# Time-fractional Moore–Gibson–Thompson equations

Barbara Kaltenbacher1 and Vanja Nikolić2 Department of Mathematics
Alpen-Adria-Universität Klagenfurt
Universitätsstr. 65–67, A-9020 Klagenfurt, Austria
<EMAIL_ADDRESS>Department of Mathematics
Radboud University
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider several time-fractional generalizations of the
Jordan–Moore–Gibson–Thompson (JMGT) equations in nonlinear acoustics as well
as their linear Moore–Gibson–Thompson (MGT) versions. Following the procedure
described in Jordan (2014), these time-fractional acoustic equations are
derived from four fractional versions of the Maxwell–Cattaneo law in Compte
and Metzler (1997). Additionally to providing well-posedness results for each
of them, we also study the respective limits as the fractional order tends to
one, leading to the classical third order in time (J)MGT equation.

###### Key words and phrases:

fractional MGT equations, nonlinear acoustics, well-posedness, limiting
behavior

###### 2010 Mathematics Subject Classification:

35L05, 35L72

1Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Austria
2Department of Mathematics, Radboud University, The Netherlands

##  1\. Introduction

It is well-known that using the Fourier temperature flux law, given by

(1.1) $\boldsymbol{q}=-\kappa\nabla\theta,$

in the derivation of second-order models of nonlinear acoustics may lead to
the so-called paradox of infinite speed of propagation; see [33, 26, 31, 23,
25]. As a remedy, the Maxwell–Cattaneo law may be used instead

$\boldsymbol{q}+\tau\boldsymbol{q}_{t}=-\kappa\nabla\theta,$

whereby a time lag $\tau>0$ is introduced between the heat flux and the
temperature induced by it. This change within the governing equations leads to
the third-order in time sound propagation described by a family of
Moore–Gibson–Thompson (MGT) equations in linear acoustics:

(1.2) $\displaystyle\tau\psi_{ttt}+\psi_{tt}-c^{2}\Delta\psi-(\tau
c^{2}+\delta)\Delta\psi_{t}=0$

or Jordan–Moore–Gibson–Thompson (JMGT) equations in nonlinear acoustics:

(1.3) $\displaystyle\tau\psi_{ttt}+\psi_{tt}-c^{2}\Delta\psi-(\tau
c^{2}+\delta)\Delta\psi_{t}=f(\psi_{t},\psi_{tt},\nabla\psi,\nabla\psi_{t});$

see the works of Moore and Gibson [35], Thompson [42], and Jordan [24, 23] for
a detailed insight into their derivation and physical background and [27, 29,
6, 28, 36] for a selection of results on their mathematical analysis.
However, a drawback of using the hyperbolic heat equation is that it may
violate the second law of thermodynamics; see, for example, [45, 17, 18].
Fractional generalizations of the heat flux law have emerged in the literature
as a way of interpolating between the properties of the two flux laws; see,
e.g., [38, 11, 16, 4] and the references contained therein. In [11], Compte
and Metzler proposed several generalized time-fractional heat-flux laws in the
following form:

(1.4)
$\displaystyle(1+\tau^{\alpha_{1}}\textup{D}_{t}^{\alpha_{1}})\boldsymbol{q}(t)=-\kappa\textup{D}_{t}^{\alpha_{2}}\nabla\theta,$

where the choice of $(\alpha_{1},\alpha_{2})$ arises from a particular
anomalous diffusion process in complex media. In the present work, we derive
and analyze the time-fractional (J)MGT equations that arise from the use of
fractional temperature laws (1.4) in place of the standard heat-flux law
within the governing equations.
One such model coming from the choice of fractional orders
$(\alpha_{1},\alpha_{2})=(1,1-\alpha)$ in the generalized Cattaneo law (1.4)
is given by

(1.5) $\tau\psi_{ttt}+\psi_{tt}-c^{2}\Delta\psi-(\tau
c^{2}+\delta{\textup{D}}_{t}^{1-\alpha})\Delta\psi_{t}=f(\psi_{t},\psi_{tt},\nabla\psi,\nabla\psi_{t}),\
0<\alpha\leq 1,$

whereas the choice $(\alpha_{1},\alpha_{2})=(\alpha,1-\alpha)$ leads to

(1.6)
$\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta{\textup{D}}_{t}^{2-\alpha}\Delta\psi=f(\psi_{t},\psi_{tt},\nabla\psi,\nabla\psi_{t}),$

where we assume that $\alpha\in(1/2,1)$. We refer to Section 2 below for the
definition of $\textup{D}_{t}^{\alpha}$ and details on the modeling and to
Tables 1 and 2 for a complete list of the fractional models that are
considered in this work. In particular, we analyze the time-fractional JMGT
equations in terms of local-in-time solvability and the limiting behavior of
their solutions as $\alpha\rightarrow 1^{-}$.
To the best of our knowledge, this is the first work dealing with the
mathematical analysis of time-fractional MGT models. We point out that, on the
other hand, (J)MGT equations with memories that involve smooth kernel
functions represent an active field of research; see, e.g., [34, 9, 13, 3, 12,
7] and the references contained therein.
Our exposition is organized as follows. In Section 2 we derive four fractional
versions of JMGT based on the four instances of (1.4) elaborated on in [11].
After a short Section 3 with mathematical notation and tools, we first in
Section 4 focus on the version (1.5) of fixed highest order three. We prove
its well-posedness in the linear as well as in the nonlinear case without
gradient nonlinearity and justify the limit $\alpha\to 1^{-}$. Next, in
Section 5 we provide a similar analysis for the other models, that have in
common a $2+\alpha$ leading derivative. This analysis works out with one
exception, where the damping term is too weak to allow for varying
coefficients or nonlinearities and whose linear version is analyzed separately
in Section 7 based on a reformulation as a second-order wave equation. Before
doing so, we return to (1.5) in Section 6 and provide well-posedness and the
limit $\alpha\to 1^{-}$ in its full version, including the gradient
nonlinearity, which requires higher-order energy estimates.

##  2\. Modeling with generalized heat-flux equations

In this section, we consider the four general versions of the constitutive
equation (1.1) proposed by Compte and Metzler in [11] and discuss the
resulting acoustic equations. These time-fractional general flux equations
(GFE) are as follows:

(2.1) (GFE)
$\displaystyle\qquad\qquad(1+\tau^{\alpha}{\textup{D}}_{t}^{\alpha})\boldsymbol{q}(t)=$
$\displaystyle\,-\kappa\nabla\theta;\hphantom{{\textup{D}}_{t}^{1-\alpha}}$
(2.2) (GFE I)
$\displaystyle\qquad\qquad(1+\tau^{\alpha}{\textup{D}}_{t}^{\alpha})\boldsymbol{q}(t)=$
$\displaystyle\,-\kappa{\textup{D}}_{t}^{1-\alpha}\nabla\theta;$ (2.3) (GFE
II)
$\displaystyle\qquad\qquad(1+\tau^{\alpha}{\textup{D}}_{t}^{\alpha})\boldsymbol{q}(t)=$
$\displaystyle\,-\kappa\textup{D}_{t}^{\alpha-1}\nabla\theta;$ (2.4) (GFE III)
$\displaystyle\qquad\qquad(1+\tau\partial_{t})\boldsymbol{q}(t)=$
$\displaystyle\,-\kappa{\textup{D}}_{t}^{1-\alpha}\nabla\theta,$

where $\boldsymbol{q}$ denotes the flux vector, $\theta$ the absolute
temperature, and $\kappa$ is the thermal conductivity. A numerical study and
comparison of the four resulting fractional heat equations has been performed
in [45] in a one-dimensional setting. Although they can all predict negative
temperatures, the fractional heat equation based on using (GFE I) appears to
avoid this nonphysical behavior for $\alpha\in(1/2,1)$ close enough to $1/2$.
Note that while Compte and Metzler [11] state the equations using the
Riemann–Liouville fractional derivative, in the present work
$\textup{D}_{t}^{\gamma}$ always denotes the Caputo–Djrbashian fractional
derivative:

$\textup{D}_{t}^{\gamma}w(t)=\frac{1}{\Gamma(1-\gamma)}\int_{0}^{t}(t-s)^{-\gamma}\textup{D}_{t}^{\lceil\gamma\rceil}w(s)\,\textup{d}s,\quad-1<\gamma<1;$

see, for example, [32, §1] and [37, §2.4.1] for its definition. Here
$n=\lceil\gamma\rceil$, $n\in\\{0,1\\}$ is the integer obtained by rounding up
$\gamma$ and $\textup{D}_{t}^{n}$ is the zeroth or first derivative operator.

(fJMGT) We begin by discussing the modeling with the first option; that is

(2.5)
$\displaystyle(1+\tau^{\alpha}{\textup{D}}_{t}^{\alpha})\boldsymbol{q}(t)=-\kappa\nabla\theta;$

cf. [45, Eq. (9)]. We note that this modification of the heat-flux law is
introduced _ad hoc_ in [11] and then disregarded, however numerical studies of
the resulting heat equation in [45] incorporate it as well, and so we include
it here.
The derivation of the acoustic equation follows the steps taken in [24, §4]
with now (2.5) in place of the Maxwell–Cattaneo law. This derivation employs a
weakly-nonlinear approximation, which for our purposes can be restated as

(2.6)
$\epsilon<<1,\quad\theta=O(\epsilon),\quad\tilde{K}=O(\epsilon),\quad\tau^{\alpha}=O(\epsilon),\quad|\mathfrak{e}|=O(\epsilon^{2}).$

Here $\epsilon$ is the Mach number, $\tilde{K}$ is the dimensionless thermal
diffusivity, and $\mathfrak{e}$ the dimensionless entropy. Note that, compared
to [24], the condition $\tau^{\alpha}=O(\epsilon)$ replaces $\tau=O(\epsilon)$
here.
It is assumed that the sound wave propagates through a thermally conductive
and relaxing liquid or gas with negligible viscosity. Starting from a one-
dimensional setting, the governing system is first approximated by

(2.7)
$\displaystyle\psi_{tt}+\tfrac{1}{2}\epsilon\partial_{t}(\psi_{x})^{2}-(1+(\gamma-2)s)[\psi_{x}s_{x}+(1+s)\psi_{xx}]=-\epsilon^{-1}\mathfrak{e}_{t},$

where $\psi$ is the acoustic velocity potential, $\gamma$ the adiabatic index,
and $s$ is known as the condensation; see [24, Eq. (44)–(49) and Eq. (53)].
Upon employing $s\approx-\epsilon\psi_{t}$, one arrives at

(2.8)
$\displaystyle\psi_{tt}+\tfrac{1}{2}\epsilon\partial_{t}(\psi_{x})^{2}-(1-(\gamma-2)\epsilon\psi_{t})[-\epsilon\psi_{x}\psi_{tx}+(1-\epsilon\psi_{t})\psi_{xx}]=-\epsilon^{-1}\mathfrak{e}_{t};$

cf. [24, Eq. (49)]. From the entropy production law

$\tilde{\kappa}{\mathfrak{e}_{t}}=-\tilde{K}\boldsymbol{q}_{x},$

with $\tilde{\kappa}$ being the dimensionless thermal conductivity, and the
general heat flux law (2.5) in a dimensionless version

$(1+\lambda_{\alpha}{\textup{D}}_{t}^{\alpha})\boldsymbol{q}(t)=-\tilde{\kappa}\nabla\theta,$

we then have the following entropy equation:

$(1+\lambda_{\alpha}{\textup{D}}_{t}^{\alpha})\mathfrak{e}_{t}=\tilde{K}\theta_{xx}.$

After utilizing that $\theta\approx-\epsilon(\gamma-1)\psi_{t}$, we can
rewrite it as

(2.9)
$\displaystyle(1+\lambda_{\alpha}{\textup{D}}_{t}^{\alpha})\mathfrak{e}_{t}=-\epsilon\tilde{K}(\gamma-1)\psi_{txx};$

cf. [23, Eq. (57) and (58)]. Applying the relaxation operator
$(1+\lambda_{\alpha}{\textup{D}}_{t}^{\alpha})$ to (2.8) and using (2.9) to
eliminate $\mathfrak{e}$ then leads to

(2.10)
$\displaystyle\begin{multlined}(1+\lambda_{\alpha}{\textup{D}}_{t}^{\alpha})\left\\{\psi_{tt}+\tfrac{1}{2}\epsilon\partial_{t}(\psi_{x})^{2}-(1-(\gamma-2)\epsilon\psi_{t})[-\epsilon\psi_{x}\psi_{tx}+(1-\epsilon\psi_{t})\psi_{xx}]\right\\}\\\
=\,\tilde{K}(\gamma-1)\psi_{txx}.\end{multlined}(1+\lambda_{\alpha}{\textup{D}}_{t}^{\alpha})\left\\{\psi_{tt}+\tfrac{1}{2}\epsilon\partial_{t}(\psi_{x})^{2}-(1-(\gamma-2)\epsilon\psi_{t})[-\epsilon\psi_{x}\psi_{tx}+(1-\epsilon\psi_{t})\psi_{xx}]\right\\}\\\
=\,\tilde{K}(\gamma-1)\psi_{txx}.$

With $\lambda_{\alpha}=O(\epsilon)$, by neglecting the $O(\epsilon^{2})$ terms
in the equation above, we arrive at

(2.11)
$\displaystyle(1+\lambda_{\alpha}{\textup{D}}_{t}^{\alpha})\psi_{tt}-\lambda_{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{xx}-(1-\epsilon(\gamma-1)\psi_{t})\psi_{xx}+\epsilon\partial_{t}(\psi_{x})^{2}=\tilde{K}(\gamma-1)\psi_{txx}.$

Dividing this equation by $(1-\epsilon(\gamma-1)\psi_{t})$, using
$(1-\epsilon(\gamma-1)\psi_{t})^{-1}\approx 1+\epsilon(\gamma-1)\psi_{t}$ for
$\epsilon<<1$ and neglecting all $O(\epsilon^{2})$ terms, yields

(2.12)
$\displaystyle\begin{multlined}\lambda_{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+\epsilon(\gamma-1)\psi_{t})\psi_{tt}-\psi_{xx}-\lambda_{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{xx}\\\
\hskip
142.26378pt-\tilde{K}(\gamma-1)\psi_{txx}+\epsilon\partial_{t}(\psi_{x})^{2}=0.\end{multlined}\lambda_{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+\epsilon(\gamma-1)\psi_{t})\psi_{tt}-\psi_{xx}-\lambda_{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{xx}\\\
\hskip
142.26378pt-\tilde{K}(\gamma-1)\psi_{txx}+\epsilon\partial_{t}(\psi_{x})^{2}=0.$

Extrapolating to a dimensionalized 3D model in a mathematically general form
gives

(2.13)
$\displaystyle\begin{multlined}\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta\Delta\psi_{t}+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=0.\end{multlined}\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta\Delta\psi_{t}+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=0.$

Since the quadratic gradient nonlinearity present in this model corresponds to
the one in the second-order Kuznetsov equation [33], we will henceforth refer
to (2.13) as the fractional JMGT–Kuznetsov equation, or the fJMGT–K equation
for short.
Assuming local nonlinear effects can be neglected so that

(2.14) $|\nabla\psi|^{2}\approx c^{-2}\psi_{t}^{2},$

we obtain

(2.15)
$\displaystyle\begin{multlined}\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta\Delta\psi_{t}=0.\end{multlined}\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta\Delta\psi_{t}=0.$

The above approximation corresponds to the one commonly used when reducing the
Kuznetsov equation to the Westvervelt second-order model of nonlinear
acoustics; cf. [25, §2.3]. For this reason, we will refer to (2.15) as the
fractional Jordan–Moore–Gibson–Thompson–Westervelt equation, or the fJMGT–W
equation for short. This approximation is appropriate when cumulative
nonlinear effects dominate the local ones, which is the case, e.g., for sound
propagation sufficiently far from the source in terms of wavelengths; see the
discussion in [22, Ch. 3, Section 6].

(fJMGT I) As a second option, we employ the general heat-flux model given by

(2.16)
$\displaystyle(1+\tau^{\alpha}{\textup{D}}_{t}^{\alpha})\boldsymbol{q}(t)=-\kappa{\textup{D}}_{t}^{1-\alpha}\nabla\theta;$

see [11, Eq. (14)] and [45, Eq. (10)]. The use of this flux law is motivated
in [11] stochastically by fractal time random walks. Retracing the derivation
steps from before leads to the following equation:

(2.17)
$\displaystyle\begin{multlined}(1+\lambda_{\alpha}{\textup{D}}_{t}^{\alpha})\left\\{\psi_{tt}+\tfrac{1}{2}\epsilon\partial_{t}(\psi_{x})^{2}-(1-(\gamma-2)\epsilon\psi_{t})[-\epsilon\psi_{x}\psi_{tx}+(1-\epsilon\psi_{t})\psi_{xx}]\right\\}\\\
=\,\tilde{K}(\gamma-1){\textup{D}}_{t}^{1-\alpha}\psi_{txx}\end{multlined}(1+\lambda_{\alpha}{\textup{D}}_{t}^{\alpha})\left\\{\psi_{tt}+\tfrac{1}{2}\epsilon\partial_{t}(\psi_{x})^{2}-(1-(\gamma-2)\epsilon\psi_{t})[-\epsilon\psi_{x}\psi_{tx}+(1-\epsilon\psi_{t})\psi_{xx}]\right\\}\\\
=\,\tilde{K}(\gamma-1){\textup{D}}_{t}^{1-\alpha}\psi_{txx}$

in place of (2.10). Neglecting the $O(\epsilon^{2})$ terms then yields

(2.18)
$\displaystyle(1+\lambda_{\alpha}{\textup{D}}_{t}^{\alpha})\psi_{tt}-(1-\epsilon(\gamma-1)\psi_{t})\psi_{xx}-\lambda_{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{xx}+\epsilon\partial_{t}(\psi_{x})^{2}=\tilde{K}(\gamma-1){\textup{D}}_{t}^{1-\alpha}\psi_{txx}.$

Analogously to before, dividing by $(1-\epsilon(\gamma-1)\psi_{t})$ leads to

(2.19)
$\displaystyle\begin{multlined}\lambda_{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+\epsilon(\gamma-1)\psi_{t})\psi_{tt}-\psi_{xx}-\lambda_{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{xx}-\tilde{K}(\gamma-1){\textup{D}}_{t}^{1-\alpha}\psi_{txx}\\\
+\epsilon\partial_{t}(\psi_{x})^{2}=0\end{multlined}\lambda_{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+\epsilon(\gamma-1)\psi_{t})\psi_{tt}-\psi_{xx}-\lambda_{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{xx}-\tilde{K}(\gamma-1){\textup{D}}_{t}^{1-\alpha}\psi_{txx}\\\
+\epsilon\partial_{t}(\psi_{x})^{2}=0$

in place of (2.12). Then extrapolating to a general 3D equation gives

(2.20)
$\displaystyle\begin{multlined}\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=0,\end{multlined}\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=0,$

which we will call the fractional Jordan–Moore–Gibson–Thompson–Kuznetsov
equation of type I, or the fJMGT–K I equation for short. By assuming local
nonlinear effects can be neglected as before, we arrive at

(2.21)
$\displaystyle\begin{multlined}\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}=0,\end{multlined}\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}=0,$

which we will refer to as the fractional
Jordan–Moore–Gibson–Thompson–Westervelt equation of type I, or just the
fJMGT–W I equation.

(fJMGT II) Thirdly, we employ the heat-flux model given by

(2.22)
$\displaystyle(1+\tau^{\alpha}{\textup{D}}_{t}^{\alpha})\boldsymbol{q}(t)=-\kappa\textup{D}_{t}^{\alpha-1}\nabla\theta;$

cf. [11, Eq. (14)] and [45, Eq. (11)]. This flux law is motivated in [11] by
nonlocal transport theory with memory effects; that is, a nonlocal relation
between the flux $\boldsymbol{q}$ and temperature $\theta$:

$\boldsymbol{q}(x,t)=\int_{0}^{t}\mathfrak{k}(t-s)\nabla\theta(x,s)\,\textup{d}s,$

with a suitable choice of the kernel. Analogously to before, we can derive the
following general fractional model:

(2.23)
$\displaystyle\begin{multlined}\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta{\textup{D}}_{t}^{\alpha}\Delta\psi+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=0,\end{multlined}\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta{\textup{D}}_{t}^{\alpha}\Delta\psi+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=0,$

which we will from now on refer to as the fractional
Jordan–Moore–Gibson–Thompson–Kuznetsov equation of type II, or the fJMGT–K II
equation for short. If the local nonlinear effects can be neglected, we obtain

(2.24)
$\displaystyle\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-(\tau^{\alpha}c^{2}+\delta){\textup{D}}_{t}^{\alpha}\Delta\psi=0,$

which we will refer to as the fractional
Jordan–Moore–Gibson–Thompson–Westervelt equation of type II, or the fJMGT–W II
equation for short.

(fJMGT III) Finally, we consider the wave-like acoustic models resulting from
using the following flux law:

(2.25)
$\displaystyle(1+\tau\partial_{t})\boldsymbol{q}(t)=-\kappa{\textup{D}}_{t}^{1-\alpha}\nabla\theta;$

see [11, Eq. (18)] and [45, Eq. (12)]. In [11], this law is motivated by a
delayed equation that may connect the flux to a generalized force

$\boldsymbol{q}(t+\tau)=-\kappa{\textup{D}}_{t}^{1-\alpha}\nabla\theta.$

Here, weakly-nonlinear acoustic approximation is based on assuming that

(2.26)
$\epsilon<<1,\quad\theta=O(\epsilon),\quad\tilde{K}=O(\epsilon),\quad\tau=O(\epsilon),\quad|\mathfrak{e}|=O(\epsilon^{2}).$

Retracing our previous derivation steps then quickly leads to

(2.27)
$\displaystyle\begin{multlined}\tau\psi_{ttt}+(1+\epsilon(\gamma-1)\psi_{t})\psi_{tt}-\psi_{xx}-\tau\psi_{txx}-\tilde{K}(\gamma-1){\textup{D}}_{t}^{1-\alpha}\psi_{txx}\\\
+\epsilon\partial_{t}(\psi_{x})^{2}=0.\end{multlined}\tau\psi_{ttt}+(1+\epsilon(\gamma-1)\psi_{t})\psi_{tt}-\psi_{xx}-\tau\psi_{txx}-\tilde{K}(\gamma-1){\textup{D}}_{t}^{1-\alpha}\psi_{txx}\\\
+\epsilon\partial_{t}(\psi_{x})^{2}=0.$

Extrapolating to a dimensionalized 3D model yields

(2.28)
$\displaystyle\begin{multlined}\tau\psi_{ttt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau
c^{2}\Delta\psi_{t}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=0,\end{multlined}\tau\psi_{ttt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau
c^{2}\Delta\psi_{t}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=0,$

which we will henceforth refer to it as the fractional
Jordan–Moore–Gibson–Thompson–Kuznetsov equation of type III, or the fJMGT–K
III equation for short. If the local nonlinear effects can be neglected, we
obtain

(2.29)
$\displaystyle\begin{multlined}\tau\psi_{ttt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau
c^{2}\Delta\psi_{t}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}=0.\end{multlined}\tau\psi_{ttt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau
c^{2}\Delta\psi_{t}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}=0.$

We will refer to this model as the fractional
Jordan–Moore–Gibson–Thompson–Westervelt equation of type III, or the fJMGT–W
III equation for short.

We collect all discussed time-fractional acoustic equations in Table 1 for
convenience and state them with a general source function $f$. Note that the
constant $\delta>0$ for models I–III no longer has the dimension of usual
sound diffusivity.
We assume that $\alpha\in(0,1]$ in the fJMGT II and III equations, whereas we
perform the analysis of the fJMGT and fJMGT I equations under the assumption
that $\alpha\in(1/2,1]$. Formally letting $\alpha\rightarrow 1^{-}$ in these
equations leads to the Jordan–Moore–Gibson–Thompson equations, either in the
Westervelt or Kuznetsov forms; cf. [24].

fJMGT– |  Nonlinear time-fractional acoustic equations
---|---
K |  $\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta\Delta\psi_{t}+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=f$
W |  $\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta\Delta\psi_{t}=f$
K I |  $\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta{\textup{D}}_{t}^{2-\alpha}\Delta\psi+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=f$
W I |  $\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta{\textup{D}}_{t}^{2-\alpha}\Delta\psi=f$
K II |  $\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta{\textup{D}}_{t}^{\alpha}\Delta\psi+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=f$
W II |  $\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-(\tau^{\alpha}c^{2}+\delta){\textup{D}}_{t}^{\alpha}\Delta\psi=f$
K III |  $\tau\psi_{ttt}+(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau c^{2}\Delta\psi_{t}-\delta{\textup{D}}_{t}^{2-\alpha}\Delta\psi+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=f$
W III |  $\tau\psi_{ttt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau c^{2}\Delta\psi_{t}-\delta{\textup{D}}_{t}^{2-\alpha}\Delta\psi=f$
Table 1. Nonlinear fJMGT models in the Kuznetsov and Westervelt forms.

We will also study the linearizations of these equations (obtained by setting
$k=\tilde{k}=\tilde{\ell}=0$), which we will refer to as fractional
Moore–Gibson–Thompson (fMGT) equations; cf. Table 2 below.

##  3\. Theoretical preliminaries

In this section, we gather several theoretical results from fractional
calculus that will be useful later on. To simplify the notation, we often omit
the spatial domain and the time interval when writing norms; for example,
$\|\cdot\|_{W_{t}^{p,q}(L^{r})}$ denotes the norm on
$W^{p,q}(0,t;L^{r}(\Omega))$ and $\|\cdot\|_{W^{p,q}(L^{r})}$ denotes the norm
on $W^{p,q}(0,T;L^{r}(\Omega))$.
Throughout the paper, we assume that $\Omega\subset\mathbb{R}^{n}$ is an open,
bounded, and sufficiently smooth set, where $n\in\\{1,2,3\\}$. When writing
solution spaces for $\psi$, we use the following notational convention:

(3.1) $\displaystyle{H_{\diamondsuit}^{2}(\Omega)}=$
$\displaystyle\,H_{0}^{1}(\Omega)\cap H^{2}(\Omega),$
$\displaystyle{H_{\diamondsuit}^{3}(\Omega)}=$ $\displaystyle\,\left\\{\psi\in
H^{3}(\Omega)\,:\,\mbox{tr}_{\partial\Omega}\psi=0,\
\mbox{tr}_{\partial\Omega}\Delta\psi=0\right\\}.$

In the analysis, we will rely on the continuous embeddings
$H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$ and $H^{2}(\Omega)\hookrightarrow
L^{\infty}(\Omega)$:

(3.2) $\displaystyle\|v\|_{L^{4}(\Omega)}\leq C_{H^{1},L^{4}}\|\nabla
v\|_{L^{2}(\Omega)},\quad$ $\displaystyle v\in H^{1}_{0}(\Omega)$
$\displaystyle\|v\|_{L^{\infty}(\Omega)}\leq C_{H^{2},L^{\infty}}\|\Delta
v\|_{L^{2}(\Omega)},\quad$ $\displaystyle v\in{H_{\diamondsuit}^{2}(\Omega)}.$

We often write $x\lesssim y$ instead of $x\leq Cy$. In such cases, $C>0$
represents a generic constant that may depend on the medium parameters and the
final time $T$, but does not depend on the order of differentiation $\alpha$.
Throughout the paper we make the following assumptions on the (constant)
medium parameters:

(3.3) $\tau>0,\quad c>0,\quad\delta>0,\quad
k,\,\tilde{k},\,\tilde{\ell}\,\in\mathbb{R}.$

### Coercivity estimates

When performing energy analysis, we will rely on the following two coercivity
estimates.

  * •

[2, Lemma 1]: For any absolutely continuous function $w$,

(3.4)
${w(t)}\textup{D}_{t}^{\gamma}w(t)\geq\tfrac{1}{2}(\textup{D}_{t}^{\gamma}w^{2})(t).$

  * •

[14, Lemma 2.3]; see also [43, Theorem 1]: For any $w\in
H^{-(1-\alpha)/2}(0,t)$,

(3.5)
$\int_{0}^{t}\langle\textup{I}^{1-\alpha}w(s),w(s)\rangle\,\textup{d}s\geq\cos(\tfrac{\pi(1-\alpha)}{2})\|w\|_{H^{-(1-\alpha)/2}(0,t)}^{2},$

where $\textup{I}^{\gamma}$ denotes the Abel integral operator:

$\textup{I}^{\gamma}w(t)=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}(t-s)^{\gamma-1}w(s)\,\textup{d}s,\quad\gamma\in(0,1)$

and the negative norm is defined by

$\|w\|_{H^{-\gamma}(0,t)}^{2}=\int_{\mathbb{R}}(1+\omega^{2})^{-\gamma}|\hat{w}(\omega)|^{2}\,\textup{d}\omega,\quad\gamma>0,$

with $\hat{w}$ being the Fourier transform of the extension by zero of $w$ to
all of $\mathbb{R}$.

### The Kato–Ponce inequality

The following product rule estimate holds:

(3.6)
$\|fg\|_{W^{\rho,r}(0,T)}\lesssim\|f\|_{W^{\rho,p_{1}}(0,T)}\|g\|_{L^{q_{1}}(0,T)}+\|f\|_{L^{p_{2}}(0,T)}\|g\|_{W^{\rho,q_{2}}(0,T)}$

for $0\leq\rho\leq\overline{\rho}<1$, $1<r<\infty$, $p_{1}$, $p_{2}$, $q_{1}$,
$q_{2}\in(1,\infty]$, with $\frac{1}{r}=\frac{1}{p_{i}}+\frac{1}{q_{i}}$,
$i=1,2$; see, e.g., [20]. We note that the fractional derivative
$\partial_{t}^{\rho}=(I-D_{tt})^{\rho/2}$ used in [20] is an isomorphism
between $H^{\rho}(0,T)$ and $L^{2}(0,T)$ for any $\rho\in[0,1]$.

### Limits with respect to the order of differentiation

Let $w:[0,T]\mapsto X$, where $X$ is a Banach space equipped with the norm
$\|\cdot\|_{X}$. It is known that the mapping
$\alpha\mapsto{\textup{D}}_{t}^{\alpha}w(t)$ is left-sided continuous at any
integer $\alpha$, while it is discontinuous from the right side unless
$w(0)=0$. This property is due to the following two identities that follow by
integration by parts:

(3.7) $\displaystyle\lim_{\alpha\rightarrow
1^{-}}\left\|{\textup{D}}_{t}^{\alpha}w(t)-w_{t}(t)\right\|_{X}$
$\displaystyle=$ $\displaystyle\lim_{\alpha\rightarrow
1^{-}}\left\|\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-s)^{-\alpha}w_{t}(s)\,\textup{d}s-w_{t}(t)\right\|_{X}$
$\displaystyle=$ $\displaystyle\lim_{\alpha\rightarrow
1^{-}}\left\|\frac{t^{1-\alpha}w_{t}(0)}{\Gamma(2-\alpha)}+\frac{1}{\Gamma(2-\alpha)}\int_{0}^{t}(t-s)^{1-\alpha}w_{tt}(s)\,\textup{d}s-w_{t}(t)\right\|_{X}$
$\displaystyle=$
$\displaystyle\,\left\|w_{t}(0)+\int_{0}^{t}w_{tt}(s)\,\textup{d}s-w_{t}(t)\right\|_{X}=\,0,$

as well as

(3.8) $\displaystyle\lim_{\alpha\rightarrow
0^{+}}\left\|{\textup{D}}_{t}^{\alpha}w(t)-w(t)\right\|_{X}=$
$\displaystyle\,\lim_{\alpha\rightarrow
0^{+}}\left\|\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-s)^{-\alpha}w_{t}(s)\,\textup{d}s-w(t)\right\|_{X}$
$\displaystyle=$ $\displaystyle\,\left\|w(0)\right\|_{X};$

see, for example, [37, §2.4.1]. These limits extend to $L^{p}(0,T;X)$ under
relaxed smoothness assumptions on $w$. Here and below we use the notation

(3.9)
$\displaystyle\mathfrak{g}_{\alpha}(t)=\frac{t^{-\alpha}}{\Gamma(1-\alpha)},\quad\mathfrak{g}_{0}(t)=1.$

######  Lemma 3.1.

Let $1\leq p<\infty$. For any $w\in W^{1,p}(0,T;X){\,\cap\,W^{2,1}(0,T;X)}$,
it holds that

(3.10) $\limsup_{\alpha\rightarrow
1^{-}}\left\|{\textup{D}}_{t}^{\alpha}w-w_{t}\right\|_{L^{p}(0,T;X)}=0.$

For any $w\in W^{1,1}(0,T;X)$,

(3.11) ${\textup{D}}_{t}^{\alpha}w\rightharpoonup w_{t}\mbox{ in
}L^{1}(0,T;X)\mbox{ as }\alpha\rightarrow 1^{-},$

and

(3.12) $\limsup_{\alpha\rightarrow
0^{+}}\left\|{\textup{D}}_{t}^{\alpha}w-w+w(0)\right\|_{L^{p}(0,T;X)}=0\,.$

###### Proof.

To prove (3.12), we rely on Young’s convolution inequality and Lebesgue’s
Dominated Convergence theorem, which yields

$\|\mathfrak{g}_{\alpha}-\mathfrak{g}_{0}\|_{L^{p}(0,T)}\rightarrow 0\
\text{as}\ \alpha\to 0^{+},$

with the $L^{1}$ dominating function
$t\mapsto\dfrac{t^{-p\alpha_{0}}}{\Gamma(1-\alpha_{0})^{p}}$ for
$\alpha\leq\alpha_{0}<1/p$ (without loss of generality since $\alpha$ tends to
zero). Therefore, (3.12) holds via

$\displaystyle\limsup_{\alpha\rightarrow
0^{+}}\left\|{\textup{D}}_{t}^{\alpha}w-w+w(0)\right\|_{L^{p}(0,T;X)}=$
$\displaystyle\,\limsup_{\alpha\rightarrow
0^{+}}\left\|(\mathfrak{g}_{\alpha}-\mathfrak{g}_{0})*w_{t}\right\|_{L^{p}(0,T;X)}$
$\displaystyle\leq$ $\displaystyle\,\limsup_{\alpha\rightarrow
0^{+}}\left\|\mathfrak{g}_{\alpha}-\mathfrak{g}_{0}\right\|_{L^{p}(0,T)}\left\|w_{t}\right\|_{L^{1}(0,T;X)}\,=0.$

To show (3.11), we use the fact that for any $w\in W^{1,1}(0,T;X)$ and $v\in
L^{\infty}(0,T;X^{*})$ where $X^{*}$ is the dual space of $X$,

(3.13)
$\displaystyle\int_{0}^{T}\left\langle\int_{0}^{t}\mathfrak{g}_{\alpha}(t-s)w_{t}(s)\,\textup{d}s-w_{t}(t),v(t)\right\rangle_{X}\,\textup{d}t$
$\displaystyle=$
$\displaystyle\,\int_{0}^{T}\int_{0}^{t}\langle\mathfrak{g}_{\alpha}(t-s)w_{t}(s)\,\textup{d}s,v(t)\rangle_{X}\,\textup{d}t-\int_{0}^{T}\langle
w_{t}(t),v(t)\rangle_{X}\,\textup{d}t$ $\displaystyle=$
$\displaystyle\,\int_{0}^{T}\left\langle
w_{t}(s),\int_{s}^{T}\mathfrak{g}_{\alpha}(t-s)v(t)\,\textup{d}t\right\rangle_{X}\,\textup{d}s-\int_{0}^{T}\langle
w_{t}(r),v(r)\rangle_{X}\,\textup{d}t$ $\displaystyle=$
$\displaystyle\,\int_{0}^{T}\left\langle
w_{t}(r),\Bigl{(}\int_{r}^{T}\mathfrak{g}_{\alpha}(t-r)v(t)\,\textup{d}t-v(r)\Bigr{)}\right\rangle_{X}\textup{d}r\
=:\int_{0}^{T}\psi(r)\textup{d}r,$

where $\langle\cdot,\cdot\rangle_{X}$ denotes the dual pairing. From the
identity

(3.14)
$\displaystyle\int_{0}^{T}\left\langle\phi(r),\Bigl{(}\int_{r}^{T}\mathfrak{g}_{\alpha}(t-r)v(t)\,\textup{d}t-v(r)\Bigr{)}\right\rangle_{X}$
$\displaystyle=$
$\displaystyle\,\int_{0}^{T}\left\langle\int_{0}^{t}\mathfrak{g}_{\alpha}(t-s)\phi(s)\,\textup{d}s-\phi(t),v(t)\right\rangle_{X}\,\textup{d}t$
$\displaystyle=$
$\displaystyle\,\int_{0}^{T}\frac{1}{\Gamma(2-\alpha)}\left\langle
t^{1-\alpha}\phi(0)+\int_{0}^{t}(t-s)^{1-\alpha}\phi_{t}(s)\,\textup{d}s-\phi(t),v(t)\right\rangle_{X}\,\textup{d}t\
\to 0\mbox{ as }\alpha\to 1^{-}$

for any $\phi\in C^{1}(0,T;X)$, we conclude that
$\Bigl{(}\int_{r}^{T}\mathfrak{g}_{\alpha}(t-r)v(t)\,\textup{d}t-v(r)\Bigr{)}$
tends to zero pointwise almost everywhere on $(0,T)$ and therefore so does the
integrand $\psi$ in (3.13).
On the other hand, $\psi$ can be bounded by the $L^{1}(0,T)$ function

$r\mapsto(C_{T}+1)\|w_{t}(r)\|_{X}\|v\|_{L^{\infty}(0,T;X^{*})},$

where we have estimated

(3.15)
$\displaystyle\left\|\int_{r}^{T}\mathfrak{g}_{\alpha}(t-r)v(t)\,\textup{d}t\right\|_{X^{*}}$
$\displaystyle\leq$
$\displaystyle\,\|v\|_{L^{\infty}(0,T;X^{*})}\int_{r}^{T}\mathfrak{g}_{\alpha}(t-r)\,\textup{d}r=\|v\|_{L^{\infty}(0,T;X^{*})}\frac{(T-r)^{1-\alpha}}{\Gamma(2-\alpha)}$

and set

$C_{T}:=\sup_{\alpha\in(\frac{1}{2},1]}\frac{T^{1-\alpha}}{\Gamma(2-\alpha)}<\infty.$

Thus by Lebesgue’s Dominated Convergence theorem, (3.13) tends to zero as
$\alpha\to 1^{-}$. Since $v\in L^{\infty}(0,T;X^{*})$ is arbitrary, we obtain
weak convergence of ${\textup{D}}_{t}^{\alpha}w-w_{t}$ to zero in
$L^{1}(0,T;X)$. If $X$ is a Hilbert space, this computation can be repeated
replacing dual pairings with inner products and yields weak convergence in the
Hilbert space sense.
Finally, in case $w\in W^{1,p}(0,T;X)\cap W^{2,1}(0,T;X)$, (3.10) follows from

$\displaystyle\limsup_{\alpha\rightarrow
1^{-}}\left\|{\textup{D}}_{t}^{\alpha}w-w_{t}\right\|_{L^{p}(0,T;X)}$
$\displaystyle=$ $\displaystyle\,\limsup_{\alpha\rightarrow
1^{-}}\Bigl{(}\int_{0}^{T}{\|(\mathfrak{g}_{\alpha}*w_{t})(t)-w_{t}(t)\|_{X}}^{p}\,\textup{d}t\Bigr{)}^{1/p}=0\,,$

where we have again used Lebesgue’s Dominated Convergence theorem. Indeed,
$(\mathfrak{g}_{\alpha}*w_{t})-w_{t}$ (and therefore the $p$th power of its
norm) tends to zero almost everywhere on $(0,T)$ as $\alpha\to 1^{-}$, on
account of the previous discussion. Furthermore,

$\|(\mathfrak{g}_{\alpha}*w_{t})-w_{t}\|_{X}^{p}\leq
2^{p-1}\|\mathfrak{g}_{\alpha}*w_{t}\|_{X}^{p}+2^{p-1}\|w_{t}\|_{X}^{p}$

can be further dominated by the $L^{1}(0,T)$ function

$2^{p}\max\\{1,T\\}^{p}\Bigl{(}\|w_{t}(0)\|_{X}+\|w_{tt}\|_{L^{1}(0,T;X)}\Bigr{)}^{p}+2^{p-1}\|w_{t}\|_{X}^{p}$

due to

$\displaystyle\|(\mathfrak{g}_{\alpha}*w_{t})(t)\|_{X}=\frac{t^{1-\alpha}}{\Gamma(2-\alpha)}\left\|w_{t}(0)+\int_{0}^{t}(1-\tfrac{s}{t})^{1-\alpha}w_{tt}(s)\,\textup{d}s\right\|_{X}$
$\displaystyle\leq
C_{T}\left(\|w_{t}(0)\|_{X}+\|w_{tt}\|_{L^{1}(0,T;X)}\right);$

cf. (3.7) with $C_{T}$ as above. ∎

##  4\. Analysis of the equation with the third-order leading term

We begin our analytical considerations by looking at the fJMGT–W III equation

(4.1) $\tau\psi_{ttt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau
c^{2}\Delta\psi_{t}-\delta{\textup{D}}_{t}^{2-\alpha}\Delta\psi=f,$

which has an integer-order leading term; cf. Table 1. We intend to analyze it
by setting up a fixed-point mapping. To this end, we, first of all, study the
following linearization:

(4.2) $\tau\psi_{ttt}+(1+\sigma(x,t))\psi_{tt}-c^{2}\Delta\psi-\tau
c^{2}\Delta\psi_{t}-\delta{\textup{D}}_{t}^{2-\alpha}\Delta\psi=f,\quad
0<\alpha\leq 1.$

This is the linear fMGT III equation with a variable coefficient, which we
assume is uniformly bounded; cf. Table 2 below. More precisely, we assume that
there exist $\underline{\sigma}$, $\overline{\sigma}>0$, such that

(4.3) $\underline{\sigma}\leq\sigma(x,t)\leq\overline{\sigma}\ \mbox{ for all
}\ x\in\Omega,\,t\in(0,T).$

Note that since we study the local-in-time behavior in this work, we do not
impose a non-degeneracy condition on $1+\sigma$. In the upcoming analysis, the
crucial estimate involving fractional derivatives will be the following:

(4.4)
$\displaystyle\int_{0}^{t}({\textup{D}}_{t}^{2-\alpha}w,w_{tt})_{L^{2}}\,\textup{d}s\geq$
$\displaystyle\,\cos(\pi\alpha/2)\|w_{tt}\|_{H^{-\alpha/2}(0,t;L^{2}(\Omega))}^{2},$

which follows by (3.5). To formulate the first well-posedness result, we
introduce the solution space

(4.5) $\displaystyle\quad X^{\textup{low}}_{\textup{fMGT
III}}=\,\left\\{\vphantom{H^{-\alpha/2}(0,T;H_{0}^{1}(\Omega))}\right.$
$\displaystyle\psi\in L^{\infty}(0,T;H_{0}^{1}(\Omega)):\psi_{t}\in
L^{\infty}(0,T;H_{0}^{1}(\Omega)),$ $\displaystyle\,\psi_{tt}\in
L^{\infty}(0,T;L^{2}(\Omega))\cap H^{-\alpha/2}(0,T;H_{0}^{1}(\Omega)),$
$\displaystyle\left.\,\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}\|^{2}_{L^{2}}\in
L^{\infty}(0,T),\ \psi_{ttt}\in
L^{2}(0,T;H^{-1}(\Omega))\vphantom{H^{-\alpha/2}(0,T;H_{0}^{1}(\Omega))}\right\\}$

for $\alpha\in(0,1)$, and

(4.6) $\displaystyle X^{\textup{low}}_{\textup{fMGT
III}}=W^{1,\infty}(0,T;H_{0}^{1}(\Omega))\cap
W^{2,\infty}(0,T;L^{2}(\Omega))\cap H^{3}(0,T;H^{-1}(\Omega))$

in case $\alpha=1$. We denote by $\|\cdot\|_{X^{\textup{low}}_{\textup{fMGT
III}}}$ the corresponding norm on this space. We claim that the fMGT III
equation (4.2) has a unique solution in this space under suitable assumptions
on the data and the variable coefficient.

######  Proposition 4.1 (Well-posedness of the fMGT III equation).

Let $\alpha\in(0,1]$ and $\sigma\in L^{\infty}(0,T;L^{\infty}(\Omega))$. Given
$f\in L^{2}(0,T;L^{2}(\Omega))$ and

$(\psi_{0},\psi_{1},\psi_{2})\in(H_{0}^{1}(\Omega),H_{0}^{1}(\Omega),L^{2}(\Omega)),$

there exists a unique $\psi$ in $X^{\textup{low}}_{\textup{fMGT III}}$, such
that

(4.7)
$\displaystyle\begin{multlined}\langle\tau\psi_{ttt},v\rangle_{H^{-1},H_{0}^{1}}+((1+\sigma)\psi_{tt},v)_{L^{2}}\\\
+(c^{2}\nabla\psi+\tau
c^{2}\nabla\psi_{t}+\delta{\textup{D}}_{t}^{2-\alpha}\nabla\psi,\nabla
v)_{L^{2}}=(f,v)_{L^{2}}\end{multlined}\langle\tau\psi_{ttt},v\rangle_{H^{-1},H_{0}^{1}}+((1+\sigma)\psi_{tt},v)_{L^{2}}\\\
+(c^{2}\nabla\psi+\tau
c^{2}\nabla\psi_{t}+\delta{\textup{D}}_{t}^{2-\alpha}\nabla\psi,\nabla
v)_{L^{2}}=(f,v)_{L^{2}}$

for all $v\in H_{0}^{1}(\Omega)$, a.e. in $(0,T)$, with
$(\psi,\psi_{t},\psi_{tt})|_{t=0}=(\psi_{0},\psi_{1},\psi_{2})$. Furthermore,
the solution satisfies the following estimate:

(4.8)
$\displaystyle\,\begin{multlined}\|\psi\|^{2}_{W^{1,\infty}(H^{1})}+\|\psi_{tt}\|_{L^{\infty}(L^{2})}^{2}+\|\psi_{ttt}\|^{2}_{L^{2}(H^{-1})}\\\\[5.69054pt]
\hskip
28.45274pt+\cos(\alpha\pi/2)\|\psi_{tt}\|^{2}_{H^{-\alpha/2}(H^{1})}+\sup_{t\in(0,T)}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}\|^{2}_{L^{2}}\end{multlined}\|\psi\|^{2}_{W^{1,\infty}(H^{1})}+\|\psi_{tt}\|_{L^{\infty}(L^{2})}^{2}+\|\psi_{ttt}\|^{2}_{L^{2}(H^{-1})}\\\\[5.69054pt]
\hskip
28.45274pt+\cos(\alpha\pi/2)\|\psi_{tt}\|^{2}_{H^{-\alpha/2}(H^{1})}+\sup_{t\in(0,T)}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}\|^{2}_{L^{2}}$
$\displaystyle\lesssim$
$\displaystyle\,\|f\|_{L^{2}(L^{2})}^{2}+\|\nabla\psi_{0}\|_{L^{2}}^{2}+\|\nabla\psi_{1}\|_{L^{2}}^{2}+\|\psi_{2}\|_{L^{2}}^{2},$

where for $\alpha=1$, the $\cos(\alpha\pi/2)$ term should be omitted.

###### Proof.

We focus in the proof on the case $\alpha\in(0,1)$ since the case $\alpha=1$
follows in a more straightforward manner. We perform the analysis by employing
the standard Galerkin procedure to discretize the problem in space [15, §7],
with alterations needed to accommodate the third-order derivative and the
fractional term. We approximate the solution by

(4.9) $\displaystyle\psi^{n}(x,t)=\sum_{i=1}^{n}\xi^{n}_{i}(t)\phi_{i}(x),$

where $\\{\phi_{i}\\}_{i=1}^{\infty}$ are the eigenfunctions of the Dirichlet-
Laplacian operator:

$-\Delta\phi_{i}=\lambda_{i}\phi_{i}\ \text{ in }\Omega,\qquad\phi_{i}=0\
\text{ on }\partial\Omega.$

Denote $V_{n}=\textup{span}\\{\phi_{1},\ldots,\phi_{n}\\}$. The semi-discrete
problem is given by

(4.10)
$\displaystyle\begin{multlined}(\tau\psi_{ttt}^{n},\phi_{j})_{L^{2}}+((1+\sigma)\psi_{tt}^{n},\phi_{j})_{L^{2}}\\\
+(c^{2}\nabla\psi^{n}+\tau
c^{2}\nabla\psi_{t}^{n}+\delta{\textup{D}}_{t}^{2-\alpha}\nabla\psi^{n},\nabla\phi_{j})_{L^{2}}=(f,\phi_{j})_{L^{2}},\end{multlined}(\tau\psi_{ttt}^{n},\phi_{j})_{L^{2}}+((1+\sigma)\psi_{tt}^{n},\phi_{j})_{L^{2}}\\\
+(c^{2}\nabla\psi^{n}+\tau
c^{2}\nabla\psi_{t}^{n}+\delta{\textup{D}}_{t}^{2-\alpha}\nabla\psi^{n},\nabla\phi_{j})_{L^{2}}=(f,\phi_{j})_{L^{2}},$

for all $j=1,\ldots,n$, with approximate initial conditions
$(\psi^{n},\psi_{t}^{n},\psi_{tt}^{n})|_{t=0}=(\psi_{0}^{n},\psi_{1}^{n},\psi_{2}^{n})$
chosen as $L^{2}$ projections of $(\psi_{0},\psi_{1},\psi_{2})$ onto $V_{n}$.
In other words,

(4.11)
$\displaystyle\psi_{0}^{n}=\sum_{i=1}^{n}\xi^{n}_{0,i}\phi_{i}(x),\quad\psi_{1}^{n}=\sum_{i=1}^{n}\xi^{n}_{1,i}\phi_{i}(x),\quad\psi_{2}^{n}=\sum_{i=1}^{n}\xi^{n}_{2,i}\phi_{i}(x)$

with

(4.12)
$\xi^{n}_{0,i}=(\psi_{0},\phi_{i})_{L^{2}},\quad\xi^{n}_{1,i}=(\psi_{1},\phi_{i})_{L^{2}},\quad\xi^{n}_{2,i}=(\psi_{2},\phi_{i})_{L^{2}}.$

Then we know that

(4.13) $\displaystyle\|\psi^{n}_{0}\|_{H^{1}}\leq$
$\displaystyle\,\|\psi_{0}\|_{H^{1}}\ \text{ and }\ $
$\displaystyle\psi^{n}_{0}\rightarrow\psi_{0}\ \text{ in }\
H_{0}^{1}(\Omega),$ $\displaystyle\|\psi^{n}_{1}\|_{H^{1}}\leq$
$\displaystyle\,\|\psi_{1}\|_{H^{1}}\ \text{ and }\ $
$\displaystyle\psi^{n}_{1}\rightarrow\psi_{1}\ \text{ in }\
H_{0}^{1}(\Omega),$ $\displaystyle\|\psi^{n}_{2}\|_{L^{2}}\leq$
$\displaystyle\,\|\psi_{2}\|_{L^{2}}\ \,\text{ and }\ $
$\displaystyle\psi^{n}_{2}\rightarrow\psi_{2}\ \text{ in }\ L^{2}(\Omega);$

cf. [39, §7, Lemma 7.5].

(I) _Existence of an approximate solution._ We first show that for a given
$n$, a unique approximate solution exists. With
$\boldsymbol{\xi}=[\xi^{n}_{1}\ \xi^{n}_{2}\ \ldots\ \xi^{n}_{n}]^{T}$, the
approximate problem can be rewritten in matrix form

(4.14) $\displaystyle\tau
M\boldsymbol{\xi}_{ttt}+M_{\sigma}(t)\boldsymbol{\xi}_{tt}+c^{2}K\boldsymbol{\xi}+\tau
c^{2}K\boldsymbol{\xi}_{t}+\delta
K{\textup{D}}_{t}^{2-\alpha}\boldsymbol{\xi}=\boldsymbol{f}$

with the entries of matrices $M=[M_{ij}]$, $M_{\sigma}(t)=[M_{\sigma,ij}(t)]$,
$K=[K_{ij}]$, and the vector $\boldsymbol{f}(t)=[f_{i}(t)]$ given by

(4.15) $\displaystyle M_{ij}=(\phi_{i},\phi_{j})_{L^{2}},\quad$ $\displaystyle
M_{\sigma,ij}(t)=((1+\sigma(t))\phi_{i},\phi_{j})_{L^{2}},$ $\displaystyle
K_{ij}=(\nabla\phi_{i},\nabla\phi_{j})_{L^{2}},\quad$ $\displaystyle
f_{i}(t)=(f(t),\phi_{i})_{L^{2}}.$

We also introduce the vectors of coordinates of the approximate initial data
in the basis:

$\boldsymbol{\xi}_{0}=[\xi^{n}_{0,1}\ \xi^{n}_{0,2}\ \ldots\
\xi^{n}_{0,n}]^{T},\quad\boldsymbol{\xi}_{1}=[\xi^{n}_{1,1}\ \xi^{n}_{1,2}\
\ldots\ \xi^{n}_{1,n}]^{T},\quad\boldsymbol{\xi}_{2}=[\xi^{n}_{2,1}\
\xi^{n}_{2,2}\ \ldots\ \xi^{n}_{2,n}]^{T}.$

Then by setting $\boldsymbol{\mu}=\boldsymbol{\xi}_{ttt}$, we have

(4.16)
$\boldsymbol{\xi}(t)=\boldsymbol{\xi}_{0}+t\boldsymbol{\xi}_{1}+\frac{1}{2}t^{2}\boldsymbol{\xi}_{2}+\int_{0}^{t}\frac{1}{2}(t-s)^{2}\boldsymbol{\mu}(s)\,\textup{d}s,$

and we can restate the semi-discrete problem as a system of Volterra integral
equations given by

(4.17) $\displaystyle\begin{multlined}\tau
M\boldsymbol{\mu}(t)+M_{\sigma}(t)\left(\int_{0}^{t}\boldsymbol{\mu}(s)\,\textup{d}s+\boldsymbol{\xi}_{2}\right)+c^{2}K\left(\boldsymbol{\xi}_{0}+t\boldsymbol{\xi}_{1}+\frac{t^{2}}{2}\boldsymbol{\xi}_{2}+\int_{0}^{t}\frac{1}{2}(t-s)^{2}\boldsymbol{\mu}(s)\,\textup{d}s\right)\\\
+\tau
c^{2}K\left(\boldsymbol{\xi}_{1}+t\boldsymbol{\xi}_{2}+\int_{0}^{t}(t-s)\boldsymbol{\mu}(s)\,\textup{d}s\right)+\frac{\delta}{\Gamma(\alpha)}K\int_{0}^{t}(t-s)^{\alpha-1}\left(\int_{0}^{s}\boldsymbol{\mu}(r)\,\textup{d}r+\boldsymbol{\xi}_{2}\right)\textup{d}s\\\
=\boldsymbol{f}(t)\hphantom{fill}\end{multlined}\tau
M\boldsymbol{\mu}(t)+M_{\sigma}(t)\left(\int_{0}^{t}\boldsymbol{\mu}(s)\,\textup{d}s+\boldsymbol{\xi}_{2}\right)+c^{2}K\left(\boldsymbol{\xi}_{0}+t\boldsymbol{\xi}_{1}+\frac{t^{2}}{2}\boldsymbol{\xi}_{2}+\int_{0}^{t}\frac{1}{2}(t-s)^{2}\boldsymbol{\mu}(s)\,\textup{d}s\right)\\\
+\tau
c^{2}K\left(\boldsymbol{\xi}_{1}+t\boldsymbol{\xi}_{2}+\int_{0}^{t}(t-s)\boldsymbol{\mu}(s)\,\textup{d}s\right)+\frac{\delta}{\Gamma(\alpha)}K\int_{0}^{t}(t-s)^{\alpha-1}\left(\int_{0}^{s}\boldsymbol{\mu}(r)\,\textup{d}r+\boldsymbol{\xi}_{2}\right)\textup{d}s\\\
=\boldsymbol{f}(t)\hphantom{fill}$

for $t\in(0,T)$. By using Dirichlet’s formula

(4.18)
$\displaystyle{\frac{\delta}{\Gamma(\alpha)}}\int_{0}^{t}(t-s)^{\alpha-1}\left(\int_{0}^{s}\boldsymbol{\mu}(r)\,\textup{d}r\right)\,\textup{d}s={\frac{\delta}{\Gamma(\alpha)}}\int_{0}^{t}\left(\int_{r}^{t}(t-s)^{\alpha-1}\,\textup{d}s\right)\boldsymbol{\mu}(r)\textup{d}r,$

we arrive at an equivalent reformulation

(4.19)
$\displaystyle\boldsymbol{\mu}(t)=\tilde{\boldsymbol{f}}(t)+\int_{0}^{t}K_{\alpha}(t,s)\boldsymbol{\mu}(s)\,\textup{d}s,$

where the first term on the right is defined as

(4.20) $\displaystyle\tilde{\boldsymbol{f}}(t)=$
$\displaystyle\,\begin{multlined}-\frac{1}{\tau}M^{-1}\left\\{M_{\sigma}(t)\boldsymbol{\xi}_{2}+c^{2}K(\boldsymbol{\xi}_{0}+t\boldsymbol{\xi}_{1}+\frac{1}{2}t^{2}\boldsymbol{\xi}_{2})+\tau
c^{2}K(\boldsymbol{\xi}_{1}+t\boldsymbol{\xi}_{2})-\boldsymbol{f}(t)\right.\\\
\left.+\frac{\delta}{\Gamma(\alpha)}K\int_{0}^{t}(t-s)^{\alpha-1}\boldsymbol{\xi}_{2}\,\textup{d}s\right\\}\end{multlined}-\frac{1}{\tau}M^{-1}\left\\{M_{\sigma}(t)\boldsymbol{\xi}_{2}+c^{2}K(\boldsymbol{\xi}_{0}+t\boldsymbol{\xi}_{1}+\frac{1}{2}t^{2}\boldsymbol{\xi}_{2})+\tau
c^{2}K(\boldsymbol{\xi}_{1}+t\boldsymbol{\xi}_{2})-\boldsymbol{f}(t)\right.\\\
\left.+\frac{\delta}{\Gamma(\alpha)}K\int_{0}^{t}(t-s)^{\alpha-1}\boldsymbol{\xi}_{2}\,\textup{d}s\right\\}$

and the kernel function is given by

(4.21) $\displaystyle K_{\alpha}(t,s)=$
$\displaystyle\,\begin{multlined}-\frac{1}{\tau}M^{-1}\left(M_{\sigma}(t)+\frac{1}{2}c^{2}K(t-s)^{2}+\tau
c^{2}K(t-s)\right)\\\
-\frac{1}{\tau}\frac{\delta}{\Gamma(\alpha+1)}M^{-1}K(t-s)^{\alpha}.\end{multlined}-\frac{1}{\tau}M^{-1}\left(M_{\sigma}(t)+\frac{1}{2}c^{2}K(t-s)^{2}+\tau
c^{2}K(t-s)\right)\\\
-\frac{1}{\tau}\frac{\delta}{\Gamma(\alpha+1)}M^{-1}K(t-s)^{\alpha}.$

To arrive at the kernel expression, we have employed

$\frac{\delta}{\Gamma(\alpha)}K\int_{s}^{t}(t-r)^{\alpha-1}\,\textup{d}r=\frac{\delta}{\alpha\Gamma(\alpha)}K(t-s)^{\alpha}.$

Due to the $L^{\infty}(0,T)$ regularity of the kernel $K_{\alpha}$ on
$D=\\{(t,s):\ 0\leq s\leq t\leq T\\}$ and the fact that function
$\tilde{\boldsymbol{f}}$ belongs to $L^{2}(0,T)$, vector equation (4.19) has a
unique solution $\boldsymbol{\mu}\in L^{2}(0,T)$. This claim directly follows
by considering (systems of) integral equations in $L^{2}(0,T)$ instead of
$C[0,T]$ in [8, Theorem 2.1.7]; see also [21, Theorem 4.2, p. 24 in §9]. From
(4.16), taking into account initial data, a unique $\boldsymbol{\xi}\in
H^{3}(0,T)$ and, in turn, $\psi^{n}\in H^{3}(0,T;V_{n})$ exists.

(II) _A priori energy analysis._ We next focus on the derivation of the energy
estimate, which goes through by testing the semi-discrete problem by
$\psi^{n}_{tt}(t)\in V_{n}$. More precisely, we test (4.10) with
$\xi^{n}_{tt}(t)$ and sum over $j=1,\ldots,n$. After integrating over $(0,t)$,
we at first obtain the identity

(4.22)
$\displaystyle\begin{multlined}\frac{1}{2}\tau\|\psi_{tt}^{n}(t)\|_{L^{2}}^{2}\Big{|}_{0}^{t}+\frac{1}{2}\tau
c^{2}\|\nabla\psi_{t}^{n}(s)\|_{L^{2}}^{2}\Big{|}_{0}^{t}+\delta\int_{0}^{t}(\textup{D}_{t}^{2-\alpha}\nabla\psi^{n},\nabla\psi_{tt}^{n})_{L^{2}}\,\textup{d}s\end{multlined}\frac{1}{2}\tau\|\psi_{tt}^{n}(t)\|_{L^{2}}^{2}\Big{|}_{0}^{t}+\frac{1}{2}\tau
c^{2}\|\nabla\psi_{t}^{n}(s)\|_{L^{2}}^{2}\Big{|}_{0}^{t}+\delta\int_{0}^{t}(\textup{D}_{t}^{2-\alpha}\nabla\psi^{n},\nabla\psi_{tt}^{n})_{L^{2}}\,\textup{d}s$
$\displaystyle=$
$\displaystyle\begin{multlined}-\int_{0}^{t}(1+\sigma)\|\psi_{tt}^{n}\|^{2}_{L^{2}}\,\textup{d}s+\int_{0}^{t}(f,\psi_{tt}^{n})_{L^{2}}\,\textup{d}s-c^{2}(\nabla\psi^{n},\nabla\psi_{t}^{n})_{L^{2}}\Big{|}_{0}^{t}+c^{2}\int_{0}^{t}\|\nabla\psi_{t}^{n}\|_{L^{2}}^{2}\,\textup{d}s.\end{multlined}-\int_{0}^{t}(1+\sigma)\|\psi_{tt}^{n}\|^{2}_{L^{2}}\,\textup{d}s+\int_{0}^{t}(f,\psi_{tt}^{n})_{L^{2}}\,\textup{d}s-c^{2}(\nabla\psi^{n},\nabla\psi_{t}^{n})_{L^{2}}\Big{|}_{0}^{t}+c^{2}\int_{0}^{t}\|\nabla\psi_{t}^{n}\|_{L^{2}}^{2}\,\textup{d}s.$

By Young’s $\varepsilon$-inequality, we have

(4.23) $\displaystyle
c^{2}\int_{0}^{t}(\nabla\psi^{n},\nabla\psi_{t}^{n})_{L^{2}}\,\textup{d}s\leq\frac{1}{4\varepsilon}c^{2}(C(T)\|\nabla\psi_{t}^{n}\|_{L^{2}_{t}(L^{2})}+\|\nabla\psi^{n}_{0}\|_{L^{2}})^{2}+\varepsilon
c^{2}\|\nabla\psi_{t}^{n}(t)\|_{L^{2}}^{2}.$

For $0<\varepsilon<\tau/2$, employing estimate (4.4) and Gronwall’s inequality
leads to

(4.24)
$\displaystyle\begin{multlined}\|\psi_{tt}^{n}(t)\|_{L^{2}}^{2}+\|\nabla\psi_{t}^{n}(t)\|_{L^{2}}^{2}+\cos(\pi\alpha/2)\|\nabla\psi_{tt}^{n}\|_{H^{-\alpha/2}(0,t;L^{2}(\Omega))}^{2}\end{multlined}\|\psi_{tt}^{n}(t)\|_{L^{2}}^{2}+\|\nabla\psi_{t}^{n}(t)\|_{L^{2}}^{2}+\cos(\pi\alpha/2)\|\nabla\psi_{tt}^{n}\|_{H^{-\alpha/2}(0,t;L^{2}(\Omega))}^{2}$
$\displaystyle\lesssim$
$\displaystyle\,\begin{multlined}\|f\|_{L^{2}(L^{2})}^{2}+\|\nabla\psi_{0}\|_{L^{2}}^{2}+\|\nabla\psi_{1}\|_{L^{2}}^{2}+\|\psi_{2}\|_{L^{2}}^{2}.\end{multlined}\|f\|_{L^{2}(L^{2})}^{2}+\|\nabla\psi_{0}\|_{L^{2}}^{2}+\|\nabla\psi_{1}\|_{L^{2}}^{2}+\|\psi_{2}\|_{L^{2}}^{2}.$

Note that we can also use the estimate

$\|\psi^{n}(t)\|_{L^{2}}\leq
C_{\textup{PF}}\|\nabla\psi^{n}(t)\|_{L^{2}}\lesssim
T\sup_{t\in(0,T)}\|\nabla\psi_{t}^{n}(t)\|_{L^{2}}+{\|\nabla\psi^{n}_{0}\|_{L^{2}}}.$

Additionally, standard arguments (cf. [15, §7]) lead to the bound

(4.25)
$\displaystyle\int_{0}^{t}\|\psi_{ttt}^{n}\|^{2}_{H^{-1}}\,\textup{d}s\lesssim$
$\displaystyle\,\begin{multlined}\int_{0}^{t}(1+\overline{\sigma})\|\psi_{tt}^{n}\|_{L^{2}}^{2}\,\textup{d}s+\int_{0}^{t}\|\nabla\psi^{n}\|_{L^{2}}^{2}\,\textup{d}s\\\
+\int_{0}^{t}\|\nabla\psi_{t}^{n}\|_{L^{2}}^{2}\,\textup{d}s+\int_{0}^{t}\|{\textup{D}}_{t}^{2-\alpha}\nabla\psi^{n}\|_{L^{2}}^{2}\,\textup{d}s+\int_{0}^{t}\|f\|_{L^{2}}^{2}\,\textup{d}s.\end{multlined}\int_{0}^{t}(1+\overline{\sigma})\|\psi_{tt}^{n}\|_{L^{2}}^{2}\,\textup{d}s+\int_{0}^{t}\|\nabla\psi^{n}\|_{L^{2}}^{2}\,\textup{d}s\\\
+\int_{0}^{t}\|\nabla\psi_{t}^{n}\|_{L^{2}}^{2}\,\textup{d}s+\int_{0}^{t}\|{\textup{D}}_{t}^{2-\alpha}\nabla\psi^{n}\|_{L^{2}}^{2}\,\textup{d}s+\int_{0}^{t}\|f\|_{L^{2}}^{2}\,\textup{d}s.$

We can further estimate the fractional term on the right as follows:

(4.26)
$\displaystyle\|{\textup{D}}_{t}^{2-\alpha}\nabla\psi^{n}\|^{2}_{L^{2}(0,t;L^{2})}=\|\textup{I}^{\alpha}\nabla\psi_{tt}^{n}\|^{2}_{L^{2}(0,t;L^{2})}\lesssim$
$\displaystyle\,\|\nabla\psi_{tt}^{n}\|^{2}_{X_{-\alpha}(0,t;L^{2})}$
$\displaystyle\lesssim$
$\displaystyle\,\|\nabla\psi_{tt}^{n}\|^{2}_{H^{-\alpha/2}(0,t;L^{2})},$

where the first inequality follows by [19, Theorem 1]. Note that
$\\{X_{\beta}\\}_{\beta\in\mathbb{R}}$ represents a scale of Hilbert spaces of
functions $(0,t)\mapsto L^{2}(\Omega)$; cf. [5, §5] and [19, Lemma 8]. The
second inequality follows by the fact that

$X_{\alpha}(0,t;L^{2}(\Omega))\subseteq
X_{\alpha/2}(0,t;L^{2}(\Omega))=H_{0}^{\alpha/2}(0,t;L^{2}(\Omega))$

for $\alpha<1$ and therefore $\alpha/2<\frac{1}{2}$, together with duality.
Note also that, on account of estimate (3.4), we know that

(4.27)
$\displaystyle(\nabla{\textup{D}}_{t}^{1-\alpha}\psi^{n}_{t},\nabla\psi_{tt}^{n})_{L^{2}}=$
$\displaystyle\,(\nabla{\textup{D}}_{t}^{1-\alpha}\psi^{n}_{t},\textup{D}_{t}^{\alpha}\nabla\textup{D}_{t}^{1-\alpha}\psi_{t}^{n})_{L^{2}}$
$\displaystyle\geq$
$\displaystyle\,\frac{1}{2}\textup{D}_{t}^{\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}^{n}\|^{2}_{L^{2}}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\,\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}^{n}\|^{2}_{L^{2}}.$

Employing this estimate instead of (4.4) in the above derivation yields a
uniform bound on
$\displaystyle\sup_{t\in(0,T)}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}^{n}\|^{2}_{L^{2}}$,
which will be needed in the proof of uniqueness.

(III) _Passing to the limit._ Thanks to the uniform bounds and (4.26), we have
weak convergence of a subsequence, which we do not relabel, in the following
sense:

(4.28) $\displaystyle\psi_{ttt}^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup\psi_{ttt}$ weakly in
$\displaystyle L^{2}(0,T;H^{-1}(\Omega)),$ $\displaystyle\psi_{tt}^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup\psi_{tt}$ weakly-$\star$ in
$\displaystyle L^{\infty}(0,T;L^{2}(\Omega)),$ $\displaystyle\psi_{t}^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup\psi_{t}$ weakly-$\star$ in
$\displaystyle L^{\infty}(0,T;H_{0}^{1}(\Omega)),$ $\displaystyle\psi^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup\psi$ weakly-$\star$ in
$\displaystyle L^{\infty}(0,T;H_{0}^{1}(\Omega)).$

Furthermore, we know that

(4.29) $\displaystyle\nabla\psi_{tt}^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup\nabla\psi_{tt}$ weakly in
$\displaystyle H^{-\alpha/2}(0,T;L^{2}(\Omega)),$
$\displaystyle{\textup{D}}_{t}^{2-\alpha}\nabla\psi^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup{\textup{D}}_{t}^{2-\alpha}\nabla\psi$
weakly in $\displaystyle L^{2}(0,T;L^{2}(\Omega)),$
$\displaystyle\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}^{n}\|^{2}_{L^{2}}$
$\displaystyle\relbar\joinrel\rightharpoonup\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}\|^{2}_{L^{2}}$
weakly-$\star$ in $\displaystyle L^{\infty}(0,T).$

We can thus pass to the weak limit in the usual way to conclude that $\psi$
solves (4.7). Further, weak/weak-$\star$ lower semi-continuity of norms
implies

$\displaystyle\|\nabla\psi_{tt}\|_{H^{-\alpha/2}(0,t;L^{2}(\Omega))}^{2}\leq$
$\displaystyle\,\liminf_{n\rightarrow\infty}\|\nabla\psi_{tt}^{n}\|_{H^{-\alpha/2}(0,t;L^{2}(\Omega))}^{2},$
$\displaystyle\sup_{t\in(0,T)}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}\|^{2}_{L^{2}}\leq$
$\displaystyle\,\liminf_{n\rightarrow\infty}\sup_{t\in(0,T)}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}^{n}\|^{2}_{L^{2}}.$

and thus by passing to the limit in the energy estimate for $\psi^{n}$, we
conclude that $\psi$ satisfies (4.8).

(IV) _Attainment of the initial conditions._ We next show that $\psi$ attains
its initial conditions. By (4.28) and [46, Lemma 3.1.7], we know that

$\psi^{n}(0)\relbar\joinrel\rightharpoonup\psi(0)\text{ weakly}\text{ in
}H_{0}^{1}(\Omega),$

and since $\psi^{n}(0)\rightarrow\psi_{0}$ in $H_{0}^{1}(\Omega)$, we have
$\psi(0)=\psi_{0}$. Further,

$\psi_{t}^{n}(0)\relbar\joinrel\rightharpoonup\psi_{t}(0)\text{ weakly}\text{
in }L^{2}(\Omega),$

and thus $\psi_{t}(0)=\psi_{1}$ as an equality in $L^{2}(\Omega)$. Similarly,
$\psi_{tt}(0)=\psi_{2}$ as an equality in $H^{-1}(\Omega)$; that is,

$\langle\psi_{tt}(0),v\rangle_{H^{-1},H^{1}}=\langle\psi_{2},v\rangle_{H^{-1},H^{1}},\quad\forall
v\in H_{0}^{1}(\Omega).$

(V) _Uniqueness._ To prove uniqueness, we should show that the only solution
of

(4.30) $\displaystyle\tau\psi_{ttt}+(1+\sigma)\psi_{tt}-\tau
c^{2}\Delta\psi_{t}-c^{2}\Delta\psi-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}=0$

with $\psi_{0}=\psi_{1}=\psi_{2}=0$ in $X^{\textup{low}}_{\textup{fMGT III}}$
is $\psi=0$. The issue, however, is that at this point we are not allowed to
directly test (4.7) with $\psi_{tt}$ due to its low regularity. Instead, in
the spirit of [41, §2.4], we will prove that such $\psi$ satisfies

(4.31) $\displaystyle(\tau\psi_{ttt}-\tau
c^{2}\Delta\psi_{t}-c^{2}\Delta\psi-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t},\psi_{tt})_{L^{2}}$
$\displaystyle\gtrsim$
$\displaystyle\,\begin{multlined}\frac{\textup{d}}{\textup{d}t}\left\\{\frac{1}{2}\tau\|\psi_{tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\|\nabla\psi_{t}\|_{L^{2}}^{2}+c^{2}(\nabla\psi,\nabla\psi_{t})_{L^{2}}\right.\\\
\left.\hskip
85.35826pt+\frac{\delta}{2}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}\|^{2}_{L^{2}}\right\\}-c^{2}\|\nabla\psi_{t}\|_{L^{2}}^{2}.\end{multlined}\frac{\textup{d}}{\textup{d}t}\left\\{\frac{1}{2}\tau\|\psi_{tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\|\nabla\psi_{t}\|_{L^{2}}^{2}+c^{2}(\nabla\psi,\nabla\psi_{t})_{L^{2}}\right.\\\
\left.\hskip
85.35826pt+\frac{\delta}{2}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}\|^{2}_{L^{2}}\right\\}-c^{2}\|\nabla\psi_{t}\|_{L^{2}}^{2}.$

This estimate combined with (4.30) implies that

(4.32)
$\displaystyle\begin{multlined}\frac{\textup{d}}{\textup{d}t}\left\\{\frac{1}{2}\tau\|\psi_{tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\|\nabla\psi_{t}\|_{L^{2}}^{2}+c^{2}(\nabla\psi,\nabla\psi_{t})_{L^{2}}\right.\\\
\left.\hskip
85.35826pt+\frac{\delta}{2}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}\|^{2}_{L^{2}}\right\\}-c^{2}\|\nabla\psi_{t}\|_{L^{2}}^{2}\lesssim-((1+\sigma\psi_{tt},\psi_{tt})_{L^{2}},\end{multlined}\frac{\textup{d}}{\textup{d}t}\left\\{\frac{1}{2}\tau\|\psi_{tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\|\nabla\psi_{t}\|_{L^{2}}^{2}+c^{2}(\nabla\psi,\nabla\psi_{t})_{L^{2}}\right.\\\
\left.\hskip
85.35826pt+\frac{\delta}{2}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\psi_{t}\|^{2}_{L^{2}}\right\\}-c^{2}\|\nabla\psi_{t}\|_{L^{2}}^{2}\lesssim-((1+\sigma\psi_{tt},\psi_{tt})_{L^{2}},$

after which we can proceed as in the previous energy analysis to arrive at
$\psi=0$. We note that if $\psi\in X^{\textup{low}}_{\textup{fMGT III}}$
solves (4.30), then

(4.33) $\displaystyle\psi_{t}\in L^{2}(H_{0}^{1}(\Omega)),\quad\psi_{tt}\in
L^{2}(0,T;L^{2}(\Omega))\cap H^{-\alpha/2}(0,T;H_{0}^{1}(\Omega)).$

Furthermore, a bootstrap argument yields

(4.34) $\displaystyle\tau\psi_{ttt}-\tau
c^{2}\Delta\psi_{t}-c^{2}\Delta\psi-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}=-(1+\sigma)\psi_{tt}\in
L^{2}(L^{2}).$

We next construct a regularization of $\psi$ which satisfies (4.31), following
[41, §2.4, Lemma 4.1]. Let $\tilde{\psi}:\mathbb{R}\mapsto H_{0}^{1}(\Omega)$
be defined by

(4.35) $\displaystyle\tilde{\psi}=\begin{cases}\theta\psi,\ &\text{ on
}(0,T),\\\ 0,\ &\text{ on }\mathbb{R}\setminus[0,T],\end{cases}$

where $\theta:\mathbb{R}\mapsto[0,1]$ is a $C^{\infty}$ truncation function,
equal to $0$ on $\mathbb{R}\setminus[0,T]$ and to $1$ on some sub-interval of
$(0,T)$. Then

(4.36) $\displaystyle\tilde{\psi}_{t}\in
L^{2}(0,\infty;H_{0}^{1}(\Omega)),\quad\tilde{\psi}_{tt}\in
L^{2}(0,\infty;L^{2}(\Omega))\cap H^{-\alpha/2}(0,\infty;H_{0}^{1}(\Omega)),$
$\displaystyle\,\tau\tilde{\psi}_{ttt}-\tau
c^{2}\Delta\tilde{\psi}_{t}-c^{2}\Delta\tilde{\psi}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\tilde{\psi}_{t}\in
L^{2}(0,\infty;L^{2}(\Omega)).$

We regularize $\tilde{\psi}$ by
$\tilde{\psi}_{\varepsilon}=\varrho_{\varepsilon}*\tilde{\psi}$, with
$\varrho_{\varepsilon}$ being a $C^{\infty}$ mollifier. Then
$\tilde{\psi}_{\varepsilon}:\mathbb{R}\mapsto H_{0}^{1}(\Omega)$ is a
$C^{\infty}$ function, which satisfies

(4.37) $\displaystyle(\tau\tilde{\psi}_{\varepsilon,ttt}-\tau
c^{2}\Delta\tilde{\psi}_{\varepsilon,t}-c^{2}\Delta\tilde{\psi}_{\varepsilon}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\tilde{\psi}_{\varepsilon,t},\tilde{\psi}_{\,\varepsilon,tt})_{L^{2}}$
$\displaystyle=$
$\displaystyle\,\begin{multlined}\frac{\textup{d}}{\textup{d}t}\left\\{\frac{1}{2}\tau\|\tilde{\psi}_{\varepsilon,tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\|\nabla\tilde{\psi}_{\varepsilon,t}\|_{L^{2}}^{2}+c^{2}(\nabla\tilde{\psi}_{\varepsilon},\nabla\tilde{\psi}_{\varepsilon,t})_{L^{2}}\right\\}\\\
-c^{2}\|\nabla\tilde{\psi}_{\varepsilon,t}\|_{L^{2}}^{2}+\delta(\nabla{\textup{D}}_{t}^{1-\alpha}\tilde{\psi}_{\varepsilon,t},\nabla\tilde{\psi}_{\varepsilon,tt})_{L^{2}}.\end{multlined}\frac{\textup{d}}{\textup{d}t}\left\\{\frac{1}{2}\tau\|\tilde{\psi}_{\varepsilon,tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\|\nabla\tilde{\psi}_{\varepsilon,t}\|_{L^{2}}^{2}+c^{2}(\nabla\tilde{\psi}_{\varepsilon},\nabla\tilde{\psi}_{\varepsilon,t})_{L^{2}}\right\\}\\\
-c^{2}\|\nabla\tilde{\psi}_{\varepsilon,t}\|_{L^{2}}^{2}+\delta(\nabla{\textup{D}}_{t}^{1-\alpha}\tilde{\psi}_{\varepsilon,t},\nabla\tilde{\psi}_{\varepsilon,tt})_{L^{2}}.$

Similarly to (4.27), we have

$\displaystyle(\nabla{\textup{D}}_{t}^{1-\alpha}\tilde{\psi}_{\varepsilon,t},\nabla\tilde{\psi}_{\varepsilon,tt})_{L^{2}}=\,(\nabla{\textup{D}}_{t}^{1-\alpha}\tilde{\psi}_{\varepsilon,t},\textup{D}_{t}^{\alpha}\nabla\textup{D}_{t}^{1-\alpha}\tilde{\psi}_{\varepsilon,t})_{L^{2}}\geq\frac{1}{2}\frac{\textup{d}}{\textup{d}t}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\tilde{\psi}_{\varepsilon,t}\|^{2}_{L^{2}}.$

Therefore,

(4.38) $\displaystyle(\tau\tilde{\psi}_{\varepsilon,ttt}-\tau
c^{2}\Delta\tilde{\psi}_{\varepsilon,t}-c^{2}\Delta\tilde{\psi}_{\varepsilon}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\tilde{\psi}_{\varepsilon,t},\tilde{\psi}_{\,\varepsilon,tt})_{L^{2}}$
$\displaystyle\gtrsim$
$\displaystyle\,\begin{multlined}\frac{\textup{d}}{\textup{d}t}\left\\{\frac{1}{2}\tau\|\tilde{\psi}_{\varepsilon,tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\|\nabla\tilde{\psi}_{\varepsilon,t}\|_{L^{2}}^{2}+c^{2}(\nabla\tilde{\psi}_{\varepsilon},\nabla\tilde{\psi}_{\varepsilon,t})_{L^{2}}\right.\\\
\left.+\frac{\delta}{2}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\tilde{\psi}_{\varepsilon,t}\|^{2}_{L^{2}}\right\\}-c^{2}\|\nabla\tilde{\psi}_{\varepsilon,t}\|_{L^{2}}^{2}.\end{multlined}\frac{\textup{d}}{\textup{d}t}\left\\{\frac{1}{2}\tau\|\tilde{\psi}_{\varepsilon,tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\|\nabla\tilde{\psi}_{\varepsilon,t}\|_{L^{2}}^{2}+c^{2}(\nabla\tilde{\psi}_{\varepsilon},\nabla\tilde{\psi}_{\varepsilon,t})_{L^{2}}\right.\\\
\left.+\frac{\delta}{2}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\tilde{\psi}_{\varepsilon,t}\|^{2}_{L^{2}}\right\\}-c^{2}\|\nabla\tilde{\psi}_{\varepsilon,t}\|_{L^{2}}^{2}.$

Thanks to (4.36), we know that

$\displaystyle\lim_{\varepsilon\rightarrow 0}\
(\tau\tilde{\psi}_{\varepsilon,ttt}-\tau
c^{2}\Delta\tilde{\psi}_{\varepsilon,t}-c^{2}\Delta\tilde{\psi}_{\varepsilon}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\tilde{\psi}_{\varepsilon,t},\tilde{\psi}_{\varepsilon,tt})_{L^{2}}$
$\displaystyle=$ $\displaystyle\,(\tau\tilde{\psi}_{ttt}-\tau
c^{2}\Delta\tilde{\psi}_{t}-c^{2}\Delta\tilde{\psi}-\delta{\textup{D}}_{t}^{\alpha}\Delta\tilde{\psi}_{t},\tilde{\psi}_{tt})_{L^{2}}.$

We can thus pass to the limit $\varepsilon\rightarrow 0$ in (4.38) to arrive
at

(4.39) $\displaystyle(\tau\tilde{\psi}_{ttt}-\tau
c^{2}\Delta\tilde{\psi}_{t}-c^{2}\Delta\tilde{\psi}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\tilde{\psi}_{t},\tilde{\psi}_{tt})_{L^{2}}$
$\displaystyle\gtrsim$
$\displaystyle\,\begin{multlined}\frac{\textup{d}}{\textup{d}t}\left\\{\frac{1}{2}\tau\|\tilde{\psi}_{tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\|\nabla\tilde{\psi}_{t}\|_{L^{2}}^{2}+c^{2}(\nabla\tilde{\psi},\nabla\tilde{\psi}_{t})_{L^{2}}+\frac{\delta}{2}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\tilde{\psi}_{t}\|^{2}_{L^{2}}\right\\}\\\
-c^{2}\|\nabla\tilde{\psi}_{t}\|_{L^{2}}^{2}.\end{multlined}\frac{\textup{d}}{\textup{d}t}\left\\{\frac{1}{2}\tau\|\tilde{\psi}_{tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\|\nabla\tilde{\psi}_{t}\|_{L^{2}}^{2}+c^{2}(\nabla\tilde{\psi},\nabla\tilde{\psi}_{t})_{L^{2}}+\frac{\delta}{2}\textup{I}^{1-\alpha}\|\nabla{\textup{D}}_{t}^{1-\alpha}\tilde{\psi}_{t}\|^{2}_{L^{2}}\right\\}\\\
-c^{2}\|\nabla\tilde{\psi}_{t}\|_{L^{2}}^{2}.$

By restriction to $(0,T)$, the same holds for $\theta\psi$, from which the
claim follows. ∎

To formulate the second well-posedness result, we introduce a higher-order
solution space:

(4.40) $\displaystyle X^{\textup{high}}_{\textup{fMGT
III}}=W^{1,\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\cap
W^{2,\infty}(0,T;H_{0}^{1}(\Omega))\cap H^{3}(0,T;L^{2}(\Omega)).$

We denote by $\|\cdot\|_{X^{\textup{high}}_{\textup{fMGT III}}}$ the
corresponding norm on this space. Under stronger regularity assumptions on the
data and the coefficient $\sigma$, the fMGT III equation (4.2) has a unique
solution in this space.

######  Proposition 4.2 (Higher regularity for the fMGT III equation).

Let $\alpha\in(0,1]$ and $\sigma\in L^{\infty}(0,T;L^{\infty}(\Omega)\cap
W^{1,4}(\Omega))$. Given $f\in L^{2}(0,T;H_{0}^{1}(\Omega))$ and

$(\psi_{0},\psi_{1},\psi_{2})\in{H_{\diamondsuit}^{2}(\Omega)}\times{H_{\diamondsuit}^{2}(\Omega)}\times
H_{0}^{1}(\Omega),$

there exists a unique solution $\psi\in X^{\textup{high}}_{\textup{fMGT
III}}$, which solves (4.7) in the $L^{2}(0,T;L^{2}(\Omega))$ sense, and
satisfies

(4.41) $\displaystyle\|\psi\|^{2}_{X^{\textup{high}}_{\textup{fMGT
III}}}\lesssim\,\begin{multlined}\|\nabla
f\|_{L^{2}(L^{2})}^{2}+\|\Delta\psi_{0}\|_{L^{2}}^{2}+\|\Delta\psi_{1}\|_{L^{2}}^{2}+\|\nabla\psi_{2}\|_{L^{2}}^{2}.\end{multlined}\|\nabla
f\|_{L^{2}(L^{2})}^{2}+\|\Delta\psi_{0}\|_{L^{2}}^{2}+\|\Delta\psi_{1}\|_{L^{2}}^{2}+\|\nabla\psi_{2}\|_{L^{2}}^{2}.$

###### Proof.

The statement when $\alpha=1$ follows analogously to [29, Theorem 3.1]. The
proof in the case $\alpha\in(0,1)$ can again be conducted by employing a
Galerkin analysis in space. We only outline the derivation of the energy
estimate, which follows by testing the semi-discrete problem by
$-\Delta\psi^{n}_{tt}$. We omit the superscript $n$ in the notation below.
After integrating over $(0,t)$, we first obtain the identity

(4.42)
$\displaystyle\begin{multlined}\frac{1}{2}\tau\|\nabla\psi_{tt}(t)\|_{L^{2}}^{2}\Big{|}_{0}^{t}+\frac{1}{2}\tau
c^{2}\|\Delta\psi_{t}(t)\|_{L^{2}}^{2}\Big{|}_{0}^{t}+\delta\int_{0}^{t}(\textup{D}_{t}^{2-\alpha}\Delta\psi,\Delta\psi_{tt})_{L^{2}}\,\textup{d}s\end{multlined}\frac{1}{2}\tau\|\nabla\psi_{tt}(t)\|_{L^{2}}^{2}\Big{|}_{0}^{t}+\frac{1}{2}\tau
c^{2}\|\Delta\psi_{t}(t)\|_{L^{2}}^{2}\Big{|}_{0}^{t}+\delta\int_{0}^{t}(\textup{D}_{t}^{2-\alpha}\Delta\psi,\Delta\psi_{tt})_{L^{2}}\,\textup{d}s$
$\displaystyle=$ $\displaystyle\,\begin{multlined}\int_{0}^{t}(\nabla
f,\nabla\psi_{tt})_{L^{2}}\,\textup{d}s-\int_{0}^{t}((1+\sigma)\nabla\psi_{tt},\nabla\psi_{tt})_{L^{2}}\,\textup{d}s-\int_{0}^{t}(\psi_{tt}\nabla\sigma,\nabla\psi_{tt})\,\textup{d}s\\\
-c^{2}(\Delta\psi,\Delta\psi_{t})_{L^{2}}\Big{|}_{0}^{t}+c^{2}\int_{0}^{t}\|\Delta\psi_{t}\|_{L^{2}}^{2}\,\textup{d}s.\end{multlined}\int_{0}^{t}(\nabla
f,\nabla\psi_{tt})_{L^{2}}\,\textup{d}s-\int_{0}^{t}((1+\sigma)\nabla\psi_{tt},\nabla\psi_{tt})_{L^{2}}\,\textup{d}s-\int_{0}^{t}(\psi_{tt}\nabla\sigma,\nabla\psi_{tt})\,\textup{d}s\\\
-c^{2}(\Delta\psi,\Delta\psi_{t})_{L^{2}}\Big{|}_{0}^{t}+c^{2}\int_{0}^{t}\|\Delta\psi_{t}\|_{L^{2}}^{2}\,\textup{d}s.$

We can rely on the following estimate:

(4.43) $\displaystyle\int_{0}^{t}(\nabla
f,\nabla\psi_{tt})_{L^{2}}\,\textup{d}s-\int_{0}^{t}(\psi_{tt}\nabla\sigma,\nabla\psi_{tt})\,\textup{d}s-c^{2}(\Delta\psi,\Delta\psi_{t})_{L^{2}}\Big{|}_{0}^{t}$
$\displaystyle\leq$ $\displaystyle\,\begin{multlined}\|\nabla
f\|_{L^{2}(L^{2})}\|\nabla\psi_{tt}\|_{L^{2}_{t}(L^{2})}+\|\psi_{tt}\|_{L^{2}_{t}(L^{4})}\|\nabla\sigma\|_{L^{\infty}(L^{4})}\|\nabla\psi_{tt}\|_{L_{t}^{2}(L^{2})}\\\
+c^{2}\|\Delta\psi(t)\|_{L^{2}}\|\Delta\psi_{t}(t)\|_{L^{2}}+c^{2}\|\Delta\psi_{0}\|_{L^{2}}\|\Delta\psi_{1}\|_{L^{2}};\end{multlined}\|\nabla
f\|_{L^{2}(L^{2})}\|\nabla\psi_{tt}\|_{L^{2}_{t}(L^{2})}+\|\psi_{tt}\|_{L^{2}_{t}(L^{4})}\|\nabla\sigma\|_{L^{\infty}(L^{4})}\|\nabla\psi_{tt}\|_{L_{t}^{2}(L^{2})}\\\
+c^{2}\|\Delta\psi(t)\|_{L^{2}}\|\Delta\psi_{t}(t)\|_{L^{2}}+c^{2}\|\Delta\psi_{0}\|_{L^{2}}\|\Delta\psi_{1}\|_{L^{2}};$

see also [30, Theorem 3.1]. We further note that

(4.44)
$\displaystyle\|\psi_{tt}\|_{L^{2}_{t}(L^{4})}\|\nabla\sigma\|_{L^{\infty}(L^{4})}\|\nabla\psi_{tt}\|_{L_{t}^{2}(L^{2})}\leq
C_{H^{1},L^{4}}\|\nabla\sigma\|_{L^{\infty}(L^{4})}\|\nabla\psi_{tt}\|_{L_{t}^{2}(L^{2})}^{2}$

and that

(4.45)
$\displaystyle\|\Delta\psi(t)\|_{L^{2}}\|\Delta\psi_{t}(t)\|_{L^{2}}\leq\frac{1}{\epsilon}(\sqrt{T}\|\Delta\psi_{t}\|_{L^{2}_{t}(L^{2})}+\|\Delta\psi_{0}\|_{L^{2}})^{2}+\epsilon\|\Delta\psi_{t}(t)\|_{L^{2}}^{2}.$

For fixed, small enough $\epsilon>0$, an application of Gronwall’s inequality
thus yields (4.50), at first in a discrete setting. Additionally, we obtain

(4.46) $\displaystyle\|\psi_{ttt}\|_{L_{t}^{2}(L^{2})}^{2}\lesssim$
$\displaystyle\,\begin{multlined}\|\psi_{tt}\|_{L_{t}^{2}(L^{2})}^{2}+\|\Delta\psi\|_{L_{t}^{2}(L^{2})}^{2}+\|\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}^{2}\\\
+\|{\textup{D}}_{t}^{2-\alpha}\Delta\psi\|_{L_{t}^{2}(L^{2})}^{2}+\|f\|_{L^{2}(L^{2})}^{2},\end{multlined}\|\psi_{tt}\|_{L_{t}^{2}(L^{2})}^{2}+\|\Delta\psi\|_{L_{t}^{2}(L^{2})}^{2}+\|\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}^{2}\\\
+\|{\textup{D}}_{t}^{2-\alpha}\Delta\psi\|_{L_{t}^{2}(L^{2})}^{2}+\|f\|_{L^{2}(L^{2})}^{2},$

where, similarly to (4.26), we can further estimate the fractional term as
follows:

(4.47)
$\|{\textup{D}}_{t}^{2-\alpha}\Delta\psi\|_{L_{t}^{2}(L^{2})}\lesssim\|\Delta\psi_{tt}\|_{H_{t}^{-\alpha/2}(L^{2})}.$

The rest of the arguments follow analogously to the proof of Proposition 4.1.
We point out that in this higher-order setting, we are allowed to test the
homogeneous problem ($f=\psi_{0}=\psi_{1}=\psi_{2}=0$) directly with
$\psi_{tt}$ to prove uniqueness.
Note that for $\psi\in X^{\textup{high}}_{\textup{fMGT III}}$, thanks to the
embedding $W^{1,\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\hookrightarrow
C([0,T];{H_{\diamondsuit}^{2}(\Omega)})$, we know that $\psi\in
C([0,T];{H_{\diamondsuit}^{2}(\Omega)})$. Likewise, we have

$\psi_{t}\in L^{\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\cap
C([0,T];H^{1}_{0}(\Omega)).$

According to [41, §2, Lemma 3.3], this implies that $\psi_{t}$ is weakly
continuous from $[0,T]$ into ${H_{\diamondsuit}^{2}(\Omega)}$. Similarly, we
can prove that $\psi_{tt}\in C_{w}([0,T];H_{0}^{1}(\Omega))$. ∎

We are now ready to prove a well-posedness result for the nonlinear fJMGT–W
III equation.

######  Theorem 4.1 (Local well-posedness of the fJMGT–W III equation).

Let $\alpha\in(0,1]$, $\tilde{T}>0$, and $\varrho>0$. Further, assume that
$f\in L^{2}(0,\tilde{T};H_{0}^{1}(\Omega))$ and that

(4.48)
$\|f\|^{2}_{L^{2}(H^{1})}+\|\psi_{0}\|_{H^{2}}^{2}+\|\psi_{1}\|_{H^{2}}^{2}+\|\psi_{2}\|_{H^{1}}^{2}\leq\varrho^{2}.$

Then there exists $T=T(\varrho)\leq\tilde{T}$, such that the initial boundary-
value problem

(4.49) $\left\\{\begin{aligned}
\tau\psi_{ttt}+&(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau
c^{2}\Delta\psi_{t}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}=f&&\text{in
}\Omega\times(0,T),\\\\[2.84526pt] &\psi=\,0&&\text{on
}\partial\Omega\times(0,T),\\\\[2.84526pt]
&(\psi,\psi_{t},\psi_{tt})=\,(\psi_{0},\psi_{1},\psi_{2})&&\mbox{in
}\Omega\times\\{0\\},\end{aligned}\right.$

has a unique solution $\psi\in X^{\textup{high}}_{\textup{fMGT III}}$, which
satisfies

(4.50)
$\displaystyle\begin{multlined}\|\psi\|^{2}_{X^{\textup{high}}_{\textup{fMGT
III}}}\end{multlined}\|\psi\|^{2}_{X^{\textup{high}}_{\textup{fMGT
III}}}\lesssim\,\begin{multlined}\|f\|^{2}_{L^{2}(H^{1})}+\|\psi_{0}\|_{H^{2}}^{2}+\|\psi_{1}\|_{H^{2}}^{2}+\|\psi_{2}\|_{H^{1}}^{2}.\end{multlined}\|f\|^{2}_{L^{2}(H^{1})}+\|\psi_{0}\|_{H^{2}}^{2}+\|\psi_{1}\|_{H^{2}}^{2}+\|\psi_{2}\|_{H^{1}}^{2}.$

###### Proof.

The proof follows by applying the Banach Fixed-point theorem to the mapping
$\mathcal{T}:w\mapsto\psi$, where $\psi$ solves the linearized equation (4.7)
with $\sigma=2kw_{t}$ and

(4.51) $w\in B_{R}:=\\{w\in X^{\textup{high}}_{\textup{fMGT
III}}\,:\|w\|_{X^{\textup{high}}_{\textup{fMGT III}}}\leq R,\
w(0)=\psi_{0},\,w_{t}(0)=\psi_{1},\,w_{tt}(0)=\psi_{2}\\},$

with $R>0$ specified below. Note that

$\displaystyle\|\sigma\|_{L^{\infty}(L^{\infty})}+\|\sigma\|_{L^{\infty}(W^{1,4})}\leq$
$\displaystyle\,2C_{H^{2},L^{\infty}}|k|\|w_{t}\|_{L^{\infty}(H^{2})}+2C_{H^{1},L^{4}}|k|\|w_{t}\|_{L^{\infty}(H^{2})}$
$\displaystyle\lesssim$ $\displaystyle\,R.$

Thus by employing estimate (4.41), where the hidden constant has the form
$C_{1}\exp(C_{2}(R+1)\tilde{T})$, it immediately follows that $\mathcal{T}$ is
a well-defined self-mapping on $B_{R}^{\textup{W}}$, provided $R>0$ is chosen
so that

$\sqrt{C_{1}\exp(C_{2}(R+1)\tilde{T})}\,\varrho\leq R.$

Next, we prove that $\mathcal{T}$ is strictly contractive. Note that we will
prove contractivity with respect to the weaker norm
$\|\cdot\|_{X^{\textup{low}}_{\textup{fMGT III}}}$; recall the definition of
the space $X^{\textup{low}}_{\textup{fMGT III}}$ in (4.5) for $\alpha\in(0,1)$
and (4.6) for $\alpha=1$.
We take any $w^{(1)}$ and $w^{(2)}$ in $B_{R}^{W}$ and set
$\psi^{(1)}=\mathcal{T}w^{(1)}$ and $\psi^{(2)}=\mathcal{T}w^{(2)}$. We also
introduce the short-hand notation for the differences

$\overline{\psi}=\psi^{(1)}-\psi^{(2)},\qquad\overline{w}=w^{(1)}-w^{(2)}.$

Then we know that $\overline{\psi}$ solves the linear equation

(4.52)
$\tau\overline{\psi}_{ttt}+(1+2kw_{t}^{(1)})\overline{\psi}_{tt}-c^{2}\Delta\overline{\psi}-\tau
c^{2}\Delta\overline{\psi}_{t}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\overline{\psi}_{t}+2k\overline{w}_{t}\psi^{(2)}_{tt}=0$

and has zero initial conditions. Employing the lower-order estimate (4.8) with
$\sigma=2kw_{t}^{(1)}$ and $f=-2k\overline{w}_{t}\psi^{(2)}_{tt}$ yields the
bound

(4.53) $\displaystyle\|\overline{\psi}\|_{X^{\textup{low}}_{\textup{fMGT
III}}}\leq$
$\displaystyle\,\sqrt{C_{1}\exp(C_{2}(R+1)\tilde{T})}\|f\|_{L^{2}(L^{2})}$
(4.55) $\displaystyle\leq$
$\displaystyle\,\begin{multlined}2\sqrt{C_{1}\exp(C_{2}(R+1)\tilde{T})}|k|\|\psi_{tt}^{(2)}\|_{L^{\infty}(L^{4})}\sqrt{\tilde{T}}\|\overline{w}_{t}\|_{L^{\infty}(L^{4})}\end{multlined}2\sqrt{C_{1}\exp(C_{2}(R+1)\tilde{T})}|k|\|\psi_{tt}^{(2)}\|_{L^{\infty}(L^{4})}\sqrt{\tilde{T}}\|\overline{w}_{t}\|_{L^{\infty}(L^{4})}$
(4.56) $\displaystyle\leq$
$\displaystyle\,\theta\|\overline{w}\|_{X^{\textup{low}}_{\textup{fMGT
III}}}.$

Thus we can guarantee that $\theta\in(0,1)$ and obtain strict contractivity of
$\mathcal{T}$ by decreasing $\tilde{T}$.
We note that the space $B_{R}$ with the metric induced by the norm
$\|\cdot\|_{X^{\textup{low}}_{\textup{fMGT III}}}$ is a closed subset of a
complete normed space; cf. [30, Theorem 4.1]. Existence of a unique solution
in $B_{R}$ then follows by Banach’s Fixed-point theorem. ∎

###  4.1. Limiting behavior of the fJMGT–W III equation

We next discuss the limit with respect to the order of differentiation. Given
$\alpha\in(0,1)$, under the assumptions of Theorem 4.1, let $\psi^{\alpha}$ be
the solution of the fJMGT–W III equation:

$\tau\psi^{\alpha}_{ttt}+(1+2k\psi^{\alpha}_{t})\psi^{\alpha}_{tt}-c^{2}\Delta\psi^{\alpha}-\tau
c^{2}\Delta\psi^{\alpha}_{t}-\delta{\textup{D}}_{t}^{2-\alpha}\Delta\psi^{\alpha}=f.$

Let $\psi$ solve the corresponding JMGT–Westervelt equation obtained by
setting $\alpha=1$ above. Then the difference
$\overline{\psi}=\psi^{\alpha}-\psi$ solves

(4.57)
$\displaystyle\tau\overline{\psi}_{ttt}+(1+2k\psi_{t}^{\alpha})\overline{\psi}_{tt}-c^{2}\Delta\overline{\psi}-\tau
c^{2}\Delta\overline{\psi}_{t}-\delta\textup{D}_{t}^{1-\alpha}\Delta\overline{\psi}_{t}+2k\overline{\psi}_{t}\psi_{tt}$
$\displaystyle=$
$\displaystyle\,\delta(\textup{D}_{t}^{1-\alpha}\Delta\psi_{t}-\Delta\psi_{t}).$

Similarly to the proof of Proposition 4.1, testing with $\overline{\psi}_{tt}$
(which we are allowed to do in this higher-regularity setting) leads to

(4.58)
$\displaystyle\|\overline{\psi}_{tt}(t)\|_{L^{2}}^{2}+\|\nabla\overline{\psi}_{t}(t)\|_{L^{2}}^{2}\lesssim\,\|\textup{D}_{t}^{1-\alpha}\nabla\psi_{t}-\nabla\psi_{t}\|_{L^{2}(L^{2})}^{2}.$

By recalling Lemma 3.1, we find that if $\psi_{t}(0)=0$, then

(4.59) $\lim_{\alpha\rightarrow
1^{-}}\|\textup{D}_{t}^{1-\alpha}\nabla\psi_{t}-\nabla\psi_{t}\|_{L^{2}(L^{2})}=0,$

and thus arrive at the following result.

######  Proposition 4.3.

Let the assumptions of Theorem 4.1 hold with $\psi_{1}=0$. Let
$\\{\psi^{\alpha}\\}_{\alpha\in(0,1)}$ be the family of solutions to the
fJMGT–W III equation and let $\psi$ solve the corresponding JMGT equation with
$\alpha=1$. Then $\psi^{\alpha}$ converges to $\psi$ in
$W^{1,\infty}(0,T;H_{0}^{1}(\Omega))\cap W^{2,\infty}(0,T;L^{2}(\Omega))$ as
$\alpha\rightarrow 1^{-}$.

##  5\. Analysis of the equations with the fractional leading term

We next discuss to what extent the analysis we have performed for the fJMGT–W
III equation carries over to the other versions. To this end, the crucial
question is whether the Galerkin approximation procedure based on energy
estimates is feasible for a linearized equation:

(5.1)
$\tau^{\alpha}\textup{D}_{t}^{2+\alpha}\psi+(1+\sigma)\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}\textup{D}_{t}^{\alpha}\Delta\psi-\delta\textup{D}_{t}^{\beta}\Delta\psi=f\,,$

with

(5.2) $\beta=\begin{cases}1&\mbox{ for fMGT},\\\ 2-\alpha&\mbox{ for fMGT
I},\\\ \alpha&\mbox{ for fMGT II},\end{cases}$

where for the respective nonlinear versions of Westervelt type we have
$\sigma=2k\psi_{t}$ in mind. These three cases have in common the fact that,
unlike with the fMGT III equation, the leading-order time derivative is
fractional and varies with $\alpha$.
The analysis of the fMGT and fMGT I equations follows similar lines of
reasoning. We thus present the proof for the fMGT I equation with details and
only outline the main arguments in the analysis of the fMGT model. To
facilitate the analysis, we assume that $\alpha\in(1/2,1)$, which, at least
for the equation based on employing (GFE I), appears to be the physically
justified range according to the numerical experiments performed in [45].
The analysis of the fMGT II equation cannot be carried out in the same manner;
we explain why in Remark 1 below and a offer different way of analyzing it
when $\sigma=0$ in Section 7.

###  5.1. Analysis of the fJMGT–W I equation

In this section we consider the equation

(5.3)
$\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta{\textup{D}}_{t}^{2-\alpha}\Delta\psi=f.$

To carry out the analysis starting from a linearization, we need to assume
that the coefficient $\sigma$ is small enough in a suitable norm. To this end,
for $\alpha\in(1/2,1)$, we introduce the space

$X^{\sigma}_{\textup{fMGT I}}=L^{2}(0,T;(W^{1,3}\cap L^{\infty})(\Omega))$

equipped with the norm $\|\cdot\|_{X^{\sigma}_{\textup{fMGT I}}}$. Further,
the solution space for $\psi$ will be

(5.4) $\displaystyle\quad X_{\textup{fMGT I}}=\,\left\\{\psi\in
H^{2+\alpha}(0,T;L^{2}(\Omega)):\right.$
$\displaystyle\left.{\textup{D}}_{t}^{\alpha}\psi\in
L^{\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)}),\right.$
$\displaystyle\left.\textup{D}_{t}^{1+\alpha}\psi\in
L^{\infty}(0,T;H_{0}^{1}(\Omega))\right\\},$

equipped with the norm $\|\cdot\|_{X_{\textup{fMGT I}}}$, which is the space
induced by the part of the energy that can be bounded by a uniform constant
independent of $\alpha$; see estimate (5.5) in Proposition 5.1 below.

Since the coefficient $\sigma$ acts as a placeholder for $2k\psi_{t}$, we note
that

$\|2k\psi_{t}\|_{X^{\sigma}_{\textup{fMGT
I}}}\lesssim\|\psi\|_{X_{\textup{fMGT I}}},$

due to interpolation

$\displaystyle H^{\alpha}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\cap
H^{1+\alpha}(0,T;H_{0}^{1}(\Omega))\subseteq$
$\displaystyle\,H^{\alpha+\theta}(0,T;H^{1+(1-\theta)}(\Omega))$
$\displaystyle=$ $\displaystyle\,H^{1}(0,T;H^{1+\alpha}(\Omega))$

with $\theta=1-\alpha$ and continuity of the embedding
$H^{1+\alpha}(\Omega)\hookrightarrow(W^{1,3}\cap L^{\infty})(\Omega)$.

We are now ready to analyze equation (5.1) for $\beta=2-\alpha$.

######  Proposition 5.1 (Well-posedness of the fMGT I equation).

Let $\alpha\in[\alpha_{0},1)$ for some $\alpha_{0}>1/2$. Assume that $f\in
H^{\alpha-1/2}(0,T;L^{2}(\Omega))$, $\sigma\in X^{\sigma}_{\textup{fMGT I}}$,
and

$(\psi_{0},\psi_{1},\psi_{2})\in({H_{\diamondsuit}^{2}(\Omega)},{H_{\diamondsuit}^{2}(\Omega)},H_{0}^{1}(\Omega)).$

There exists $\varrho>0$, independent of $\alpha$, such that if

$\|\sigma\|_{X^{\sigma}_{\textup{fMGT I}}}\leq\varrho,$

then there is a unique $\psi\in X_{\textup{fMGT I}}$, which satisfies the fMGT
I equation in the $L^{2}(0,T;L^{2}(\Omega))$ sense with
$(\psi,\psi_{t},\psi_{tt})|_{t=0}=(\psi_{0},\psi_{1},\psi_{2})$. Furthermore,
this solution fulfills the following estimate:

(5.5)
$\displaystyle\begin{multlined}\|\textup{D}_{t}^{2+\alpha}\psi\|_{L^{2}_{t}(L^{2})}^{2}+\|\nabla\textup{D}_{t}^{1+\alpha}\psi(t)\|_{L^{2}}^{2}+\|\Delta\textup{D}_{t}^{\alpha}\psi(t)\|_{L^{2}}^{2}\\\
+C(\alpha)\Bigl{(}\|\nabla\psi_{tt}\|_{{H_{t}^{-(1-\alpha)/2}}(L^{2})}^{2}+\|\textup{D}_{t}^{3/2}\Delta\psi\|_{L^{2}_{t}(L^{2})}^{2}\Bigr{)}\end{multlined}\|\textup{D}_{t}^{2+\alpha}\psi\|_{L^{2}_{t}(L^{2})}^{2}+\|\nabla\textup{D}_{t}^{1+\alpha}\psi(t)\|_{L^{2}}^{2}+\|\Delta\textup{D}_{t}^{\alpha}\psi(t)\|_{L^{2}}^{2}\\\
+C(\alpha)\Bigl{(}\|\nabla\psi_{tt}\|_{{H_{t}^{-(1-\alpha)/2}}(L^{2})}^{2}+\|\textup{D}_{t}^{3/2}\Delta\psi\|_{L^{2}_{t}(L^{2})}^{2}\Bigr{)}$
$\displaystyle\lesssim$
$\displaystyle\,\|f\|_{H^{\alpha-1/2}(L^{2})}^{2}+\|\Delta\psi_{0}\|_{L^{2}}^{2}+\|\Delta\psi_{1}\|_{L^{2}}^{2}+\|\nabla\psi_{2}\|_{L^{2}}^{2},$

where $C(\alpha)\to 0$ as $\alpha\to 1^{-}$.

###### Proof.

The proof follows by discretizing the problem with respect to the spatial
variable, using smooth eigenfunctions of the Dirichlet-Laplacian as the basis.

(I) _Existence of an approximate solution._ For $n\in\mathbb{N}$ fixed, we
first prove that the semi-discrete problem has a unique solution. We employ
the same notation as in the proof of Proposition 4.1; that is,

(5.6)
$\displaystyle\psi^{n}(x,t)=\sum_{i=1}^{n}\xi^{n}_{i}(t)\phi_{i}(x),\quad\psi_{j}^{n}(x)=\sum_{i=1}^{n}\xi^{n}_{j,i}\phi_{i}(x),\quad
j\in\\{0,1,2\\}.$

Using the mass matrices $M$ and $M_{\sigma}=M_{\sigma}(t)$, the stiffness
matrix $K$, and the source vector $\boldsymbol{f}$ defined in (4.15), the next
step is to rewrite the discretized problem as a system of integral Volterra
equations. To this end, let

$\boldsymbol{\mu}=\textup{D}_{t}^{2+\alpha}\boldsymbol{\xi}\qquad\text{and}\qquad
p^{\gamma}(t)=\frac{1}{\Gamma(\gamma+1)}t^{\gamma}.$

We can rely on the following identities:

${\textup{I}}^{\gamma}w=p^{\gamma-1}*w,\qquad{\textup{I}}^{\gamma}p^{i}=p^{\gamma+i},\qquad{\textup{I}}^{\gamma}{\textup{I}}^{s}={\textup{I}}^{\gamma+s}$

to rewrite the vector solution and its derivatives as

$\displaystyle\boldsymbol{\xi}_{tt}$
$\displaystyle={\textup{I}}^{1}\boldsymbol{\xi}_{ttt}+p^{0}\boldsymbol{\xi}_{2}={\textup{I}}^{\alpha}\boldsymbol{\mu}+p^{0}\boldsymbol{\xi}_{2}=p^{\alpha-1}*\boldsymbol{\mu}+p^{0}\boldsymbol{\xi}_{2},$
$\displaystyle\boldsymbol{\xi}_{t}$
$\displaystyle={\textup{I}}^{1}\boldsymbol{\xi}_{tt}+p^{0}\boldsymbol{\xi}_{1}=\textup{I}^{1+\alpha}\boldsymbol{\mu}+p^{1}\boldsymbol{\xi}_{2}+p^{0}\boldsymbol{\xi}_{1}=p^{\alpha}*\boldsymbol{\mu}+p^{1}\boldsymbol{\xi}_{2}+p^{0}\boldsymbol{\xi}_{1},$
$\displaystyle\boldsymbol{\xi}$
$\displaystyle={\textup{I}}^{1}\boldsymbol{\xi}_{t}+p^{0}\boldsymbol{\xi}_{0}=\textup{I}^{2+\alpha}\boldsymbol{\mu}+p^{2}\boldsymbol{\xi}_{2}+p^{1}\boldsymbol{\xi}_{1}+p^{0}\boldsymbol{\xi}_{0}=p^{\alpha+1}*\boldsymbol{\mu}+p^{2}\boldsymbol{\xi}_{2}+p^{1}\boldsymbol{\xi}_{1}+p^{0}\boldsymbol{\xi}_{0}.$

Furthermore, we can rewrite the fractional derivatives as

$\displaystyle\textup{D}_{t}^{2-\alpha}\boldsymbol{\xi}$
$\displaystyle={\textup{I}}^{\alpha}\boldsymbol{\xi}_{tt}={\textup{I}}^{2\alpha}\boldsymbol{\mu}+{\textup{I}}^{\alpha}p^{0}\boldsymbol{\xi}_{2}=p^{2\alpha-1}*\boldsymbol{\mu}+p^{\alpha}\boldsymbol{\xi}_{2},$
$\displaystyle\textup{D}_{t}^{\alpha}\boldsymbol{\xi}$
$\displaystyle=\textup{I}^{1-\alpha}\boldsymbol{\xi}_{t}=I^{2}\boldsymbol{\mu}+\textup{I}^{1-\alpha}p^{1}\boldsymbol{\xi}_{2}+\textup{I}^{1-\alpha}p^{0}\boldsymbol{\xi}_{1}=p^{1}*\boldsymbol{\mu}+p^{2-\alpha}\boldsymbol{\xi}_{2}+p^{1-\alpha}\boldsymbol{\xi}_{1}.$

Therefore, the semi-discrete problem can be equivalently rewritten as a system
of Volterra integral equations:

(5.7)
$\displaystyle\begin{multlined}\tau^{\alpha}M\boldsymbol{\mu}+M_{\sigma}(t)\Bigl{(}p^{\alpha-1}*\boldsymbol{\mu}+p^{0}\boldsymbol{\xi}_{2}\Bigr{)}\\\
+c^{2}K\Bigl{(}p^{\alpha+1}*\boldsymbol{\mu}+p^{2}\boldsymbol{\xi}_{2}+p^{1}\boldsymbol{\xi}_{1}+p^{0}\boldsymbol{\xi}_{0}\Bigr{)}\\\
+\tau^{\alpha}c^{2}K\Bigl{(}p^{1}*\boldsymbol{\mu}+p^{2-\alpha}\boldsymbol{\xi}_{2}+p^{1-\alpha}\boldsymbol{\xi}_{1}\Bigr{)}+\delta
K(p^{2\alpha-1}*\boldsymbol{\mu}+p^{\alpha}\boldsymbol{\xi}_{2})=f.\end{multlined}\tau^{\alpha}M\boldsymbol{\mu}+M_{\sigma}(t)\Bigl{(}p^{\alpha-1}*\boldsymbol{\mu}+p^{0}\boldsymbol{\xi}_{2}\Bigr{)}\\\
+c^{2}K\Bigl{(}p^{\alpha+1}*\boldsymbol{\mu}+p^{2}\boldsymbol{\xi}_{2}+p^{1}\boldsymbol{\xi}_{1}+p^{0}\boldsymbol{\xi}_{0}\Bigr{)}\\\
+\tau^{\alpha}c^{2}K\Bigl{(}p^{1}*\boldsymbol{\mu}+p^{2-\alpha}\boldsymbol{\xi}_{2}+p^{1-\alpha}\boldsymbol{\xi}_{1}\Bigr{)}+\delta
K(p^{2\alpha-1}*\boldsymbol{\mu}+p^{\alpha}\boldsymbol{\xi}_{2})=f.$

Thus, unique solvability of this system in $L^{2}(0,T)$ follows from [21,
Theorem 4.2, p. 241 in §9]. Then from

$\left\\{\begin{aligned}
&\textup{D}_{t}^{\alpha}\boldsymbol{\xi}_{tt}=\boldsymbol{\mu}\in
L^{2}(0,T),\quad\alpha\in(\tfrac{1}{2},1)\\\
&\boldsymbol{\xi}_{tt}(0)=\boldsymbol{\xi}_{2}\end{aligned}\right.$

we have a unique $\boldsymbol{\xi}_{tt}\in H^{\alpha}(0,T)$; cf. [32, §3.3].
Combined with the initial conditions
$(\boldsymbol{\xi}_{0},\boldsymbol{\xi}_{1})$, this yields a unique
$\boldsymbol{\xi}\in H^{2+\alpha}(0,T)$ and further implies the existence of a
unique $\psi^{n}\in H^{2+\alpha}(0,T;V_{n})$.

(II) _A priori energy analysis._ We next focus on deriving a uniform energy
estimate for $\psi^{n}$. We will make use of estimate (3.5) to treat the
fractional terms; that is,

(5.8)
$\int_{0}^{t}\langle\textup{I}^{\rho}w(s),w(s)\rangle\,\textup{d}s\geq\cos(\tfrac{\pi\rho}{2})\|w\|_{H^{-\rho/2}(0,t)}^{2}$

for $\rho\in(0,1)$, as well as the identity $\int_{0}^{t}\langle
w_{t}(s),w(s)\rangle\,\textup{d}s=\frac{1}{2}|w|^{2}\,\big{|}_{0}^{t}$. Thus,
the rule of thumb is that for a coercivity estimate on
$\int_{0}^{t}\textup{D}_{t}^{r}(s)w\,\textup{D}_{t}^{\rho}w(s)\,\textup{d}s$
to yield a non-negative lower bound (up to initial data), the difference
$|r-\rho|$ between the fractional orders must not exceed one. We will consider
the multiplier

$-\Delta\textup{D}_{t}^{1+\alpha}\psi^{n}(t)=\sum_{i=1}^{n}\textup{D}_{t}^{1+\alpha}\xi^{n}_{i}(t)\,\lambda_{i}\phi_{i}(x)\in
V_{n},$

for which this rule applies and yields non-negative contributions on the left-
hand side for the terms containing $\textup{D}_{t}^{2+\alpha}\psi^{n}$,
$\psi_{tt}^{n}$, $-\textup{D}_{t}^{\alpha}\Delta\psi^{n}$, and
$-\textup{D}_{t}^{2-\alpha}\Delta\psi^{n}$. Multiplying the semi-discrete
equation with $-\Delta\textup{D}_{t}^{1+\alpha}\psi^{n}$ and integrating over
space and $(0,t)$ at first leads to

(5.9)
$\displaystyle\,\begin{multlined}-\tau^{\alpha}\int_{0}^{t}(\textup{D}_{t}^{2+\alpha}\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s-\int_{0}^{t}(\psi_{tt}^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s\\\
+\tau^{\alpha}c^{2}\int_{0}^{t}({\textup{D}}_{t}^{\alpha}\Delta\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s+\delta\int_{0}^{t}(\textup{D}_{t}^{2-\alpha}\Delta\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s\end{multlined}-\tau^{\alpha}\int_{0}^{t}(\textup{D}_{t}^{2+\alpha}\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s-\int_{0}^{t}(\psi_{tt}^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s\\\
+\tau^{\alpha}c^{2}\int_{0}^{t}({\textup{D}}_{t}^{\alpha}\Delta\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s+\delta\int_{0}^{t}(\textup{D}_{t}^{2-\alpha}\Delta\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s$
$\displaystyle=$
$\displaystyle\,\begin{multlined}-\int_{0}^{t}(f-\sigma\psi_{tt}^{n}+c^{2}\Delta\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})\,\textup{d}s.\end{multlined}-\int_{0}^{t}(f-\sigma\psi_{tt}^{n}+c^{2}\Delta\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})\,\textup{d}s.$

We next exchange the order of differentiation in the first and third term on
the left as follows:

(5.10)
$\textup{D}_{t}^{2+\alpha}\psi^{n}=\textup{D}_{t}\textup{D}_{t}^{1+\alpha}\psi^{n}-p^{-\alpha}\psi^{n}_{tt}(0)\,,\qquad\textup{D}_{t}^{1+\alpha}\psi^{n}=\textup{D}_{t}\textup{D}_{t}^{\alpha}\psi^{n}-p^{-\alpha}\psi^{n}_{t}(0)\,,$

and integrate by parts to obtain

$\displaystyle\int_{0}^{t}(\textup{D}_{t}^{\alpha}\Delta\psi^{n}(s),p^{-\alpha}(s)\Delta\psi_{t}^{n}(0))_{L^{2}}\,\textup{d}s$
$\displaystyle=$
$\displaystyle\,\begin{multlined}-\int_{0}^{t}(\textup{D}_{t}^{1+\alpha}\Delta\psi^{n}(s)+p^{-\alpha}(s)\Delta\psi_{t}^{n}(0),p^{1-\alpha}\Delta\psi_{t}^{n}(0))_{L^{2}}\,\textup{d}s\\\
+p^{1-\alpha}(t)(\textup{D}_{t}^{\alpha}\Delta\psi^{n}(t),\Delta\psi_{t}^{n}(0))_{L^{2}}\end{multlined}-\int_{0}^{t}(\textup{D}_{t}^{1+\alpha}\Delta\psi^{n}(s)+p^{-\alpha}(s)\Delta\psi_{t}^{n}(0),p^{1-\alpha}\Delta\psi_{t}^{n}(0))_{L^{2}}\,\textup{d}s\\\
+p^{1-\alpha}(t)(\textup{D}_{t}^{\alpha}\Delta\psi^{n}(t),\Delta\psi_{t}^{n}(0))_{L^{2}}$
$\displaystyle=$
$\displaystyle\,-\int_{0}^{t}(\textup{D}_{t}^{1+\alpha}\Delta\psi^{n}(s),p^{1-\alpha}(s)\Delta\psi_{t}^{n}(0))_{L^{2}}\,\textup{d}s+h_{0}(t),$

where we have introduced

$h_{0}(t)=\frac{t^{2-2\alpha}}{2\Gamma(2-2\alpha)^{2}}\|\Delta\psi_{t}^{n}(0)\|_{L^{2}}^{2}+p^{1-\alpha}(t)(\textup{D}_{t}^{\alpha}\Delta\psi^{n}(t),\Delta\psi_{t}^{n}(0))_{L^{2}}.$

Thus, we arrive at the following identity:

(5.11)
$\displaystyle\,\begin{multlined}-\tau^{\alpha}\int_{0}^{t}(\textup{D}_{t}\textup{D}_{t}^{1+\alpha}\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s-\int_{0}^{t}(\psi_{tt}^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s\\\
+\tau^{\alpha}c^{2}\int_{0}^{t}({\textup{D}}_{t}^{\alpha}\Delta\psi^{n},\Delta\textup{D}_{t}\textup{D}_{t}^{\alpha}\psi^{n})_{L^{2}}\,\textup{d}s+\delta\int_{0}^{t}(\textup{D}_{t}^{2-\alpha}\Delta\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s\end{multlined}-\tau^{\alpha}\int_{0}^{t}(\textup{D}_{t}\textup{D}_{t}^{1+\alpha}\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s-\int_{0}^{t}(\psi_{tt}^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s\\\
+\tau^{\alpha}c^{2}\int_{0}^{t}({\textup{D}}_{t}^{\alpha}\Delta\psi^{n},\Delta\textup{D}_{t}\textup{D}_{t}^{\alpha}\psi^{n})_{L^{2}}\,\textup{d}s+\delta\int_{0}^{t}(\textup{D}_{t}^{2-\alpha}\Delta\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s$
$\displaystyle=$
$\displaystyle\,\begin{multlined}-\int_{0}^{t}(f-\sigma\psi_{tt}^{n}+c^{2}\Delta\psi^{n}+\tau^{\alpha}p^{-\alpha}\psi_{tt}^{n}(0)-\tau^{\alpha}c^{2}p^{1-\alpha}\Delta\psi_{t}^{n}(0),\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s\\\
+\tau^{\alpha}c^{2}h_{0}(t).\end{multlined}-\int_{0}^{t}(f-\sigma\psi_{tt}^{n}+c^{2}\Delta\psi^{n}+\tau^{\alpha}p^{-\alpha}\psi_{tt}^{n}(0)-\tau^{\alpha}c^{2}p^{1-\alpha}\Delta\psi_{t}^{n}(0),\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s\\\
+\tau^{\alpha}c^{2}h_{0}(t).$

We note that

$\textup{D}_{t}^{1+\alpha}\psi^{n}=\psi_{tt}^{n}=0\text{ on }\partial\Omega$

with our choice of the basis functions. Additionally using
$\textup{D}_{t}^{1+\alpha}\psi^{n}=\textup{I}^{1-\alpha}\psi^{n}_{tt}$ in the
second term on the left and integrating by parts in space and time yields

(5.12)
$\displaystyle\begin{multlined}\,\frac{\tau^{\alpha}}{2}\|\nabla\textup{D}_{t}^{1+\alpha}\psi^{n}(s)\|_{L^{2}}^{2}\big{|}_{0}^{t}+\int_{0}^{t}(\nabla\psi_{tt}^{n}(s),\textup{I}^{1-\alpha}\nabla\psi_{tt}^{n}(s))_{L^{2}}\,\textup{d}s\\\
+\frac{\tau^{\alpha}c^{2}}{2}\|\Delta\textup{D}_{t}^{\alpha}\psi^{n}(s)\|_{L^{2}}^{2}\Big{|}_{0}^{t}+\delta
d\end{multlined}\,\frac{\tau^{\alpha}}{2}\|\nabla\textup{D}_{t}^{1+\alpha}\psi^{n}(s)\|_{L^{2}}^{2}\big{|}_{0}^{t}+\int_{0}^{t}(\nabla\psi_{tt}^{n}(s),\textup{I}^{1-\alpha}\nabla\psi_{tt}^{n}(s))_{L^{2}}\,\textup{d}s\\\
+\frac{\tau^{\alpha}c^{2}}{2}\|\Delta\textup{D}_{t}^{\alpha}\psi^{n}(s)\|_{L^{2}}^{2}\Big{|}_{0}^{t}+\delta
d$ $\displaystyle=$
$\displaystyle\,-\int_{0}^{t}(\tilde{f}(s),\Delta\textup{D}_{t}^{1+\alpha}\psi^{n}(s))_{L^{2}}\,\textup{d}s+\tau^{\alpha}c^{2}h_{0}(t),$

where we have introduced the short-hand notation

(5.13)
$\tilde{f}=f+c^{2}\Delta\psi^{n}-\sigma\psi_{tt}^{n}+\tau^{\alpha}\left(p^{-\alpha}\psi_{tt}^{n}(0)-c^{2}p^{1-\alpha}\Delta\psi_{t}^{n}(0)\right)$

and

$d=\int_{0}^{t}(\textup{D}_{t}^{2-\alpha}\Delta\psi^{n},\Delta\textup{D}_{t}^{1+\alpha}\psi^{n})_{L^{2}}\,\textup{d}s.$

By the identity

(5.14) $\displaystyle\textup{D}_{t}^{2-\alpha}\psi^{n}=$
$\displaystyle\,{\textup{I}}^{\alpha}\psi_{tt}=\textup{I}^{2\alpha-1}\textup{I}^{1-\alpha}\psi_{tt}^{n}=\textup{I}^{2\alpha-1}\textup{D}_{t}^{1+\alpha}\psi^{n},$

we know that

(5.15) $\displaystyle
d=\int_{0}^{t}(\textup{I}^{2\alpha-1}\textup{D}_{t}^{1+\alpha}\Delta\psi^{n}(s),\textup{D}_{t}^{1+\alpha}\Delta\psi^{n}(s))_{L^{2}}\,\textup{d}s.$

Since $\alpha>1/2$, this term can be estimated from below using the fact that
$\textup{I}^{\gamma}:H^{-\gamma}(0,t)\to L^{2}(0,T)$ is an isomorphism for
$\gamma\in[0,1/2)$; see [19, Theorem 1]. Therefore,

$\displaystyle\frac{d}{\cos(\pi(\alpha-1/2))}\geq$
$\displaystyle\,\|\textup{D}_{t}^{1+\alpha}\Delta\psi^{n}\|_{H_{t}^{1/2-\alpha}(L^{2})}^{2}$
$\displaystyle\sim$
$\displaystyle\,\|\textup{I}^{\alpha-1/2}\textup{D}_{t}^{1+\alpha}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}^{2}\sim\|\textup{D}_{t}^{3/2}\Delta\psi^{n}\|_{L_{t}^{2}(L^{2})}^{2}.$

We estimate the $\tilde{f}$ term on the right-hand side of (5.12) with a view
on the possibility of bounding it by $d$ as follows:

$\displaystyle\left|\int_{0}^{t}(\tilde{f}(s),\textup{D}_{t}^{1+\alpha}\Delta\psi^{n}(s))_{L^{2}}\,\textup{d}s\right|\leq\frac{1}{2\epsilon}\|\tilde{f}\|_{H^{\alpha-1/2}_{t}(L^{2})}^{2}+\frac{\epsilon}{2}\|\textup{D}_{t}^{1+\alpha}\Delta\psi^{n}\|_{H^{1/2-\alpha}_{t}(L^{2})}^{2}.$

It remains to estimate the terms within $\tilde{f}$ in this norm; that is, to
bound

(5.16)
$\|f+c^{2}\Delta\psi^{n}-\sigma\psi_{tt}^{n}+\tau^{\alpha}\left(p^{-\alpha}\psi_{tt}^{n}(0)-c^{2}p^{1-\alpha}\Delta\psi_{t}^{n}(0)\right)\|_{H^{\alpha-1/2}_{t}(L^{2})}.$

For the $c^{2}$ term, it is readily checked that the respective contribution
of $c^{2}\Delta\psi^{n}$ to the above norm of $\tilde{f}$ (cf. (5.13)) can be
bounded by means of
$c^{2}\|\textup{D}_{t}^{\alpha}\Delta\psi^{n}\|_{L^{2}(L^{2})}$ as follows:

$\displaystyle\|\Delta\psi^{n}\|_{H^{\alpha-1/2}_{t}(L^{2})}\lesssim$
$\displaystyle\,\|\textup{D}_{t}^{\alpha-1/2}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}+\|\Delta\psi^{n}_{0}\|_{L^{2}}$
$\displaystyle=$
$\displaystyle\,\|\mathfrak{g}_{1/2}*\textup{D}_{t}^{\alpha}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}+\|\Delta\psi^{n}_{0}\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\,\|\mathfrak{g}_{1/2}\|_{L^{1}(0,T)}\|\textup{D}_{t}^{\alpha}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}+\|\Delta\psi^{n}_{0}\|_{L^{2}}\,,$

and therefore tackled by the second term on the left-hand side of (5.12)
together with Gronwall’s inequality.
By the Kato–Ponce inequality (3.6) with

$\rho=\alpha-1/2,\qquad(p_{1},q_{1})=\left(\frac{2}{2\alpha-1},\frac{1}{1-\alpha}\right),\qquad(p_{2},q_{2})=(2,\infty),$

we obtain

(5.17) $\displaystyle\|\sigma\psi_{tt}^{n}\|_{H_{t}^{\alpha-1/2}(L^{2})}$
$\displaystyle\lesssim$
$\displaystyle\,\|\sigma\|_{W_{t}^{\alpha-1/2,\frac{2}{2\alpha-1}}(L^{\infty})}\|\psi_{tt}^{n}\|_{L_{t}^{\frac{1}{1-\alpha}}(L^{2})}+\|\sigma\|_{L_{t}^{2}(L^{\infty})}\|\psi_{tt}^{n}\|_{W_{t}^{\alpha-1/2,\infty}(L^{2})}.$

By the Sobolev embedding $L^{2}(0,T)\hookrightarrow
W^{\alpha-1/2,\frac{2}{2\alpha-1}}(0,T)$, we have

$\|\sigma\|_{W_{t}^{\alpha-1/2,\frac{2}{2\alpha-1}}(L^{\infty})}\leq
C_{H^{2},L^{\infty}}^{\Omega}\|\sigma\|_{X^{\sigma}_{\textup{fMGT
I}}}\,,\quad\|\sigma\|_{L_{t}^{2}(L^{\infty})}\leq
C_{H^{2},L^{\infty}}^{\Omega}\|\sigma\|_{X^{\sigma}_{\textup{fMGT I}}}\,.$

To further estimate the norms of $\psi_{tt}^{n}$ in (5.17), we will use the
leading time derivative term $\textup{D}_{t}^{2+\alpha}\psi^{n}$ as well as
its representation via the PDE. That is, we rely on the following Sobolev
embeddings:

(5.18)
$\displaystyle\|\psi_{tt}^{n}\|_{L_{t}^{\frac{1}{1-\alpha}}(L^{2})}\lesssim$
$\displaystyle\,\|\psi_{tt}^{n}\|_{H^{\alpha-1/2}_{t}(L^{2})}\lesssim\|\textup{D}_{t}^{2+\alpha}\psi^{n}\|_{H^{-1/2}_{t}(L^{2})}+\|\psi_{tt}^{n}(0)\|_{L^{2}}$

and

(5.19)
$\displaystyle\|\psi_{tt}^{n}\|_{W_{t}^{\alpha-1/2,\infty}(L^{2})}\lesssim$
$\displaystyle\,\|\psi_{tt}^{n}\|_{H^{\alpha}_{t}(L^{2})}\lesssim\|\textup{D}_{t}^{2+\alpha}\psi^{n}\|_{L^{2}_{t}(L^{2})}+\|\psi_{tt}^{n}(0)\|_{L^{2}},$

where $\psi_{tt}^{n}$ satisfies the fractional ODE

$\tau^{\alpha}\textup{D}_{t}^{\alpha}\psi_{tt}^{n}+\psi_{tt}^{n}=-r\mbox{ with
}r=\sigma\psi_{tt}^{n}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}\textup{D}_{t}^{\alpha}\Delta\psi^{n}-\delta\textup{D}_{t}^{2-\alpha}\Delta\psi^{n}-f$

and therefore

(5.20)
$\textup{D}_{t}^{\alpha}\psi_{tt}^{n}=\tau^{-\alpha}\Bigl{(}-E_{\alpha,1}(-(\tfrac{t}{\tau})^{\alpha})\psi_{tt}^{n}(0)+\int_{0}^{t}E_{\alpha,\alpha}(-(\tfrac{t-s}{\tau})^{\alpha})r(s)\,\textup{d}s-r(t)\Bigr{)};$

see, e.g., [32, §3]. Thus, we have

(5.21)
$\displaystyle\|\textup{D}_{t}^{2+\alpha}\psi^{n}\|_{H^{-1/2}_{t}(L^{2})}\lesssim\|\textup{D}_{t}^{2+\alpha}\psi^{n}\|_{L^{2}_{t}(L^{2})}\lesssim\|r\|_{L^{2}_{t}(L^{2})}+\|\psi_{tt}^{n}(0)\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\,\begin{multlined}\|\sigma\psi_{tt}^{n}\|_{L^{2}_{t}(L^{2})}+c^{2}\|\Delta\psi\|_{L^{2}_{t}(L^{2})}+\tau^{\alpha}c^{2}\|\textup{D}_{t}^{\alpha}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}\\\
+\delta\|\textup{D}_{t}^{2-\alpha}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}+\|f\|_{L^{2}(L^{2})}+\|\psi_{tt}^{n}(0)\|_{L^{2}}.\end{multlined}\|\sigma\psi_{tt}^{n}\|_{L^{2}_{t}(L^{2})}+c^{2}\|\Delta\psi\|_{L^{2}_{t}(L^{2})}+\tau^{\alpha}c^{2}\|\textup{D}_{t}^{\alpha}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}\\\
+\delta\|\textup{D}_{t}^{2-\alpha}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}+\|f\|_{L^{2}(L^{2})}+\|\psi_{tt}^{n}(0)\|_{L^{2}}.$

In here, the terms with factors $c^{2}$, $\tau^{\alpha}c^{2}$, and $\delta$
can be controlled – in a (generalized) Gronwall inequality fashion – by left-
hand side terms in (5.12); to see this for the latter, consider

$\|\textup{D}_{t}^{2-\alpha}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}=\|\textup{I}^{\alpha-1/2}\textup{D}_{t}^{3/2}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}\lesssim\mathfrak{g}_{3/2-\alpha}*d.$

Thus, from (5.17) to (5.21), we have obtained an estimate of the form

$\displaystyle\|\sigma\psi_{tt}^{n}\|_{H_{t}^{\alpha-1/2}(L^{2})}\lesssim$
$\displaystyle\,\|\sigma\|_{X^{\sigma}_{\textup{fMGT
I}}}\left(\|\textup{D}_{t}^{2+\alpha}\psi^{n}\|_{L^{2}_{t}(L^{2})}+\|\psi_{tt}^{n}(0)\|_{L^{2}}\right)$
$\displaystyle\lesssim$ $\displaystyle\,\|\sigma\|_{X^{\sigma}_{\textup{fMGT
I}}}\left(\|\sigma\psi_{tt}^{n}\|_{H_{t}^{\alpha-1/2}(L^{2})}+\textup{rhs}\right),$

where

$\textup{rhs}:=c^{2}\|\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}+\tau^{\alpha}c^{2}\|\textup{D}_{t}^{\alpha}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}+\mathfrak{g}_{3/2-\alpha}*d+\|f\|_{L^{2}(L^{2})}+\|\psi_{tt}^{n}(0)\|_{L^{2}}.$

Thus, provided $\|\sigma\|_{X^{\sigma}_{\textup{fMGT I}}}$ is sufficiently
small (where the bound can be chosen independent of $\alpha\in[\alpha_{0},1)$
for $\alpha_{0}>1/2$), the term
$\|\sigma\psi_{tt}^{n}\|_{H_{t}^{\alpha-1/2}(L^{2})}$ is bounded by a multiple
of rhs. By combining this with (5.12), (5.21), and Gronwall’s inequality in
its generalized version, see, e.g., [32, Lemma 7.2], we therefore obtain the
following estimate:

(5.22)
$\displaystyle\begin{multlined}\|\textup{D}_{t}^{2+\alpha}\psi^{n}\|_{L^{2}_{t}(L^{2})}^{2}+\|\nabla\textup{D}_{t}^{1+\alpha}\psi^{n}(t)\|_{L^{2}}^{2}+\|\Delta\textup{D}_{t}^{\alpha}\psi^{n}(t)\|_{L^{2}}^{2}\\\
+C(\alpha)(\|\nabla\psi_{tt}^{n}\|_{{H_{t}^{-(1-\alpha)/2}}(L^{2})}^{2}+\|\textup{D}_{t}^{3/2}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}^{2})\end{multlined}\|\textup{D}_{t}^{2+\alpha}\psi^{n}\|_{L^{2}_{t}(L^{2})}^{2}+\|\nabla\textup{D}_{t}^{1+\alpha}\psi^{n}(t)\|_{L^{2}}^{2}+\|\Delta\textup{D}_{t}^{\alpha}\psi^{n}(t)\|_{L^{2}}^{2}\\\
+C(\alpha)(\|\nabla\psi_{tt}^{n}\|_{{H_{t}^{-(1-\alpha)/2}}(L^{2})}^{2}+\|\textup{D}_{t}^{3/2}\Delta\psi^{n}\|_{L^{2}_{t}(L^{2})}^{2})$
$\displaystyle\lesssim$
$\displaystyle\,\|f\|_{H^{\alpha-1/2}(L^{2})}^{2}+\|\Delta\psi_{0}\|_{L^{2}}^{2}+\|\Delta\psi_{1}\|_{L^{2}}^{2}+\|\nabla\psi_{2}\|_{L^{2}}^{2},$

where we have also relied on the uniform boundedness of the approximate data.
Here the constant $C(\alpha)$ tends to zero as $\alpha\to 1^{-}$, since it
contains the factor $\cos(\tfrac{\pi(1-\alpha)}{2})$ from the coercivity
estimate (3.5).

(III) _Passing to the limit._ Thanks to the uniform bound (5.22), there exists
a subsequence, which we do not relabel, such that

(5.23) $\displaystyle\textup{D}_{t}^{2+\alpha}\psi^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup\textup{D}_{t}^{2+\alpha}\psi$
weakly in $\displaystyle L^{2}(0,T;L^{2}(\Omega)),$
$\displaystyle{\textup{D}}_{t}^{\alpha}\Delta\psi^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup{\textup{D}}_{t}^{\alpha}\Delta\psi$
weakly-$\star$ in $\displaystyle L^{\infty}(0,T;L^{2}(\Omega)),$
$\displaystyle\textup{D}_{t}^{1+\alpha}\nabla\psi^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup\textup{D}_{t}^{1+\alpha}\nabla\psi$
weakly-$\star$ in $\displaystyle L^{\infty}(0,T;L^{2}(\Omega)),$
$\displaystyle\textup{D}_{t}^{3/2}\Delta\psi^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup\textup{D}_{t}^{3/2}\Delta\psi$
weakly in $\displaystyle L^{2}(0,T;L^{2}(\Omega)).$

Furthermore,

(5.24) $\displaystyle(1+\sigma)\psi_{tt}^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup(1+\sigma)\psi_{tt}$ weakly in
$\displaystyle L^{2}(0,T;L^{2}(\Omega)),$ $\displaystyle\Delta\psi^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup\Delta\psi$ weakly in
$\displaystyle L^{2}(0,T;L^{2}(\Omega)),$
$\displaystyle{\textup{D}}_{t}^{2-\alpha}\Delta\psi^{n}$
$\displaystyle\relbar\joinrel\rightharpoonup{\textup{D}}_{t}^{2-\alpha}\Delta\psi$
weakly in $\displaystyle L^{2}(0,T;L^{2}(\Omega)).$

Thus, we can pass to the limit in the usual way in the semi-discrete problem.
Further, weak/weak-$\star$ lower semi-continuity of norms implies that the
solution we constructed satisfies (5.5) a.e. in time.

(IV) _Attainment of the initial conditions._ Similarly to step (IV) in the
proof of Proposition 4.1, we show that $\psi$ attains its initial conditions
by, on one hand concluding from (5.23) that

$\displaystyle\psi^{n}(0)\relbar\joinrel\rightharpoonup\psi(0)\text{
weakly}\text{ in }{H_{\diamondsuit}^{2}(\Omega)},$
$\displaystyle\psi_{t}^{n}(0)\relbar\joinrel\rightharpoonup\psi_{t}(0)\text{
weakly}\text{ in }H_{0}^{1}(\Omega),$
$\displaystyle\psi_{tt}^{n}(0)\relbar\joinrel\rightharpoonup\psi_{tt}(0)\text{
weakly}\text{ in }L^{2}(\Omega),$

and, on the other hand, $\psi^{n}(0)\rightarrow\psi_{0}$ in
${H_{\diamondsuit}^{2}(\Omega)}$, $\psi_{t}^{n}(0)\rightarrow\psi_{1}$ in
$H_{0}^{1}(\Omega)$, $\psi_{tt}^{n}(0)\rightarrow\psi_{1}$ in $L^{2}(\Omega)$
based on our choice of the approximate data. Thus the initial data are
attained in an ${H_{\diamondsuit}^{2}(\Omega)}\times H_{0}^{1}(\Omega)\times
L^{2}(\Omega)$ sense.

(IV) _Uniqueness._ The fact that the obtained solution is unique follows by
testing the homogeneous problem

$\tau^{\alpha}\textup{D}_{t}^{2+\alpha}\psi+(1+\sigma)\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}\textup{D}_{t}^{\alpha}\Delta\psi-\delta\textup{D}_{t}^{2-\alpha}\Delta\psi=0$

(with zero initial data) with $\textup{D}_{t}^{1+\alpha}\psi_{t}$. Analogously
to above, but replacing $\Delta\to\nabla$ and $\nabla\to\mbox{id}$, we obtain

$\displaystyle\begin{multlined}\frac{\tau^{\alpha}}{2}\|\textup{D}_{t}^{1+\alpha}\psi(t)\|_{L^{2}}^{2}+\frac{\tau^{\alpha}c^{2}}{2}\|\nabla\textup{D}_{t}^{\alpha}\psi(t)\|_{L^{2}}^{2}+C(\alpha)(\|\psi_{tt}\|_{H_{t}^{-(1-\alpha)}(L^{2})}^{2}+\|\textup{D}_{t}^{3/2}\nabla\psi\|_{L^{2}_{t}(L^{2})}^{2})\end{multlined}\frac{\tau^{\alpha}}{2}\|\textup{D}_{t}^{1+\alpha}\psi(t)\|_{L^{2}}^{2}+\frac{\tau^{\alpha}c^{2}}{2}\|\nabla\textup{D}_{t}^{\alpha}\psi(t)\|_{L^{2}}^{2}+C(\alpha)(\|\psi_{tt}\|_{H_{t}^{-(1-\alpha)}(L^{2})}^{2}+\|\textup{D}_{t}^{3/2}\nabla\psi\|_{L^{2}_{t}(L^{2})}^{2})$
$\displaystyle\leq$
$\displaystyle\,\left|\int_{0}^{t}(c^{2}\Delta\psi-\sigma\psi_{tt},\textup{D}_{t}^{1+\alpha}\psi)_{L^{2}}\,\textup{d}s\right|$
$\displaystyle\leq$
$\displaystyle\,\frac{1}{2\epsilon}\Bigl{(}\|c^{2}\nabla\psi\|_{H_{t}^{\alpha-1/2}(L^{2})}+\|\sigma\psi_{tt}\|_{H_{t}^{\alpha-1/2}(H^{-1})}\Bigr{)}^{2}+\frac{\epsilon}{2}\|\textup{D}_{t}^{3/2}\nabla\psi\|_{L^{2}_{t}(L^{2})}^{2},$

where

$\displaystyle\|c^{2}\nabla\psi\|_{H^{\alpha-1/2}(L^{2})}$
$\displaystyle\lesssim\|\nabla\textup{D}_{t}^{\alpha}\psi\|_{L^{2}(L^{2})}.$

Further, on account of the following estimate:

(5.25) $\displaystyle\|ab\|_{H^{-1}(\Omega)}=$
$\displaystyle\,\|a\|_{H^{-1}(\Omega)}\sup_{v\in
H_{0}^{1}(\Omega)\setminus\\{0\\}}\|v\|_{H_{0}^{1}(\Omega)}^{-1}\|v\nabla
b+b\nabla v\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\,\|a\|_{H^{-1}(\Omega)}(C_{H^{1},L^{6}}\|\nabla
b\|_{L^{3}}+\|b\|_{L^{\infty}}),$

we have, similarly to (5.17),

(5.26)
$\displaystyle\|\sigma\psi_{tt}\|_{H_{t}^{\alpha-1/2}(H^{-1})}\lesssim\,\|\sigma\|_{X^{\sigma}_{\textup{fMGT
I}}}\|\psi_{tt}\|_{L_{t}^{\frac{1}{1-\alpha}}(H^{-1})}+\|\sigma\|_{X^{\sigma}_{\textup{fMGT
I}}}\|\psi_{tt}\|_{W_{t}^{\alpha-1/2,\infty}(H^{-1})}.$

Again using (5.20) with $\psi$ in place of $\psi^{n}$ yields

(5.27)
$\displaystyle\|\psi_{tt}\|_{L_{t}^{\frac{1}{1-\alpha}}(H^{-1})}+\|\psi_{tt}\|_{W_{t}^{\alpha-1/2,\infty}(H^{-1})}\lesssim\,\|\textup{D}_{t}^{2+\alpha}\psi\|_{L_{t}^{2}(H^{-1})}$
$\displaystyle\lesssim$
$\displaystyle\,\|\sigma\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}\textup{D}_{t}^{\alpha}\Delta\psi-\delta\textup{D}_{t}^{2-\alpha}\Delta\psi\|_{L_{t}^{2}(H^{-1})},$

where
$\|\sigma\psi_{tt}\|_{L_{t}^{2}(H^{-1})}\lesssim\|\sigma\psi_{tt}\|_{H_{t}^{\alpha-1/2}(H^{-1})}$.
Therefore, these terms can be absorbed for small enough $\sigma$ by the left-
hand side or handled by Gronwall’s inequality to conclude that $\psi=0$. ∎

We next prove a well-posedness result for the corresponding nonlinear problem.
To guarantee that the coefficient $\sigma$ is small enough in the fixed-point
iteration, we impose a smallness condition on the data.

######  Theorem 5.1 (Local well-posedness of the fJMGT–W I equation).

Let $\alpha\in[\alpha_{0},1)$ for some $\alpha_{0}>1/2$ and $T>0$. Further,
assume that $f\in H^{\alpha-1/2}(0,{T};L^{2}(\Omega))$. There exists
$\varrho=\varrho(\alpha,T)>0$, such that if

(5.28)
$\|f\|^{2}_{H^{\alpha-1/2}(L^{2})}+\|\psi_{0}\|_{H^{2}}^{2}+\|\psi_{1}\|_{H^{2}}^{2}+\|\psi_{2}\|_{H^{1}}^{2}\leq\varrho^{2},$

then the initial boundary-value problem

(5.29) $\left\\{\begin{aligned}
\tau^{\alpha}\textup{D}_{t}^{2+\alpha}\psi+&(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}\textup{D}_{t}^{\alpha}\Delta\psi-\delta\textup{D}_{t}^{2-\alpha}\Delta\psi=f&&\text{in
}\Omega\times(0,T),\\\\[2.84526pt] &\psi=\,0&&\text{on
}\partial\Omega\times(0,T),\\\\[2.84526pt]
&(\psi,\psi_{t},\psi_{tt})=\,(\psi_{0},\psi_{1},\psi_{2})&&\mbox{in
}\Omega\times\\{0\\},\end{aligned}\right.$

has a unique solution $\psi\in X_{\textup{fMGT I}}$, which satisfies

(5.30) $\displaystyle\begin{multlined}\|\psi\|^{2}_{X_{\textup{fMGT
I}}}\end{multlined}\|\psi\|^{2}_{X_{\textup{fMGT
I}}}\lesssim\,\begin{multlined}\|f\|^{2}_{H^{\alpha-1/2}(L^{2})}+\|\psi_{0}\|_{H^{2}}^{2}+\|\psi_{1}\|_{H^{2}}^{2}+\|\psi_{2}\|_{H^{1}}^{2}.\end{multlined}\|f\|^{2}_{H^{\alpha-1/2}(L^{2})}+\|\psi_{0}\|_{H^{2}}^{2}+\|\psi_{1}\|_{H^{2}}^{2}+\|\psi_{2}\|_{H^{1}}^{2}.$

###### Proof.

The proof follows by setting up a fixed-point mapping
$\mathcal{T}:w\mapsto\psi$, which associates

$w\in B_{R}:=\\{w\in X_{\textup{fMGT I}}\,:\|w\|_{X_{\textup{fMGT I}}}\leq R,\
w(0)=\psi_{0},\,w_{t}(0)=\psi_{1},\,w_{tt}(0)=\psi_{2}\\}$

with the solution $\psi$ of the linearized problem (5.1) with
$\sigma=2kw_{t}$. We recall that

$X^{\sigma}_{\textup{fMGT I}}=L^{2}(0,T;(W^{1,3}\cap L^{\infty})(\Omega)),$

and so

$\|\sigma\|_{X^{\sigma}_{\textup{fMGT
I}}}=2|k|\|w_{t}\|_{X^{\sigma}_{\textup{fMGT I}}}\leq C\|w\|_{X_{\textup{fMGT
I}}}\lesssim R.$

Thus, $\sigma$ can be made small enough by decreasing $R>0$. The self-mapping
is thus an immediate consequence of the energy estimate (5.5), provided we
choose $\varrho$ small enough, so that

$C(R,T)(\|f\|^{2}_{H^{\alpha-1/2}(L^{2})}+\|\psi_{0}\|_{H^{2}}^{2}+\|\psi_{1}\|_{H^{2}}^{2}+\|\psi_{2}\|_{H^{1}}^{2})\leq
C(R,T)\varrho^{2}\leq R^{2}.$

We prove strict contractivity of this mapping next. Let $w^{(1)}$, $w^{(2)}\in
B_{R}$. Denote $\psi^{(1)}=\mathcal{T}w^{(1)}$ and
$\psi^{(2)}=\mathcal{T}w^{(2)}$. Contractivity of $\mathcal{T}$ follows by
considering the difference equation for
$\overline{\psi}=\psi^{(1)}-\psi^{(2)}$:

(5.31)
$\displaystyle\tau^{\alpha}\textup{D}_{t}^{2+\alpha}\overline{\psi}+(1+2kw_{t}^{(1)})\overline{\psi}_{tt}-c^{2}\Delta\overline{\psi}-\tau^{\alpha}c^{2}\Delta\textup{D}_{t}^{\alpha}\overline{\psi}-\delta\textup{D}_{t}^{2-\alpha}\Delta\overline{\psi}_{t}+2k\psi^{(2)}_{tt}\overline{w}_{t}=0,$

which is supplemented by zero initial conditions. Similarly to the proof of
uniqueness in Proposition 5.1, testing with $\overline{\psi}_{tt}$ yields

$\displaystyle\,\begin{multlined}\frac{\tau^{\alpha}}{2}\|\textup{D}_{t}^{1+\alpha}\overline{\psi}(t)\|_{L^{2}}^{2}+\frac{\tau^{\alpha}c^{2}}{2}\|\nabla\textup{D}_{t}^{\alpha}\overline{\psi}(t)\|_{L^{2}}^{2}\\\
\hskip
56.9055pt+C(\alpha)(\|\psi_{tt}\|_{H_{t}^{-(1-\alpha)}(L^{2})}^{2}+\|\textup{D}_{t}^{3/2}\nabla\overline{\psi}\|_{L^{2}_{t}(L^{2})}^{2})\end{multlined}\frac{\tau^{\alpha}}{2}\|\textup{D}_{t}^{1+\alpha}\overline{\psi}(t)\|_{L^{2}}^{2}+\frac{\tau^{\alpha}c^{2}}{2}\|\nabla\textup{D}_{t}^{\alpha}\overline{\psi}(t)\|_{L^{2}}^{2}\\\
\hskip
56.9055pt+C(\alpha)(\|\psi_{tt}\|_{H_{t}^{-(1-\alpha)}(L^{2})}^{2}+\|\textup{D}_{t}^{3/2}\nabla\overline{\psi}\|_{L^{2}_{t}(L^{2})}^{2})$
$\displaystyle\leq$
$\displaystyle\,\left|\int_{0}^{t}(-2k\psi^{(2)}_{tt}\overline{w}_{t}+c^{2}\Delta\overline{\psi}-2kw_{t}^{(1)}\overline{\psi}_{tt},\textup{D}_{t}^{1+\alpha}\overline{\psi})_{L^{2}}\,\textup{d}s\right|$
$\displaystyle\leq$
$\displaystyle\,\begin{multlined}\frac{\epsilon}{2}\|\textup{D}_{t}^{1+\alpha}\overline{\psi}\|_{L_{t}^{\infty}(L^{2})}^{2}+\frac{1}{2\epsilon}\|-2k\psi^{(2)}_{tt}\overline{w}_{t}\|_{L_{t}^{1}(L^{2})}^{2}+\frac{\epsilon}{2}\|\textup{D}_{t}^{3/2}\nabla\overline{\psi}\|_{L^{2}_{t}(L^{2})}^{2}\\\
+\frac{1}{2\epsilon}\Bigl{(}\|c^{2}\nabla\overline{\psi}\|_{H_{t}^{\alpha-1/2}(L^{2})}+\|2kw_{t}^{(1)}\overline{\psi}_{tt}\|_{H_{t}^{\alpha-1/2}(H^{-1})}\Bigr{)}^{2}.\end{multlined}\frac{\epsilon}{2}\|\textup{D}_{t}^{1+\alpha}\overline{\psi}\|_{L_{t}^{\infty}(L^{2})}^{2}+\frac{1}{2\epsilon}\|-2k\psi^{(2)}_{tt}\overline{w}_{t}\|_{L_{t}^{1}(L^{2})}^{2}+\frac{\epsilon}{2}\|\textup{D}_{t}^{3/2}\nabla\overline{\psi}\|_{L^{2}_{t}(L^{2})}^{2}\\\
+\frac{1}{2\epsilon}\Bigl{(}\|c^{2}\nabla\overline{\psi}\|_{H_{t}^{\alpha-1/2}(L^{2})}+\|2kw_{t}^{(1)}\overline{\psi}_{tt}\|_{H_{t}^{\alpha-1/2}(H^{-1})}\Bigr{)}^{2}.$

We can then rely on the following bound:

$\displaystyle\|c^{2}\nabla\overline{\psi}\|_{H^{\alpha-1/2}(L^{2})}$
$\displaystyle\lesssim\|\nabla\textup{D}_{t}^{\alpha}\overline{\psi}\|_{L^{2}(L^{2})}$

and, by (5.17)–(5.21) with $2kw_{t}^{(1)}$ in place of $\sigma$, we have

$\displaystyle\|2kw_{t}^{(1)}\psi_{tt}\|_{H^{\alpha-1/2}(H^{-1})}\lesssim\|w_{t}^{(1)}\|_{X^{\sigma}_{\textup{fMGT
I}}}\|\textup{D}_{t}^{2+\alpha}\psi\|_{L^{2}(H^{-1})}$ $\displaystyle\lesssim$
$\displaystyle\,\|w_{t}^{(1)}\|_{X^{\sigma}_{\textup{fMGT
I}}}\|2kw_{t}^{(1)}\psi_{tt}-c^{2}\Delta\overline{\psi}-\tau^{\alpha}c^{2}\Delta\textup{D}_{t}^{\alpha}\overline{\psi}-\delta\textup{D}_{t}^{2-\alpha}\Delta\overline{\psi}_{t}\|_{L^{2}(H^{-1})}.$

Thus for $\|w_{t}^{(1)}\|_{X_{\sigma}}\lesssim R$ small enough, similarly to
the proof of Proposition 5.1, we obtain

(5.32)
$\displaystyle\frac{\tau^{\alpha}}{2}\|\textup{D}_{t}^{1+\alpha}\overline{\psi}(t)\|_{L^{2}}^{2}+\frac{\tau^{\alpha}c^{2}}{2}\|\nabla\textup{D}_{t}^{\alpha}\overline{\psi}(t)\|_{L^{2}}^{2}+\|\overline{\psi}_{tt}\|_{H_{t}^{-(1-\alpha)}(L^{2})}^{2}+\|\textup{D}_{t}^{3/2}\nabla\overline{\psi}\|_{L^{2}_{t}(L^{2})}^{2}$
$\displaystyle\lesssim$
$\displaystyle\,R^{2}\|2k\psi_{tt}^{(2)}\overline{w}_{t}\|_{L^{2}(H^{-1})}^{2}+\|2k\psi_{tt}^{(2)}\overline{w}_{t}\|_{L^{1}(L^{2})}.$

Furthermore,

(5.33) $\displaystyle
R^{2}\|2k\psi_{tt}^{(2)}\overline{w}_{t}\|_{L^{2}(H^{-1})}^{2}+\|2k\psi_{tt}^{(2)}\overline{w}_{t}\|_{L^{1}(L^{2})}\lesssim$
$\displaystyle\,R^{2}\|\psi_{tt}^{(2)}\|^{2}_{L^{\infty}L^{4}}\|\overline{w}_{t}\|^{2}_{L^{2}(L^{4})}$
$\displaystyle\lesssim$
$\displaystyle\,R^{2}\varrho^{2}\|\overline{w}_{t}\|^{2}_{L^{2}(L^{4})}.$

We can further bound the last term as follows:

(5.34)
$\displaystyle\|\nabla\overline{w}_{t}\|^{2}_{L^{2}(L^{2})}=\|\textup{I}^{\gamma}\textup{D}_{t}^{1+\gamma}\nabla\overline{w}\|^{2}_{L^{2}(L^{2})}\leq\|\mathfrak{g}_{1-\gamma}\|_{L^{1}(0,T)}\|\textup{D}_{t}^{1+\gamma}\nabla\overline{w}\|^{2}_{L^{2}(L^{2})},$

choosing $\gamma=1/2$. Thus, by decreasing $\varrho>0$, we can guarantee that
$\mathcal{T}$ is strictly contractive in the following norm:

$|||\psi|||=\|\textup{D}_{t}^{1+\alpha}\overline{\psi}\|_{L^{\infty}(L^{2})}^{2}+\|\nabla\textup{D}_{t}^{\alpha}\overline{\psi}\|_{L^{\infty}(L^{2})}^{2}+\|\overline{\psi}_{tt}\|_{H^{-(1-\alpha)}(L^{2})}^{2}+\|\nabla\overline{\psi}\|_{L^{2}(L^{2})}^{2}.$

The rest of the arguments follow as in Theorem 4.1 and complete the proof. ∎

###  5.2. Analysis of the fJMGT–W equation

To formulate the corresponding result for the fMGT-W equation

$\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta\Delta\psi_{t}=f$

we again need smallness of the coefficient $\sigma$ in a suitable norm. To
this end, let

$X^{\sigma}_{\textup{fMGT}}=H^{\alpha/2}(0,T;(W^{1,3}\cap
L^{\infty})(\Omega))$

for $\alpha\in(1/2,1)$, and denote the corresponding norm by
$\|\cdot\|_{X^{\sigma}_{\textup{fMGT}}}$. We also introduce the solution space
by

(5.35) $\displaystyle\quad X_{\textup{fMGT}}=\,\left\\{\psi\in
H^{2+\alpha}(0,T;L^{2}(\Omega)):\right.$
$\displaystyle\,\textup{D}_{t}^{1+\alpha/2}\psi\in
L^{\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)}),$
$\displaystyle\left.\textup{D}_{t}^{1+\alpha}\in
L^{\infty}(0,T;H_{0}^{1}(\Omega))\right\\},$

equipped with the norm $\|\cdot\|_{X_{\textup{fMGT}}}$. Note that with this
choice, again

$\|2k\psi_{t}\|_{X^{\sigma}_{\textup{fMGT}}}\lesssim\|\psi\|_{X_{\textup{fMGT}}}$

holds but the energy term
$\|\textup{D}_{t}^{1+\alpha/2}\Delta\psi\|_{L^{2}(L^{2})}$ needed for this
purpose comes with an $\alpha$-dependent coefficient in (5.36). For this
reason, while still being able to show well-posedness also of the nonlinear
fJMGT–W equation for each $\alpha\in(0,1)$, we will not obtain a uniform bound
quantifying smallness of the initial data. That is, we will not be able to
show that for fixed small enough $(\psi_{0},\psi_{1},\psi_{2})$, there exists
a family of solutions to the nonlinear problem. Hence, concerning limits as
$\alpha\to 1-$, we will restrict ourselves to the linear fMGT equation:

$\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+\sigma(x,t))\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}{\textup{D}}_{t}^{\alpha}\Delta\psi-\delta\Delta\psi_{t}=f.$

We next prove the well-posedness of the linear time-fractional problem. Note
that under the same regularity conditions on the initial and boundary data,
the fMGT equation allows us to prove slightly better regularity of the
solution as compared to fMGT I; cf. Proposition 5.1.

######  Proposition 5.2 (Well-posedness of the fMGT equation).

Let $\alpha\in[\alpha_{0},1)$ for some $\alpha_{0}>1/2$. Assume that $f\in
H^{\alpha-1/2}(0,T;L^{2}(\Omega))$, $\sigma\in X^{\sigma}_{\textup{fMGT I}}$,
and

$(\psi_{0},\psi_{1},\psi_{2})\in({H_{\diamondsuit}^{2}(\Omega)},{H_{\diamondsuit}^{2}(\Omega)},H_{0}^{1}(\Omega)).$

Then there exists $\varrho=\varrho(\alpha)>0$, such that if
$\|\sigma\|_{X^{\sigma}_{\textup{fMGT}}}\leq\varrho$, there is a unique
$\psi\in X_{\textup{fMGT}}$, which satisfies the problem in the
$L^{2}(0,T;L^{2}(\Omega))$ sense with
$(\psi,\psi_{t},\psi_{tt})|_{t=0}=(\psi_{0},\psi_{1},\psi_{2})$. Furthermore,
this solution fulfills the following estimate:

(5.36)
$\displaystyle\begin{multlined}\|\textup{D}_{t}^{2+\alpha}\psi\|_{L^{2}(L^{2})}^{2}+\|\nabla\textup{D}_{t}^{1+\alpha}\psi(t)\|_{L^{2}}^{2}+\|\Delta\textup{D}_{t}^{\alpha}\psi(t)\|_{L^{2}}^{2}\\\
+C(\alpha)\Bigl{(}\|\nabla\psi_{tt}\|_{{H^{-(1-\alpha)/2}}(L^{2})}^{2}+\|\textup{D}_{t}^{1+\alpha/2}\Delta\psi\|_{L^{2}(L^{2})}^{2}\Bigr{)}\end{multlined}\|\textup{D}_{t}^{2+\alpha}\psi\|_{L^{2}(L^{2})}^{2}+\|\nabla\textup{D}_{t}^{1+\alpha}\psi(t)\|_{L^{2}}^{2}+\|\Delta\textup{D}_{t}^{\alpha}\psi(t)\|_{L^{2}}^{2}\\\
+C(\alpha)\Bigl{(}\|\nabla\psi_{tt}\|_{{H^{-(1-\alpha)/2}}(L^{2})}^{2}+\|\textup{D}_{t}^{1+\alpha/2}\Delta\psi\|_{L^{2}(L^{2})}^{2}\Bigr{)}$
$\displaystyle\lesssim$
$\displaystyle\,\|f\|_{H^{\alpha-1/2}(L^{2})}^{2}+\|\Delta\psi_{0}\|_{L^{2}}^{2}+\|\Delta\psi_{1}\|_{L^{2}}^{2}+\|\nabla\psi_{2}\|_{L^{2}}^{2}.$

where $C(\alpha)\to 0$ as $\alpha\to 1^{-}$.

###### Proof.

The proof follows similarly to the proof of Proposition 5.1 with the main
changes contained in the energy analysis, on which we focus here. Note that
now the semi-discrete problem can be equivalently rewritten as a system of
Volterra integral equations:

$\displaystyle\begin{multlined}\tau^{\alpha}M\boldsymbol{\mu}+M_{\sigma}(t)\Bigl{(}p^{\alpha-1}*\boldsymbol{\mu}+p^{0}\boldsymbol{\xi}_{2}\Bigr{)}+c^{2}K\Bigl{(}p^{\alpha+1}*\boldsymbol{\mu}+p^{2}\boldsymbol{\xi}_{2}+p^{1}\boldsymbol{\xi}_{1}+p^{0}\boldsymbol{\xi}_{0}\Bigr{)}\\\
+\tau^{\alpha}c^{2}\Bigl{(}p^{1}*\boldsymbol{\mu}+p^{2-\alpha}\boldsymbol{\xi}_{2}+p^{1-\alpha}\boldsymbol{\xi}_{1}\Bigr{)}+\delta
K(p^{\alpha}*\boldsymbol{\mu}+p^{1}\boldsymbol{\xi}_{2}+p^{0}\boldsymbol{\xi}_{1})=f\end{multlined}\tau^{\alpha}M\boldsymbol{\mu}+M_{\sigma}(t)\Bigl{(}p^{\alpha-1}*\boldsymbol{\mu}+p^{0}\boldsymbol{\xi}_{2}\Bigr{)}+c^{2}K\Bigl{(}p^{\alpha+1}*\boldsymbol{\mu}+p^{2}\boldsymbol{\xi}_{2}+p^{1}\boldsymbol{\xi}_{1}+p^{0}\boldsymbol{\xi}_{0}\Bigr{)}\\\
+\tau^{\alpha}c^{2}\Bigl{(}p^{1}*\boldsymbol{\mu}+p^{2-\alpha}\boldsymbol{\xi}_{2}+p^{1-\alpha}\boldsymbol{\xi}_{1}\Bigr{)}+\delta
K(p^{\alpha}*\boldsymbol{\mu}+p^{1}\boldsymbol{\xi}_{2}+p^{0}\boldsymbol{\xi}_{1})=f$

in place of (5.7); the existence of an approximate solutions follows by the
same arguments. We present the energy analysis of the semi-discrete problem
here, but omit the superscript $n$ below for simplicity. Multiplying the semi-
discrete equation with $-\Delta\textup{D}_{t}^{1+\alpha}\psi$, yields the
energy identity (5.12), where now

(5.37) $\displaystyle d=$
$\displaystyle\,\int_{0}^{t}(\textup{D}_{t}\Delta\psi,\Delta\textup{D}_{t}^{1+\alpha}\psi)_{L^{2}}\,\textup{d}s$
$\displaystyle=$
$\displaystyle\,d_{0}+\int_{0}^{t}({\textup{I}}^{\alpha}\textup{D}_{t}^{1+\alpha}\Delta\psi(s),\textup{D}_{t}^{1+\alpha}\Delta\psi(s))_{L^{2}}\,\textup{d}s,$

with
$d_{0}=(\Delta\psi_{t}(0),\textup{D}_{t}^{\alpha}\Delta\psi(t))_{L^{2}}-p^{1-\alpha}(t)\|\Delta\psi_{t}(0)\|_{L^{2}}^{2}$
instead of (5.15). Here, we have used the identities

$\textup{D}_{t}\psi={\textup{I}}^{1}\psi_{tt}+\psi_{t}(0)={\textup{I}}^{\alpha}\textup{D}_{t}^{1+\alpha}\psi+\psi_{t}(0)$

and

$\displaystyle\int_{0}^{t}(\Delta\psi_{t}(0),\textup{D}_{t}^{1+\alpha}\Delta\psi(s))_{L^{2}}\,\textup{d}s=$
$\displaystyle\,(\Delta\psi_{t}(0),\int_{0}^{t}((\textup{D}_{t}\textup{D}_{t}^{\alpha}\Delta\psi)(s)-p^{-\alpha}(s)\Delta\psi_{t}(0))\,\textup{d}s)_{L^{2}}$
$\displaystyle=$
$\displaystyle\,(\Delta\psi_{t}(0),\textup{D}_{t}^{\alpha}\Delta\psi(t))_{L^{2}}-p^{1-\alpha}(t)\|\Delta\psi_{t}(0)\|_{L^{2}}^{2}.$

The damping term can now be estimated from below as follows

$\frac{d+d_{0}}{\cos(\pi\gamma/2)}\geq\|\textup{D}_{t}^{1+\alpha}\Delta\psi\|_{H_{t}^{-\alpha/2}(L^{2})}^{2}\sim\|\textup{D}_{t}^{1+\alpha/2}\Delta\psi\|_{L_{t}^{2}(L^{2})}^{2}.$

Furthermore, we have

$\displaystyle\left|\int_{0}^{t}(\tilde{f}(s),\textup{D}_{t}^{1+\alpha}\Delta\psi(s))_{L^{2}}\,\textup{d}s\right|\leq\frac{1}{2\epsilon}\|\tilde{f}\|_{H^{\alpha/2}(L^{2})}^{2}+\frac{\epsilon}{2}\|\textup{D}_{t}^{1+\alpha/2}\Delta\psi\|_{L^{2}(L^{2})}^{2}$

with

(5.38)
$\tilde{f}=f+c^{2}\Delta\psi^{n}-\sigma\psi_{tt}^{n}+\tau^{\alpha}\left(p^{-\alpha}\psi_{tt}^{n}(0)-c^{2}p^{1-\alpha}\Delta\psi_{t}^{n}(0)\right)$

as before. By the Kato–Ponce inequality (3.6) with $\rho=\alpha/2$, we then
have

(5.39) $\displaystyle\|\sigma\psi_{tt}\|_{H_{t}^{\alpha/2}(L^{2})}$
$\displaystyle\lesssim$
$\displaystyle\,\|\sigma\|_{W_{t}^{\alpha/2,p_{1}}(L^{\infty})}\|\psi_{tt}\|_{L_{t}^{q_{1}}(L^{2})}+\|\sigma\|_{L_{t}^{p_{2}}(L^{\infty})}\|\psi_{tt}\|_{W_{t}^{\alpha/2,q_{2}}(L^{2})}$

for
$\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{p_{2}}+\frac{1}{q_{2}}=\frac{1}{2}$.
Similarly to before, to further estimate the norms of $\psi_{tt}$ in (5.39) we
will use the leading time derivative term $\textup{D}_{t}^{2+\alpha}\psi$ and
its representation via the fractional ODE

$\tau^{\alpha}\textup{D}_{t}^{\alpha}\psi_{tt}+\psi_{tt}=-r,\qquad
r=\sigma\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}\textup{D}_{t}^{\alpha}\Delta\psi-\delta\textup{D}_{t}\Delta\psi-f.$

Therefore,

$\displaystyle\|\psi_{tt}\|_{L^{q_{1}}(0,t;L^{2})}+\|\psi_{tt}\|_{W^{\alpha/2,q_{2}}(0,t;L^{2})}\lesssim\|\textup{D}_{t}^{2+\alpha}\psi\|_{L^{2}(L^{2})}$
$\displaystyle\lesssim$
$\displaystyle\,\|\sigma\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}\textup{D}_{t}^{\alpha}\Delta\psi-\delta\textup{D}_{t}\Delta\psi-f\|_{L^{2}(L^{2})}+\|\psi_{tt}(0)\|_{L^{2}},$

where again we take care of the highest order term by using the damping term
$d$. Thus, we require

(5.40) $2-\frac{1}{q_{1}}\leq 2+\alpha-\frac{1}{2},\quad
2+\alpha/2-\frac{1}{q_{2}}\leq 2+\alpha-\frac{1}{2}.$

Additionally, we aim at choosing the available parameters $p_{i}$ (yielding
$q_{i}=2p_{i}/(p_{i}-2)$), such that (having in mind that $\sigma=2k\psi_{t}$
in the fixed point argument later on)

$\|\Delta\psi_{t}(s)\|_{W^{\alpha/2,p_{1}}_{t}(L^{2})}\lesssim
d,\quad\|\Delta\psi_{t}(s)\|_{L^{p_{2}}_{t}(L^{2})}\lesssim d\,,$

which leads to

(5.41) $1+\frac{\alpha}{2}-\frac{1}{p_{1}}\leq
1+\frac{\alpha}{2}-\frac{1}{2}\mbox{ and }1-\frac{1}{p_{2}}\leq
1+\frac{\alpha}{2}-\frac{1}{2}.$

It is readily checked that all conditions in (5.40), (5.41) can be satisfied
with the choice

(5.42)
$(p_{1},q_{1})=(2,\infty)\,,\qquad(p_{2},q_{2})=\left(\frac{2}{1-\alpha},\frac{2}{\alpha}\right).$

Thus analogously to the proof of Proposition 5.1, we arrive at (5.36). ∎

######  Theorem 5.2 (Local well-posedness of the fJMGT–W equation).

Let $\alpha\in[\alpha_{0},1)$ for some $\alpha_{0}>1/2$ and $T>0$. Further,
assume that $f\in H^{\alpha-1/2}(0,{T};L^{2}(\Omega))$. Then there exists
$\varrho=\varrho(\alpha,T)>0$ such that if

(5.43)
$\|f\|^{2}_{H^{\alpha-1/2}(L^{2})}+\|\psi_{0}\|_{H^{2}}^{2}+\|\psi_{1}\|_{H^{2}}^{2}+\|\psi_{2}\|_{H^{1}}^{2}\leq\varrho^{2},$

then the initial boundary-value problem

(5.44) $\left\\{\begin{aligned}
\tau^{\alpha}\textup{D}_{t}^{2+\alpha}\psi+&(1+2k\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau^{\alpha}c^{2}\textup{D}_{t}^{\alpha}\Delta\psi-\delta\Delta\psi_{t}=f&&\text{in
}\Omega\times(0,T),\\\\[2.84526pt] &\psi=\,0&&\text{on
}\partial\Omega\times(0,T),\\\\[2.84526pt]
&(\psi,\psi_{t},\psi_{tt})=\,(\psi_{0},\psi_{1},\psi_{2})&&\mbox{in
}\Omega\times\\{0\\},\end{aligned}\right.$

has a unique solution $\psi\in X_{\textup{fMGT}}$, which satisfies

(5.45)
$\displaystyle\begin{multlined}\|\psi\|^{2}_{X_{\textup{fMGT}}}\end{multlined}\|\psi\|^{2}_{X_{\textup{fMGT}}}\lesssim\,\begin{multlined}\|f\|^{2}_{H^{\alpha-1/2}(L^{2})}+\|\psi_{0}\|_{H^{2}}^{2}+\|\psi_{1}\|_{H^{2}}^{2}+\|\psi_{2}\|_{H^{1}}^{2}.\end{multlined}\|f\|^{2}_{H^{\alpha-1/2}(L^{2})}+\|\psi_{0}\|_{H^{2}}^{2}+\|\psi_{1}\|_{H^{2}}^{2}+\|\psi_{2}\|_{H^{1}}^{2}.$

###### Proof.

The proof follows in an analogous manner to the proof of Theorem 5.1, combined
with the results of Proposition 5.2. We therefore omit the details here. ∎

######  Remark 1 (On the analysis of the fMGT II equation with
$\boldsymbol{\sigma\neq 0}$).

We note that the fMGT II equation

$\tau^{\alpha}{\textup{D}}_{t}^{\alpha}\psi_{tt}+(1+\sigma(x,t))\psi_{tt}-c^{2}\Delta\psi-(\tau^{\alpha}c^{2}+\delta){\textup{D}}_{t}^{\alpha}\Delta\psi=f$

does not seem to be tractable this way with $\sigma\not=0$. In particular, we
would have

$d=\|\Delta\textup{D}_{t}^{\alpha}\psi\|_{L^{2}}^{2}\Big{|}_{0}^{t}$

in place of (5.15) and (5.37). Thus, the damping term $\delta d$ is obviously
too weak.
Note that the multiplier $-\Delta\psi_{tt}$ that we have successfully used for
the fMGT III equation in Section 4 does to work out either since then the
$\delta$ term cannot be proven to be nonnegative due to the fact that the
difference $2-\alpha$ of the differentiation orders is larger than one. We
provide an analysis of the fMGT II equation with $\sigma=0$ in Section 7.1 by
rewriting it as a second-order wave equation with memory.

###  5.3. Limiting behavior of the fMGT–W and fJMGT–W I equations

The difference $\overline{\psi}=\psi^{\alpha}-\psi$ solves

(5.46)
$\displaystyle\tau^{\alpha}\textup{D}_{t}^{2+\alpha}\overline{\psi}+\sigma\overline{\psi}_{tt}-c^{2}\Delta\overline{\psi}-\tau^{\alpha}c^{2}\Delta\textup{D}_{t}^{\alpha}\overline{\psi}-\delta\textup{D}_{t}^{\beta}\Delta\overline{\psi}_{t}$
$\displaystyle=\,(\tau\psi_{ttt}-\tau^{\alpha}\textup{D}_{t}^{2+\alpha}\psi_{tt})-c^{2}\Delta(\tau\psi_{t}-\tau^{\alpha}\textup{D}_{t}^{\alpha}\psi)-\delta\Delta(\psi_{t}-\textup{D}_{t}^{\beta}\psi_{t})=:\tilde{f}$

in the linear case and

(5.47)
$\displaystyle\tau^{\alpha}\textup{D}_{t}^{2+\alpha}\overline{\psi}+(1+2k\psi_{t}^{\alpha})\overline{\psi}_{tt}-c^{2}\Delta\overline{\psi}-\tau^{\alpha}c^{2}\Delta\textup{D}_{t}^{\alpha}\overline{\psi}-\delta\textup{D}_{t}^{\beta}\Delta\overline{\psi}+2k\psi_{tt}\overline{\psi}_{t}=\tilde{f}$

in the nonlinear case with vanishing initial data and $\beta=\beta(\alpha)$ as
in (5.2) (which implies that the $\beta$ difference term on the right-hand
side just vanishes in case of the fJMGT–W equation).

Multiplication with $\textup{D}_{t}^{1+\alpha}\overline{\psi}$ with the
abbreviations $\sigma^{\alpha}=\sigma$, $\mu=0$ in the linear case and
$\sigma^{\alpha}=2k\psi_{t}^{\alpha}$, $\mu=2k\psi_{tt}\overline{\psi}_{t}$ in
the nonlinear case yields, analogously to the proof of uniqueness in
Proposition 5.1,

(5.48) $\displaystyle\textup{lhs}:=$
$\displaystyle\,\frac{\tau^{\alpha}}{2}\|\textup{D}_{t}^{1+\alpha}\overline{\psi}(t)\|_{L^{2}}^{2}+\frac{\tau^{\alpha}c^{2}}{2}\|\nabla\textup{D}_{t}^{\alpha}\overline{\psi}(t)\|_{L^{2}}^{2}$
$\displaystyle\quad+\cos(\pi(1-\alpha)/2)\|\overline{\psi}_{tt}\|_{H_{t}^{-(1-\alpha)}(L^{2})}^{2}+\delta\cos(\pi\gamma/2)\|\textup{D}_{t}^{m}\nabla\overline{\psi}\|_{L_{t}^{2}(L^{2})}^{2}$
$\displaystyle=$
$\displaystyle\int_{0}^{t}(\tilde{f}+c^{2}\Delta\overline{\psi}-\sigma^{\alpha}\overline{\psi}_{tt}+\mu,\textup{D}_{t}^{1+\alpha}\overline{\psi})_{L^{2}}\,\textup{d}s$
$\displaystyle\leq$
$\displaystyle\,\begin{multlined}\frac{\epsilon}{2}\|\textup{D}_{t}^{1+\alpha}\overline{\psi}\|_{L_{t}^{\infty}(L^{2})}^{2}+\frac{1}{2\epsilon}\|\tilde{f}+2k\psi_{tt}\overline{\psi}_{t}\|_{L_{t}^{1}(L^{2})}^{2}+\frac{\epsilon}{2}\|\textup{D}_{t}^{m}\nabla\overline{\psi}\|_{L_{t}^{2}(L^{2})}^{2}\\\
+\frac{1}{2\epsilon}\left(\|c^{2}\nabla\overline{\psi}\|_{H_{t}^{\rho}(L^{2})}+\|\sigma^{\alpha}\overline{\psi}_{tt}\|_{H_{t}^{\rho}(H^{-1})}\right)^{2},\end{multlined}\frac{\epsilon}{2}\|\textup{D}_{t}^{1+\alpha}\overline{\psi}\|_{L_{t}^{\infty}(L^{2})}^{2}+\frac{1}{2\epsilon}\|\tilde{f}+2k\psi_{tt}\overline{\psi}_{t}\|_{L_{t}^{1}(L^{2})}^{2}+\frac{\epsilon}{2}\|\textup{D}_{t}^{m}\nabla\overline{\psi}\|_{L_{t}^{2}(L^{2})}^{2}\\\
+\frac{1}{2\epsilon}\left(\|c^{2}\nabla\overline{\psi}\|_{H_{t}^{\rho}(L^{2})}+\|\sigma^{\alpha}\overline{\psi}_{tt}\|_{H_{t}^{\rho}(H^{-1})}\right)^{2},$

where

$\begin{cases}\gamma=2\alpha-1\,,\quad m=3/2\,,\quad\rho=\alpha-1/2&\mbox{ for
fMGT I,}\\\ \gamma=\alpha\,,\quad m=1+\alpha/2\,,\quad\rho=\alpha/2&\mbox{ for
fMGT.}\end{cases}$

We know that

$\displaystyle\|2k\psi_{tt}\overline{\psi}_{t}\|_{L^{1}(L^{2})}$
$\displaystyle{\leq