frac{t}{\tau}}\left(\psi_2(x)-(\Delta\psi_0(x)+\delta\Delta\psi_1(x))\right).
\end{align*}
For $0\leqslant t\leqslant t_0$, the function $\mathrm{e}^{-\frac{t}{\tau}}$ is changed rapidly as $\tau\downarrow0$. We conjecture that Proposition \ref{Prop_01} is valid in some suitable function spaces, e.g. $L^{\infty}([0,\infty),L^2)$.

It would be interesting to rigorously justify the correction of expansion \eqref{Solution_01} with the aid of some energy methods in the Fourier space, but such a result would be beyond the purposes of the present manuscript from our prospective.





\section{Global well-posedness of the JMGT equation}\label{Section_GESDS_JMGT}
We now turn to the model of nonlinear acoustics. To begin with this section, let us state our main result for global (in time) well-posedness for the nonlinear Cauchy problem \eqref{JMGT_Dissipative}.
\begin{theorem}\label{Thm_GESDS}
	Let us consider $s>n/2-1$ for $n\geqslant 2$. Then, there exists a constant $\epsilon>0$ such that for all $(\psi_0,\psi_1,\psi_2)\in\ml{A}_s:= (H^{s+2}\cap L^1)\times(H^{s+1}\cap L^1)\times(H^s\cap L^1)$ with $\|(\psi_0,\psi_1,\psi_2)\|_{\ml{A}_s}\leqslant \epsilon$, there is a uniquely determined Sobolev solution
	\begin{align*}
		\psi\in\ml{C}([0,\infty),H^{s+2})\cap\ml{C}^1([0,\infty),H^{s+1})\cap \ml{C}^2([0,\infty),H^s)
	\end{align*}
	to the JMGT equation \eqref{JMGT_Dissipative}. Furthermore, the following estimates:
	\begin{align*}
		\|\psi(t,\cdot)\|_{L^2}&\lesssim\ml{D}_n(t)\|(\psi_0,\psi_1,\psi_2)\|_{\ml{A}_s},\\
		\|\partial_t^{\ell}\psi(t,\cdot)\|_{L^2}&\lesssim (1+t)^{\frac{1}{2}-\frac{\ell}{2}-\frac{n}{4}}\|(\psi_0,\psi_1,\psi_2)\|_{\ml{A}_s}\ \ \mbox{if}\ \ \ell=1,2,\\
		\|\partial_{t}^{\ell}\psi(t,\cdot)\|_{\dot{H}^{s+2-\ell}}&\lesssim (1+t)^{-\frac{1}{2}-\frac{s}{2}-\frac{n}{4}}\|(\psi_0,\psi_1,\psi_2)\|_{\ml{A}_s}\ \ \mbox{if}\ \ \ell=0,1,2,
	\end{align*}
	hold, where the time-dependent coefficient $\ml{D}_n(t)$ were defined in Proposition \ref{Prop_Estimate_Solution_itself}.
\end{theorem}
\begin{remark}
	The derived estimates of solutions to the JMGT equation coincide with those in Proposition \ref{Prop_Estimate_Solution_itself} for the MGT equation. That is to say we observe the effect of no loss of decay to the corresponding linear problem. One may conjecture the sharpness of the estimates in Theorem \ref{Thm_GESDS}.
\end{remark}
\begin{remark}
	Concerning the two-dimensional MGT equation \eqref{Linear_MGT_Dissipative}, the authors of \cite{Chen-Ikehata-2021} derived optimal logarithmic growth rate $(\ln t)^{\frac{1}{2}}$ (as $t\to\infty$) for the solution itself in the $L^2$ norm, where the sharpness of estimate is guaranteed by the same upper bound and lower bound estimates. It means the solution in the $L^2$ norm blows up as $t\to\infty$ with logarithmic rate. Inherited from the linearized Cauchy problem \eqref{Linear_MGT_Dissipative}, we demonstrated the estimate of solution for the two-dimensional JMGT equation \eqref{JMGT_Dissipative} with the rate $(\ln(\mathrm{e}+t))^{\frac{1}{2}}$.
\end{remark}	
\begin{remark}\label{Rem_02}
 Concerning global (in time) solution for the JMGT equation \eqref{JMGT_Dissipative} in the 3D case, Theorem 1.1 in \cite{Racke-Said-2020} required $(\psi_0,\psi_1,\psi_2)\in H^{s+2}\times H^{s+2}\times H^{s+1}$ carrying $s>5/2$. Differently from their result, in Theorem \ref{Thm_GESDS} for $n=3$, we assume $(\psi_0,\psi_1,\psi_2)\in H^{s+2}\times H^{s+1}\times H^{s}$ with additional $L^1$ regularity carrying $s>1/2$. In some sense, the assumption for the additional $L^1$ regularity is beneficial to the demand of higher-order $H^s$ regularity.  
\end{remark}
\begin{remark}\label{Rem_03}
Concerning estimates for the global (in time) solution and its derivatives for the JMGT equation \eqref{JMGT_Dissipative} in the 3D case, Theorem 1.3 in \cite{Racke-Said-2020} assumed $H^s\cap L^1$ regularity for initial data and weighted $L^1$	regularity carrying $\int_{\mb{R}^3}\psi_{tt}(0,x)\mathrm{d}x=0=\int_{\mb{R}^3}\psi_{ttt}(0,x)\mathrm{d}x$. Differently from their result, in Theorem \ref{Thm_GESDS} for $n=3$, we derived estimates for each derivative of solution separately instead of the total energy, even for the solution itself, without any special assumption for initial data.
\end{remark}

\begin{remark}\label{Rem_01}
Our approach in the proof of  Theorem \ref{Thm_GESDS} also can be applied to demonstrate global (in time) well-posedness for
\begin{itemize}
	\item the Cauchy problem for the following Westervelt's type JMGT equation:
	\begin{align*}
		\tau\psi_{ttt}+\psi_{tt}-\Delta\psi-(\delta+\tau)\Delta\psi_t=\partial_t\left(\left(1+\frac{B}{2A}\right)(\psi_t)^2\right),
	\end{align*}
where we neglected local nonlinear effects in the modeling;
\item the Cauchy problem for the following acoustic pressure equation:
\begin{align*}
	\tau p_{ttt}+p_{tt}-\Delta p-(\delta+\tau)\Delta p_t=\partial_{tt}\left(\frac{1}{2\rho_0}\left(1+\frac{B}{2A}\right)p^2\right),
\end{align*}
with the medium density (constant) $\rho_0>0$, where $p=p(t,x)\in\mb{R}$ denotes the acoustic pressure;
\end{itemize}
without additional difficulty (the absence of the gradient nonlinearity makes the problem easier), since estimates for the nonlinear parts can follow Subsection \ref{Subsec_Est_Nonlinear_Term}.
\end{remark}
\subsection{Philosophy of the proof}
For $T>0$, we introduce evolution spaces in the sense of energy that
\begin{align*}
X_s(T):=\ml{C}([0,T],H^{s+2})\cap\ml{C}^1([0,T],H^{s+1})\cap \ml{C}^2([0,T],H^s),
\end{align*}
carrying the corresponding norm
\begin{align*}
	\|\psi\|_{X_s(T)}:=\sup\limits_{t\in[0,T]}&\left((\ml{D}_n(t))^{-1}\|\psi(t,\cdot)\|_{L^2}+\sum\limits_{\ell=1,2}(1+t)^{\frac{\ell-1}{2}+\frac{n}{4}}\|\partial_t^{\ell}\psi(t,\cdot)\|_{L^2}\right.\\
	&\quad\left.+\sum\limits_{\ell=0,1,2}(1+t)^{\frac{1}{2}+\frac{s}{2}+\frac{n}{4}}\|\,|D|^{s+2-\ell}\partial_t^{\ell}\psi(t,\cdot)\|_{L^2}\right)
\end{align*}
for $s\in(n/2-1,\infty)$. Let us recall that $\ml{D}_n(t)$ for $n\geqslant 2$ has been defined in Proposition \ref{Prop_Estimate_Solution_itself}. Then, we define the operator $N$ such that
\begin{align*}
	N:\ \psi(t,x)\in X_s(T)\to N\psi(t,x):=\psi^{\lin} (t,x)+\psi^{\non}(t,x),
\end{align*}
where $\psi^{\lin} =\psi^{\lin}(t,x)$ is the solution to the linearized Cauchy problem \eqref{Linear_MGT_Dissipative}, and the integral operator is denoted by
\begin{align*}
	\psi^{\non}(t,x):=\int_0^tK_2(t-\sigma,x)\ast_{(x)}f(\psi(\sigma,x))\mathrm{d}\sigma,
\end{align*}
where $K_2(t,x)$ is the kernel for the third data of the linearized problem \eqref{Linear_MGT_Dissipative}, which is motivated by Duhamel's principle. Here, we denote by $f(\psi)=f(\psi(t,x))$ the nonlinear term on the right-hand side, precisely,
\begin{align*}
	f(\psi):=\partial_t\left(\frac{B}{2A}(\psi_t)^2+|\nabla\psi|^2\right)=\frac{B}{A}\psi_t\psi_{tt}+2\nabla\psi\cdot\nabla\psi_t.
\end{align*}

In the next parts, we will demonstrate global (in time) existence and uniqueness of small data Sobolev solutions to the JMGT equation \eqref{JMGT_Dissipative} by proving a fixed point of operator $N$ which means $N\psi\in X_s(T)$. Namely, the following two crucial inequalities:
\begin{align}
	\|N\psi\|_{X_s(T)}&\lesssim\|(\psi_0,\psi_1,\psi_2)\|_{\ml{A}_{s} }+\|\psi\|_{X_s(T)}^2,\label{Cruc-01}\\
	\|N\psi-N\bar{\psi}\|_{X_s(T)}&\lesssim\|\psi-\bar{\psi}\|_{X_s(T)}\left(\|\psi\|_{X_s(T)}+\|\bar{\psi}\|_{X_s(T)}\right),\label{Cruc-02}
\end{align}
will be proved, respectively, with initial data space
\begin{align*}
	\ml{A}_s=(H^{s+2}\cap L^1)\times(H^{s+1}\cap L^1)\times(H^s\cap L^1).
\end{align*}
In our aim inequality \eqref{Cruc-02}, $\psi$ and $\bar{\psi}$ are two solutions to the JMGT equation \eqref{JMGT_Dissipative}. Providing that we take $\|(\psi_0,\psi_1,\psi_2)\|_{\ml{A}_{s} }=\epsilon$ as a sufficiently small and positive constant, then we combine \eqref{Cruc-01} with \eqref{Cruc-02} to claim that there exists global (in time) small data Sobolev solution $\psi^{*}=\psi^{*}(t,x)\in X_s(T)$ by using Banach's fixed point theorem.

\subsection{Estimates of nonlinear terms}\label{Subsec_Est_Nonlinear_Term}
Our purpose in this part is to give some estimates to the nonlinear terms in three norms as follows:
\begin{align*}
	\|f(\psi(\sigma,\cdot))\|_{L^1},\ \ \|f(\psi(\sigma,\cdot))\|_{L^2}, \ \ \|f(\psi(\sigma,\cdot))\|_{\dot{H}^s}
\end{align*}
for any $s\in(n/2-1,\infty)$, which will be controlled by $\|\psi\|_{X_s(T)}$ with suitable time-dependent coefficients for any $\sigma\in[0,T]$.
\medskip

\noindent\underline{Part I: Estimates in the $L^1$ and $L^2$ norms.}\\
Here, let us fix $m=1,2$. Applying H\"older's inequality, we arrive at
\begin{align*}
	\|f(\psi(\sigma,\cdot))\|_{L^m}&\lesssim\|\psi_t(\sigma,\cdot)\psi_{tt}(\sigma,\cdot)\|_{L^m}+\|\nabla\psi(\sigma,\cdot)\cdot\nabla\psi_t(\sigma,\cdot)\|_{L^m}\\
	&\lesssim\|\psi_t(\sigma,\cdot)\|_{L^{p_1}}\|\psi_{tt}(\sigma,\cdot)\|_{L^{p_2}}+\|\,|D|\psi(\sigma,\cdot)\|_{L^{q_1}}\|\,|D|\psi_t(\sigma,\cdot)\|_{L^{q_2}},
\end{align*}
where $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{q_1}+\frac{1}{q_2}=\frac{1}{m}$ with $p_1,p_2,q_1,q_2\in[m,\infty]$. Then, the fractional Gagliardo-Nirenberg inequality (see Proposition \ref{fractionalgagliardonirenbergineq}) shows
\begin{align*}
	\|f(\psi(\sigma,\cdot))\|_{L^m}&\lesssim\|\psi_t(\sigma,\cdot)\|_{L^2}^{1-\beta_1}\|\psi_t(\sigma,\cdot)\|_{\dot{H}^{s+1}}^{\beta_1}\|\psi_{tt}(\sigma,\cdot)\|_{L^2}^{1-\beta_2}\|\psi_{tt}(\sigma,\cdot)\|_{\dot{H}^s}^{\beta_2}\\
	&\quad+\|\psi(\sigma,\cdot)\|_{L^2}^{1-\beta_3}\|\psi(\sigma,\cdot)\|_{\dot{H}^{s+2}}^{\beta_3}\|\psi_t(\sigma,\cdot)\|_{L^2}^{1-\beta_4}\|\psi_t(\sigma,\cdot)\|_{\dot{H}^{s+1}}^{\beta_4}\\
	&\lesssim(1+\sigma)^{-\frac{n}{4}(1-\beta_1)-(\frac{1}{2}+\frac{s}{2}+\frac{n}{4})\beta_1-(\frac{1}{2}+\frac{n}{4})(1-\beta_2)-(\frac{1}{2}+\frac{s}{2}+\frac{n}{4})\beta_2}\|\psi\|_{X_s(\sigma)}^2\\
	&\quad+(\ml{D}_n(\sigma))^{1-\beta_3}(1+\sigma)^{-(\frac{1}{2}+\frac{s}{2}+\frac{n}{4})\beta_3-\frac{n}{4}(1-\beta_4)-(\frac{1}{2}+\frac{s}{2}+\frac{n}{4})\beta_4}\|\psi\|_{X_s(\sigma)}^2,
\end{align*}
where we denoted the exponents by
\begin{align*}
	&\ \ \ \ \ \ \ \ \ \ \ \ \, \beta_1=\frac{n}{s+1}\left(\frac{1}{2}-\frac{1}{p_1}\right)\in[0,1],\ \ \beta_2=\frac{n}{s}\left(\frac{1}{2}-\frac{1}{p_2}\right)\in[0,1]\\
	&\beta_3=\frac{n}{s+2}\left(\frac{1}{2}-\frac{1}{q_1}+\frac{1}{n}\right)\in\left[\frac{1}{s+2},1\right],\ \ \beta_4=\frac{n}{s+1}\left(\frac{1}{2}-\frac{1}{q_2}+\frac{1}{n}\right)\in\left[\frac{1}{s+1},1\right].
\end{align*}
We should remark that the existence of $p_1,p_2,q_1,q_2$ will be shown in Appendix \ref{Existence_Parameter} later. We can precisely compute the time-dependent coefficients for $\|\psi\|_{X_s(\sigma)}^2$ leading to
\begin{align*}
	\|f(\psi(\sigma,\cdot))\|_{L^m}\lesssim\begin{cases}
		(1+\sigma)^{-\frac{5}{2}+\frac{1}{m}+\varepsilon_0}\|\psi\|_{X_s(\sigma)}^2&\mbox{if}\ \ n=2,\\
		(1+\sigma)^{-\frac{1}{2}-n+\frac{n}{2m}}\|\psi\|_{X_s(\sigma)}^2&\mbox{if}\ \ n\geqslant 3,
	\end{cases}
\end{align*}
for $m=1,2$, with some sufficient small constants $\varepsilon_0>0$. Since $\|\psi\|_{X_s(\sigma)}\lesssim\|\psi\|_{X_s(T)}$ for any $\sigma\in[0,T]$, we conclude
\begin{align}
	\|f(\psi(\sigma,\cdot))\|_{L^1}&\lesssim\begin{cases}
		(1+\sigma)^{-\frac{3}{2}+\varepsilon_0}\|\psi\|_{X_s(T)}^2&\mbox{if}\ \ n=2,\\
		(1+\sigma)^{-\frac{1}{2}-\frac{n}{2}}\|\psi\|_{X_s(T)}^2&\mbox{if}\ \ n\geqslant3,
	\end{cases}\label{Est_02}\\
	\|f(\psi(\sigma,\cdot))\|_{L^2}&\lesssim \begin{cases}
		(1+\sigma)^{-2+\varepsilon_0}\|\psi\|_{X_s(T)}^2&\mbox{if}\ \ n=2,\\
		(1+\sigma)^{-\frac{1}{2}-\frac{3n}{4}}\|\psi\|_{X_s(T)}^2&\mbox{if}\ \ n\geqslant3\label{Est_03}.
	\end{cases}
\end{align}

\medskip

\noindent\underline{Part II: Estimates in the $\dot{H}^s$ norms.}\\
Differently from Part I, we employ the fractional Leibniz rule (see Proposition \ref{fractionleibnizrule}) rather than H\"older's inequality because of the Riesz potential spaces in the $L^2$ basic, to deduce
\begin{align*}
	\|f(\psi(\sigma,\cdot))\|_{\dot{H}^s}&\lesssim\|\psi_t(\sigma,\cdot)\psi_{tt}(\sigma,\cdot)\|_{\dot{H}^s}+\|\nabla\psi(\sigma,\cdot)\cdot\nabla\psi_t(\sigma,\cdot)\|_{\dot{H}^s}\\
	&\lesssim \|\psi_t(\sigma,\cdot)\|_{\dot{H}^s_{p_3}}\|\psi_{tt}(\sigma,\cdot)\|_{L^{p_4}}+\|\psi_t(\sigma,\cdot)\|_{L^{\infty}}\|\psi_{tt}(\sigma,\cdot)\|_{\dot{H}^s}\\
	&\quad+\|\,|D|\psi(\sigma,\cdot)\|_{\dot{H}^{s}_{p_5}}\|\,|D|\psi_t(\sigma,\cdot)\|_{L^{p_6}}+\|\,|D|\psi(\sigma,\cdot)\|_{L^{\infty}}\|\,|D|\psi_t(\sigma,\cdot)\|_{\dot{H}^s}\\
	&=:I_1(\sigma)+I_2(\sigma)+I_3(\sigma)+I_4(\sigma),
\end{align*}
where $\frac{1}{p_3}+\frac{1}{p_4}=\frac{1}{p_5}+\frac{1}{p_6}=\frac{1}{2}$ with $p_3,\dots,p_6\in(1,\infty]$. By using the fractional Gagliardo-Nirenberg inequality again, one notices
\begin{align*}
	I_1(\sigma)&\lesssim\|\psi_t(\sigma,\cdot)\|_{L^2}^{1-\beta_5}\|\psi_t(\sigma,\cdot)\|_{\dot{H}^{s+1}}^{\beta_5}\|\psi_{tt}(\sigma,\cdot)\|_{L^2}^{1-\beta_6}\|\psi_{tt}(\sigma,\cdot)\|_{\dot{H}^s}^{\beta_6},\\
	I_3(\sigma)&\lesssim\|\psi(\sigma,\cdot)\|_{L^2}^{1-\beta_7}\|\psi(\sigma,\cdot)\|_{\dot{H}^{s+2}}^{\beta_7}\|\psi_t(\sigma,\cdot)\|_{L^2}^{1-\beta_8}\|\psi_t(\sigma,\cdot)\|_{\dot{H}^{s+1}}^{\beta_8},
\end{align*}
where the exponents are given by
\begin{align*}
	&\ \ \ \ \, \beta_5=\frac{n}{s+1}\left(\frac{1}{2}-\frac{1}{p_3}+\frac{s}{n}\right)\in\left[\frac{s}{s+1},1\right],\ \ \beta_6=\frac{n}{s}\left(\frac{1}{2}-\frac{1}{p_4}\right)\in[0,1],\\
	&\beta_7=\frac{n}{s+2}\left(\frac{1}{2}-\frac{1}{p_5}+\frac{s+1}{n}\right)\in\left[\frac{s+1}{s+2},1\right],\ \ \beta_8=\frac{n}{s+1}\left(\frac{1}{2}-\frac{1}{p_6}+\frac{1}{n}\right)\in\left[\frac{1}{s+1},1\right].
\end{align*}
The existence of parameters $p_3,\dots,p_6$ will be proved in Appendix \ref{Existence_Parameter}. As a consequence, we may find
\begin{align*}
I_1(\sigma)&\lesssim(1+\sigma)^{-\frac{n}{4}(1-\beta_5)-(\frac{1}{2}+\frac{s}{2}+\frac{n}{4})\beta_5-(\frac{1}{2}+\frac{n}{4})(1-\beta_6)-(\frac{1}{2}+\frac{s}{2}+\frac{n}{4})\beta_6}\|\psi\|_{X_s(\sigma)}^2\\
&\lesssim (1+\sigma)^{-\frac{1}{2}-\frac{s}{2}-\frac{3n}{4}}\|\psi\|_{X_s(T)}^2,
\end{align*}
and
\begin{align*}
I_3(\sigma)&\lesssim(\ml{D}_{n}(\sigma))^{1-\beta_7}(1+\sigma)^{-(\frac{1}{2}+\frac{s}{2}+\frac{n}{4})\beta_7-\frac{n}{4}(1-\beta_8)-(\frac{1}{2}+\frac{s}{2}+\frac{n}{4})\beta_8}\|\psi\|_{X_s(\sigma)}^2\\
&\lesssim\begin{cases}
	(1+\sigma)^{-2-\frac{s}{2}+\varepsilon_0}\|\psi\|_{X_s(T)}^2&\mbox{if}\ \ n=2,\\
	(1+\sigma)^{-\frac{1}{2}-\frac{s}{2}-\frac{3n}{4}}\|\psi\|_{X_s(T)}^2&\mbox{if}\ \ n\geqslant 3.
\end{cases}
\end{align*}
To estimate the rest terms, we may use Propositions \ref{fractionembedd} and \ref{fractionalgagliardonirenbergineq}. To be specific,
\begin{align*}
	\|\psi_t(\sigma,\cdot)\|_{L^{\infty}}&\lesssim\|\psi_t(\sigma,\cdot)\|_{\dot{H}^{s^*}}+\|\psi_t(\sigma,\cdot)\|_{\dot{H}^{s+1}}\\
	&\lesssim\|\psi_t(\sigma,\cdot)\|_{L^2}^{1-\frac{s^*}{s+1}}\|\psi_t(\sigma,\cdot)\|_{\dot{H}^{s+1}}^{\frac{s^*}{s+1}}+\|\psi_t(\sigma,\cdot)\|_{\dot{H}^{s+1}},
\end{align*}
as well as
	\begin{align*}
	\|\,|D|\psi(\sigma,\cdot)\|_{L^{\infty}}&\lesssim\|\psi(\sigma,\cdot)\|_{\dot{H}^{s^*+1}}+\|\psi(\sigma,\cdot)\|_{\dot{H}^{s+2}}\\
	&\lesssim\|\psi(\sigma,\cdot)\|_{L^2}^{1-\frac{s^*+1}{s+2}}\|\psi(\sigma,\cdot)\|_{\dot{H}^{s+2}}^{\frac{s^*+1}{s+2}}+\|\psi(\sigma,\cdot)\|_{\dot{H}^{s+2}},
\end{align*}
where $0<2s^*<n<2(s+1)$, which implies $s>n/2-1$. The choice of $s^*$ will be shown later, which can avoid the phenomenon of loss of decay so that the estimates for the JMGT equation can inherit from those for the linearized Cauchy problem. We summarize the previous estimate to arrive at
\begin{align*}
	I_2(\sigma)&\lesssim (1+\sigma)^{-\frac{1}{2}-\frac{s+s^*}{2}-\frac{n}{2}}\|\psi\|_{X_s(T)}^2,\\
	I_4(\sigma)&\lesssim \begin{cases}
		(1+\sigma)^{-\frac{3}{2}-\frac{s+s^*}{2}+\varepsilon_0}\|\psi\|_{X_s(T)}^2&\mbox{if}\ \ n=2,\\
		(1+\sigma)^{-\frac{1}{2}-\frac{s+s^*}{2}-\frac{n}{2}}\|\psi\|_{X_s(T)}^2&\mbox{if}\ \ n\geqslant3.
	\end{cases}
\end{align*}
Hence, recalling $0<2s^*<n$, it yields
\begin{align}\label{Est_10}
	\|f(\psi(\sigma,\cdot))\|_{\dot{H}^s}\lesssim\begin{cases}
				(1+\sigma)^{-\frac{3}{2}-\frac{s+s^*}{2}+\varepsilon_0}\|\psi\|_{X_s(T)}^2&\mbox{if}\ \ n=2,\\
		(1+\sigma)^{-\frac{1}{2}-\frac{s+s^*}{2}-\frac{n}{2}}\|\psi\|_{X_s(T)}^2&\mbox{if}\ \ n\geqslant3.
	\end{cases}
\end{align}

\subsection{Existence of global (in time) solution}
First of all, according to the derived estimates in Propositions 2.1 as well as 2.2, we claim that
\begin{align*}
	\|\psi^{\lin} \|_{X_s(T)}\lesssim\|(\psi_0,\psi_1,\psi_2)\|_{\ml{A}_s}.
\end{align*}
In other words, concerning the aim \eqref{Cruc-01}, we just need to estimate $\|\psi^{\non}\|_{X_s(T)}$ in the remaining of this subsection.

By using derived $(L^2\cap L^1)-L^2$ estimates \eqref{Est_01} for $[0,t]$ and then dividing the interval into $[0,t/2]$ as well as $[t/2,t]$, we may see
\begin{align*}
	\|\psi^{\non}(t,\cdot)\|_{L^2}&\lesssim\int_0^t\ml{D}_n(t-\sigma)\|f(\psi(\sigma,\cdot))\|_{L^2\cap L^1}\mathrm{d}\sigma,\\
	&\lesssim\|\psi\|_{X_s(T)}^2\times\begin{cases}
		\displaystyle{\int_0^t(\ln(\mathrm{e}+t-\sigma))^{\frac{1}{2}}(1+\sigma)^{-\frac{3}{2}+\varepsilon_0}\mathrm{d}\sigma}&\mbox{if}\ \ n=2,\\
		\displaystyle{\int_0^t(1+t-\sigma)^{\frac{1}{2}-\frac{n}{4}}(1+\sigma)^{-\frac{1}{2}-\frac{n}{2}}\mathrm{d}\sigma}&\mbox{if}\ \ n\geqslant 3,
	\end{cases}\\
&\lesssim\|\psi\|_{X_s(T)}^2\times\begin{cases}
	\displaystyle{(\ln(\mathrm{e}+t))^{\frac{1}{2}}+(1+t)^{-\frac{1}{2}+\varepsilon_0}(\ln(\mathrm{e}+t))^{\frac{1}{2}}}&\mbox{if}\ \ n=2,\\
	\displaystyle{(1+t)^{\frac{1}{2}-\frac{n}{4}}+(1+t)^{-\frac{1}{2}-\frac{n}{2}}\int_{t/2}^t(1+t-\sigma)^{\frac{1}{2}-\frac{n}{4}}\mathrm{d}\sigma}&\mbox{if}\ \ n\geqslant 3.
\end{cases}
\end{align*}
In the above, we used \eqref{Est_02}, \eqref{Est_03} and the facts that $(1+\sigma)^{-\frac{3}{2}+\varepsilon_0}\in L^1([0,t/2])$ with sufficiently small $\varepsilon_0>0$ and $(1+\sigma)^{-\frac{1}{2}-\frac{n}{2}}\in L^1([0,t/2])$ for any $n\geqslant 3$. Due to the fact that
\begin{align*}
	\int_{t/2}^t(1+t-\sigma)^{\frac{1}{2}-\frac{n}{4}}\mathrm{d}\sigma\lesssim\begin{cases}
		(1+t)^{\frac{3}{2}-\frac{n}{4}}&\mbox{if}\ \ n=3,4,5,\\
		\ln(\mathrm{e}+t)&\mbox{if} \ \ n=6,\\
		1&\mbox{if}\ \ n\geqslant7,
	\end{cases}
\end{align*}
one finds
\begin{align*}
(1+t)^{-\frac{1}{2}-\frac{n}{2}}\int_{t/2}^t(1+t-\sigma)^{\frac{1}{2}-\frac{n}{4}}\mathrm{d}\sigma\lesssim (1+t)^{\frac{1}{2}-\frac{n}{4}}.
\end{align*}
Namely, the upper bound estimates hold
\begin{align*}
\|\psi^{\non}(t,\cdot)\|_{L^2}\lesssim\ml{D}_n(t)\|\psi\|_{X_s(T)}^2.	
\end{align*}

For the first- and second-order time derivatives of solution, we employ $(L^2\cap L^1)-L^2$ estimates \eqref{Est_04} in $[0,t/2]$ and $L^2-L^2$ estimates \eqref{Est_05} in $[t/2,t]$. Moreover, using \eqref{Est_02} and \eqref{Est_03}, we get
\begin{align*}
	\|\partial_t^{\ell}\psi^{\non}(t,\cdot)\|_{L^2}&\lesssim\int_0^{t/2}(1+t-\sigma)^{\frac{1}{2}-\frac{\ell}{2}-\frac{n}{4}}\|f(\psi(\sigma,\cdot))\|_{L^2\cap L^1}\mathrm{d}\sigma\\
	&\quad+\int_{t/2}^t(1+t-\sigma)^{\frac{1}{2}-\frac{\ell}{2}}\|f(\psi(\sigma,\cdot))\|_{L^2}\mathrm{d}\sigma\\
	&\lesssim\|\psi\|_{X_s(T)}^2\times\begin{cases}
		\displaystyle{(1+t)^{-\frac{\ell}{2}}+(1+t)^{-2+\varepsilon_0}\int_{t/2}^t(1+t-\sigma)^{\frac{1}{2}-\frac{\ell}{2}}\mathrm{d}\sigma}&\mbox{if}\ \ n=2,\\
		\displaystyle{(1+t)^{\frac{1}{2}-\frac{\ell}{2}-\frac{n}{4}}+(1+t)^{-\frac{1}{2}-\frac{3n}{4}}\int_{t/2}^t(1+t-\sigma)^{\frac{1}{2}-\frac{\ell}{2}}\mathrm{d}\sigma}&\mbox{if}\ \ n\geqslant 3,
	\end{cases}\\
&\lesssim\|\psi\|_{X_s(T)}^2\times\begin{cases}
	\displaystyle{(1+t)^{-\frac{\ell}{2}}+(1+t)^{-\frac{1}{2}+\varepsilon_0-\frac{\ell}{2}}}&\mbox{if}\ \ n=2,\\
	\displaystyle{(1+t)^{\frac{1}{2}-\frac{\ell}{2}-\frac{n}{4}}+(1+t)^{1-\frac{\ell}{2}-\frac{3n}{4}}}&\mbox{if}\ \ n\geqslant 3,
\end{cases}\\
&\lesssim(1+t)^{\frac{1}{2}-\frac{\ell}{2}-\frac{n}{4}}\|\psi\|_{X_s(T)}^2
\end{align*}
for $\ell=1,2$.

Concerning higher-order spatial derivatives of solution, we will use \eqref{Est_06}, \eqref{Est_07} in the sub-interval $[0,t/2]$ and \eqref{Est_08}, \eqref{Est_09} in the sub-interval $[t/2,t]$. Furthermore, the derived estimates \eqref{Est_02} and \eqref{Est_10} also will be applied. Precisely, for $\ell=0,1,2$, we have
\begin{align*}
	&\|\,|D|^{s+2-\ell}\partial_t^{\ell}\psi^{\non}(t,\cdot)\|_{L^2}&\\
	&\qquad\lesssim\int_{0}^{t/2}(1+t-\sigma)^{-\frac{1}{2}-\frac{s}{2}-\frac{n}{4}}\|f(\psi(\sigma,\cdot))\|_{\dot{H}^s\cap L^1}\mathrm{d}\sigma+\int_{t/2}^t(1+t-\sigma)^{-\frac{1}{2}}\|f(\psi(\sigma,\cdot))\|_{\dot{H}^s}\mathrm{d}\sigma\\
	&\qquad\lesssim\|\psi\|_{X_s(T)}^2\times\begin{cases}
(1+t)^{-1-\frac{s}{2}}+(1+t)^{-1-\frac{s+s^*}{2}+\varepsilon_0}&\mbox{if}\ \ n=2,\\
(1+t)^{-\frac{1}{2}-\frac{s}{2}-\frac{n}{4}}+(1+t)^{-\frac{s+s^*}{2}-\frac{n}{2}}&\mbox{if}\ \ n\geqslant3,
	\end{cases}\\
&\qquad\lesssim(1+t)^{-\frac{1}{2}-\frac{s}{2}-\frac{n}{4}}\|\psi\|_{X_s(T)}^2,
\end{align*}
where we chose $s^*=2\varepsilon_0$. 

All in all, summarizing the above all derived estimates, we directly conclude
\begin{align*}
	\|\psi^{\non}\|_{X_s(T)}\lesssim\|\psi\|_{X_s(T)}^2,
\end{align*}
which says that the desired estimate \eqref{Cruc-01} holds if $s>n/2-1$ for all $n\geqslant 2$.

Let us sketch the proof of the Lipschitz condition by remarking
\begin{align*}
	&\|\,|D|^{s+2-\ell}\partial_t^{\ell}(N\psi(t,\cdot)-N\bar{\psi}(t,\cdot))\|_{L^2}\\
	&\qquad=\left\|\,|D|^{s+2-\ell}\partial_t^{\ell}\int_0^tK_2(t-\sigma,\cdot)\ast_{(x)}\left(f(\psi(\sigma,\cdot)-f(\bar{\psi}(\sigma,\cdot))\right)\mathrm{d}\sigma\right\|_{L^2}
\end{align*}
for $s\in\{\ell-2\}\cup(0,\infty)$ and $\ell=0,1,2$. By using some estimates in Propositions \ref{Prop_Estimate_Solution_itself} and \ref{Prop_Estimate_Time_Derivative} again, we just need to estimate
\begin{align*}
	\|f(\psi(\sigma,\cdot))-f(\bar{\psi}(\sigma,\cdot))\|_{L^1},\ \ \|f(\psi(\sigma,\cdot))-f(\bar{\psi}(\sigma,\cdot))\|_{L^2}, \ \ \|f(\psi(\sigma,\cdot))-f(\bar{\psi}(\sigma,\cdot))\|_{\dot{H}^s}.
\end{align*}
Actually, we may estimate the difference of the nonlinear terms by
\begin{align*}
	|f(\psi)-f(\bar{\psi})|&\lesssim|\psi_t\psi_{tt}-\bar{\psi}_t\bar{\psi}_{tt}|+|\nabla\psi\cdot\nabla\psi_t-\nabla\bar{\psi}\cdot\nabla\bar{\psi}_t|\\
	&\lesssim|\psi_t(\psi_{tt}-\bar{\psi}_{tt})|+|(\psi_t-\bar{\psi}_t)\bar{\psi}_{tt}|+|\nabla\psi\cdot\nabla(\psi_t-\bar{\psi}_t)|+|\nabla(\psi-\bar{\psi})\cdot\nabla\bar{\psi}_t|.
\end{align*}
Therefore, by repeating the same procedure as Subsection \ref{Subsec_Est_Nonlinear_Term}, we obtain
\begin{align*}
	\|f(\psi(\sigma,\cdot))-f(\bar{\psi}(\sigma,\cdot))\|_{L^m}&\lesssim\|\psi-\bar{\psi}\|_{X_s(T)}\left(\|\psi\|_{X_s(T)}+\|\bar{\psi}\|_{X_s(T)}\right)\\
	&\quad\times\begin{cases}
		(1+\sigma)^{-\frac{5}{2}+\frac{1}{m}+\varepsilon_0}&\mbox{if}\ \ n=2,\\
		(1+\sigma)^{-\frac{1}{2}-n+\frac{n}{2m}}&\mbox{if}\ \ n\geqslant3,
	\end{cases}
\end{align*}
where $m=1,2$ and $\varepsilon_0>0$ is a small constant. Furthermore, the next estimates hold:
\begin{align*}
	\|f(\psi(\sigma,\cdot))-f(\bar{\psi}(\sigma,\cdot))\|_{\dot{H}^s}&\lesssim\|\psi-\bar{\psi}\|_{X_s(T)}\left(\|\psi\|_{X_s(T)}+\|\bar{\psi}\|_{X_s(T)}\right)\\
	&\quad\times\begin{cases}
	(1+\sigma)^{-\frac{3}{2}-\frac{s+s^*}{2}+\varepsilon_0}&\mbox{if}\ \ n=2,\\
	(1+\sigma)^{-\frac{1}{2}-\frac{s+s^*}{2}-\frac{n}{2}}&\mbox{if}\ \ n\geqslant3,
\end{cases}
\end{align*}
for $0<2s^*<n<2(s+1)$. Then, following the same steps as those in the proof of \eqref{Cruc-01}, we can conclude \eqref{Cruc-02}. The proof is complete.



\section{Final remarks}\label{Sec_Final}
In Section \ref{Section_Singular_Limits}, we have derived global (in time) convergence result for the MGT equation \eqref{Linear_MGT_Dissipative} in the  $L^2$ norm. Moreover, in Section \ref{Section_GESDS_JMGT}, we have proved global (in time) existence result for the JMGT equation \eqref{JMGT_Dissipative} in the framework of $L^2$. Motivated by these two results, one may expect global (in time) convergence result for the JMGT equation \eqref{JMGT_Dissipative} with respect to the thermal relaxation $\tau\downarrow0$.
\medskip

Let $\psi^{\tau}=\psi^{\tau}(t,x)$ to be the solution for the JMGT equation 
\begin{align}\label{JMGT_tau_Dissipative}
	\begin{cases}
		\displaystyle{\tau \psi^{\tau}_{ttt}+\psi^{\tau}_{tt}-\Delta \psi^{\tau}-(\delta+\tau)\Delta \psi^{\tau}_t=\partial_t\left(\frac{B}{2A}(\psi^{\tau}_t)^2+|\nabla\psi^{\tau}|^2\right),}&x\in\mb{R}^n,\ t>0,\\
		\psi^{\tau}(0,x)=\psi^{\tau}_0(x),\ \psi^{\tau}_t(0,x)=\psi^{\tau}_1(x),\ \psi^{\tau}_{tt}(0,x)=\psi^{\tau}_2(x),&x\in\mb{R}^n,
	\end{cases}
\end{align}
with $0<\tau\ll 1$, and $\psi^{0}=\psi^{0}(t,x)$ to be  the solution for Kuznetsov's equation
 \begin{align}\label{Kuz_Dissipative}
 	\begin{cases}
 		\displaystyle{\psi^{0}_{tt}-\Delta \psi^{0}-\delta\Delta\psi^{0}_t=\partial_t\left(\frac{B}{2A}(\psi^{0}_t)^2+|\nabla\psi^{0}|^2\right),}&x\in\mb{R}^n,\ t>0,\\
 		\psi^{0}(0,x)=\psi^{0}_0(x),\ \psi^{0}_t(0,x)=\psi^{0}_1(x),&x\in\mb{R}^n.
 	\end{cases}
 \end{align}
\medskip

\noindent\textbf{Open Questions:}
\begin{enumerate}[(1)]
	\item Does $\psi^{\tau}(t,x)$ converge to $\psi^0(t,x)$ as $\tau\downarrow0$ in the framework of $L^2$ for any $t>0$? Namely, global (in time) singular limit with respect to $\tau$ for nonlinear problems.
	\item What is the outer solution (first-order profile) and higher-order profiles for solution with the small thermal relaxations? Particularly since the absence of an initial condition, does the singular layer appear for small $\tau$? Inherited by the linear problem, the next condition may be guaranteed the existence for initial layer:
	\begin{align*}
		\psi^{\tau}_0\equiv\psi^{0}_0,\ \ \psi^{\tau}_1\equiv\psi^{0}_1,\ \ \psi^{\tau}_2\not\equiv\frac{\Delta\psi^{\tau}_0+\delta\Delta\psi^{\tau}_1+2\nabla\psi^{\tau}_0\cdot\nabla\psi^{\tau}_1}{1-\frac{B}{A}\psi^{\tau}_1}.
	\end{align*}
\end{enumerate}
\medskip


However, these questions for the Cauchy problem are still open. For one thing, Kuznetsov's equation \eqref{Kuz_Dissipative} belongs to nonlinear $2$-evolution equation and the JMGT equation \eqref{JMGT_tau_Dissipative} is strictly nonlinear hyperbolic, which means the desired strong convergence as $\tau\downarrow 0$ will change the type of models. Thus, different techniques are needed to derive global (in time) existence of solution, especially, the uniform bounded estimates play a crucial role. For another, with the aim of obtaining the global (in time) convergence result, it is significant to get some ``good'' decay estimates for the limit equation \eqref{Kuz_Dissipative}, which is still not clear.

