2|k|C_{H^{1},L^{6}}^{\Omega}C_{H^{\alpha},L^{3}}^{\Omega}\|\nabla\psi_{tt}\|_{L_{t}^{2}(L^{2})}\|\overline{\psi}_{t}\|_{L_{t}^{2}(H^{\alpha})}}$
$\displaystyle{\lesssim\|\nabla\psi_{tt}\|_{L^{2}(L^{2})}\|\textup{D}_{t}^{1+\alpha}\overline{\psi}\|_{L_{t}^{2}(L^{2})}^{1-\alpha}\|\textup{D}_{t}^{\alpha}\nabla\overline{\psi}\|_{L_{t}^{2}(L^{2})}^{\alpha},}$
$\displaystyle\|c^{2}\Delta\overline{\psi}\|_{H^{\rho}(L^{2})}$
$\displaystyle\lesssim\|\nabla\textup{D}_{t}^{\alpha}\overline{\psi}\|_{L^{2}(L^{2})}.$

where we have used interpolation; cf. [1, Chapter 7]. Let $X^{\sigma}$ be
either $X^{\sigma}_{\textup{fMGT I}}$ or $X^{\sigma}_{\textup{fMGT}}$,
depending on the equation. By proceeding similarly to (5.17)–(5.21), we find
that

$\displaystyle\|\sigma^{\alpha}\overline{\psi}_{tt}\|_{H^{\rho}(H^{-1})}\lesssim$
$\displaystyle\,\begin{multlined}\|\sigma^{\alpha}\|_{X^{\sigma}}\left(\|\sigma^{\alpha}\overline{\psi}_{tt}\|_{H_{t}^{\rho}(H^{-1})}+\textup{rhs}\right),\end{multlined}\|\sigma^{\alpha}\|_{X^{\sigma}}\left(\|\sigma^{\alpha}\overline{\psi}_{tt}\|_{H_{t}^{\rho}(H^{-1})}+\textup{rhs}\right),$

where

$\displaystyle\textup{rhs}:=$ $\displaystyle
c^{2}\|\nabla\overline{\psi}\|_{L^{2}_{t}(L^{2})}+\tau^{\alpha}c^{2}\|\textup{D}_{t}^{\alpha}\nabla\overline{\psi}\|_{L^{2}_{t}(L^{2})}+\|\tilde{f}\|_{L^{2}(H^{-1})}$
$\displaystyle\qquad+\delta\begin{cases}\mathfrak{g}_{3/2-\alpha}*\|\textup{D}_{t}^{m}\nabla\overline{\psi}\|_{L^{2}_{t}(L^{2})}^{2}&\mbox{
for fMGT I,}\\\
\|\textup{D}_{t}^{m}\nabla\overline{\psi}\|_{L^{2}_{t}(L^{2})}^{2}&\mbox{ for
fMGT.}\end{cases}$

We can therefore tackle all terms by generalized Gronwall in the fMGT–I case,
whereas for fMGT we need to absorb the $\delta$ term by the lhs $\delta$ term
in (5.48) and therefore need to impose smallness of
$\|\sigma^{\alpha}\|_{X^{\sigma}}$ with an $\alpha$ dependent bound.

It remains to estimate the contribution arising from

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

which we do for each of the difference terms separately,

$\displaystyle\|\tau\psi_{ttt}-\tau^{\alpha}\textup{D}_{t}^{2+\alpha}\psi_{tt}\|_{L^{1}(L^{2})}\leq$
$\displaystyle\,|\tau-\tau^{\alpha}|\,\|\psi_{ttt}\|_{L^{1}(L^{2})}+\tau^{\alpha}\|(\textup{D}_{t}-\textup{D}_{t}^{\alpha})\psi_{tt}\|_{L^{1}(L^{2})},$
$\displaystyle\|\Delta(\tau\psi_{t}-\tau^{\alpha}\textup{D}_{t}^{\alpha}\psi)\|_{L^{1}(L^{2})}\leq$
$\displaystyle\,|\tau-\tau^{\alpha}|\,\|\Delta\psi_{t}\|_{L^{1}(L^{2})}+\tau^{\alpha}\|(\textup{D}_{t}-\textup{D}_{t}^{\alpha})\Delta\psi\|_{L^{1}(L^{2})},$

and, in case of fJMGT–W I with $\beta=2-\alpha$,

$\displaystyle\|\Delta(\psi_{t}-\textup{D}_{t}^{2-\alpha}\psi)\|_{L^{1}(L^{2})}=\|(\mbox{id}-\textup{D}_{t}^{1-\alpha})\Delta\psi_{t}\|_{L^{1}(L^{2})}.$

Thus, to be able to apply the limits (3.10) and (3.12), we need

$\psi_{tt},\,\Delta\psi{\in W^{2,1}(0,T;L^{2}(\Omega))\,,\ }\Delta\psi_{t}\in
W^{1,1}(0,T;L^{2}(\Omega)),$

and, in case of fJMGT–W I additionally $\psi_{t}(0)=0$.
Note that the required smoothness of $\psi$ follows, e.g., from Theorem 6.1
below under restrictive regularity conditions on the initial data. We expect
that these assumptions might be relaxed in view of the fact that the
$W^{1,\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\cap
W^{2,\infty}(0,T;H_{0}^{1}(\Omega))\cap W^{3,\infty}(0,T;L^{2}(\Omega))$
regularity that we obtain from Theorem 4.1 is already very close to what is
needed here. Altogether, with

$\displaystyle X^{\textup{low}}=\left\\{\psi\in
H^{1/2}(0,T;H_{0}^{1}(\Omega)):\textup{D}_{t}^{3/2}\psi\in
L^{\infty}(0,T;L^{2}(\Omega))\right\\}$

and the corresponding norm denoted by $\|\cdot\|_{X^{\textup{low}}}$, we have
the following results.

######  Proposition 5.3 (Limit of the fMGT–I equation).

Let $f\in H^{1/2}(0,T;L^{2}(\Omega))$, $\sigma\in X^{\sigma}_{\textup{fMGT
I}}$, and

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

Further, let $\varrho>0$ be as in Proposition 5.1 and
$\|\sigma\|_{X^{\sigma}_{\textup{fMGT I}}}\leq\varrho$; let
$\\{\psi^{\alpha}\\}_{\alpha\in(0,1)}$ be the family of solutions to the
fMGT–I equation, let $\psi$ solve the corresponding equation with $\alpha=1$
and assume that

(5.49) $\displaystyle\nabla\psi_{tt}\in L^{2}(0,T;L^{2}(\Omega)),\
\psi_{tt},\Delta\psi{\in W^{2,1}(0,T;L^{2}(\Omega)),}$
$\displaystyle\Delta\psi_{t}\in W^{1,1}(0,T;L^{2}(\Omega)).$

Then $\psi^{\alpha}$ converges to $\psi$ in the $\|\cdot\|_{X^{\textup{low}}}$
norm as $\alpha\rightarrow 1^{-}$.

######  Proposition 5.4 (Limit of the fJMGT–W I equation).

Let $f\in H^{1/2}(0,T;L^{2}(\Omega))$ and
$(\psi_{0},\psi_{1},\psi_{2})\in({H_{\diamondsuit}^{2}(\Omega)},\\{0\\},H_{0}^{1}(\Omega))$.
Furthermore, let $\varrho>0$ be as in Theorem 5.1 and

$\|f\|^{2}_{H^{1/2}(H^{1})}+\|\psi_{0}\|_{H^{2}}^{2}+\|\psi_{2}\|_{H^{1}}^{2}\leq\varrho^{2};$

let $\\{\psi^{\alpha}\\}_{\alpha\in(0,1)}$ be the family of solutions to the
fJMGT–W I equation, let $\psi$ solve the corresponding equation with
$\alpha=1$, and assume that (5.49) holds. Then $\psi^{\alpha}$ converges to
$\psi$ in the $\|\cdot\|_{X^{\textup{low}}}$ norm as $\alpha\rightarrow
1^{-}$.

######  Proposition 5.5 (Limit of the fMGT equation).

Assume that $f\in H^{1/2}(0,T;L^{2}(\Omega))$, $\sigma=0$, and
$(\psi_{0},\psi_{1},\psi_{2})\in({H_{\diamondsuit}^{2}(\Omega)},{H_{\diamondsuit}^{2}(\Omega)},H_{0}^{1}(\Omega))$.
Let $\\{\psi^{\alpha}\\}_{\alpha\in(0,1)}$ be the family of solutions to the
fMGT equation, let $\psi$ solve the corresponding equation with $\alpha=1$ and
assume that (5.49) holds. Then $\psi^{\alpha}$ converges to $\psi$ in the
$\|\cdot\|_{X^{\textup{low}}}$ norm as $\alpha\rightarrow 1^{-}$.

##  6\. Equations with the quadratic gradient nonlinearity

Unlike its Westervelt version, the Kuznetsov versions of these time-fractional
equations contain a quadratic gradient nonlinearity, and so their analysis
requires the use of higher-order energy estimates. We thus limit our
presentation to the analysis of the fJMGT–K III equation, which has the
integer-order leading term.
The study of well-posedness for this equation follows by combining the ideas
from the previous section concerning the fractional term with the ideas used
in the analysis of its integer-order counterpart considered in [30, 29]. To
formulate the well-posedness result we introduce the solution space

(6.1) $\displaystyle X_{\textup{fMGT--K III}}=$
$\displaystyle\,\begin{multlined}\left\\{\vphantom{H^{-\alpha/2}(0,T;{H_{\diamondsuit}^{3}(\Omega)})}\psi\in
L^{\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)}):\psi_{t}\in
L^{\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)}),\right.\\\ \left.\psi_{tt}\in
L^{\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\cap
H^{-\alpha/2}(0,T;{H_{\diamondsuit}^{3}(\Omega)}),\ \psi_{ttt}\in
L^{2}(0,T;H_{0}^{1}(\Omega))\right\\}\end{multlined}\left\\{\vphantom{H^{-\alpha/2}(0,T;{H_{\diamondsuit}^{3}(\Omega)})}\psi\in
L^{\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)}):\psi_{t}\in
L^{\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)}),\right.\\\ \left.\psi_{tt}\in
L^{\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\cap
H^{-\alpha/2}(0,T;{H_{\diamondsuit}^{3}(\Omega)}),\ \psi_{ttt}\in
L^{2}(0,T;H_{0}^{1}(\Omega))\right\\}$

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

(6.2) $\displaystyle X_{\textup{fMGT--K III}}=H^{3}(0,T;H_{0}^{1}(\Omega))\cap
W^{2,\infty}(0,T;{H_{\diamondsuit}^{2}(\Omega)})\cap
W^{1,\infty}(0,T;{H_{\diamondsuit}^{3}(\Omega)}),$

for $\alpha=1$, equipped with the norm $\|\cdot\|_{X_{\textup{fMGT--K III}}}$.

######  Theorem 6.1 (Local well-posedness of the fJMGT–K III equation).

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

(6.3)
$\|f\|^{2}_{H^{1}(H^{1})}+\|\psi_{0}\|_{H^{3}}^{2}+\|\psi_{1}\|_{H^{3}}^{2}+\|\psi_{2}\|_{H^{2}}^{2}\leq\varrho^{2}.$

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

(6.4) $\left\\{\begin{aligned}
\tau\psi_{ttt}+&(1+2\tilde{k}\psi_{t})\psi_{tt}-c^{2}\Delta\psi-\tau
c^{2}\Delta\psi_{t}&&\\\\[2.84526pt] &\hskip
71.13188pt-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}+\tilde{\ell}\partial_{t}|\nabla\psi|^{2}=f&&\text{
in }\Omega\times(0,T),\\\\[2.84526pt] &\psi=\,0&&\text{ on
}\partial\Omega\times(0,T),\\\\[2.84526pt]
&(\psi,\psi_{t},\psi_{tt})=\,(\psi_{0},\psi_{1},\psi_{2})&&\mbox{ in
}\Omega\times\\{0\\},\end{aligned}\right.$

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

(6.5) $\displaystyle\begin{multlined}\|\psi\|^{2}_{X_{\textup{fMGT--K
III}}}\end{multlined}\|\psi\|^{2}_{X_{\textup{fMGT--K
III}}}\lesssim\,\begin{multlined}\|f\|^{2}_{H^{1}(H^{1})}+\|\psi_{0}\|_{H^{3}}^{2}+\|\psi_{1}\|_{H^{3}}^{2}+\|\psi_{2}\|_{H^{2}}^{2}.\end{multlined}\|f\|^{2}_{H^{1}(H^{1})}+\|\psi_{0}\|_{H^{3}}^{2}+\|\psi_{1}\|_{H^{3}}^{2}+\|\psi_{2}\|_{H^{2}}^{2}.$

###### Proof.

The proof follows by employing the Banach Fixed-point theorem to the mapping
$\mathcal{T}:w\mapsto\psi$, where $\psi$ solves

(6.6)
$\displaystyle\tau\psi_{ttt}+(1+2\tilde{k}w_{t})\psi_{tt}-c^{2}\Delta\psi-\tau
c^{2}\Delta\psi_{t}-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}+2\tilde{\ell}\nabla
w\cdot\nabla\psi_{t}=f,$

and

(6.7) $\displaystyle w\in B_{R}:=\\{w\in X_{\textup{fMGT--K III}}\,:$
$\displaystyle\,\|w\|_{X_{\textup{fMGT--K III}}}\leq R,$ $\displaystyle
w(0)=\psi_{0},\,w_{t}(0)=\psi_{1},\,w_{tt}(0)=\psi_{2}\\}.$

(I) The energy estimates for the linear equation can be rigorously derived by
a Galerkin procedure with a sufficiently smooth basis; here we present only
the derivation of the bound for the semi-discrete solution and omit the
superscript $n$ below. We denote

$p=1+2\tilde{k}w_{t},\quad\phi=2\tilde{k}w,$

then test the semi-discrete version of (6.6) with $\Delta^{2}\psi_{tt}$ and
integrate in space. We can estimate the resulting non-fractional terms and
those not involving $f$ in a similar manner to [30, Theorem 6.1]. We include
the derivation of these bounds below for completeness.
Note that $\psi_{tt}=\Delta\psi=\Delta\psi_{tt}=0$ and
${\textup{D}}_{t}^{2-\alpha}\Delta\psi=0$ on $\partial\Omega$ for smooth
Galerkin approximations based on the eigenfunctions of the Dirichlet-
Laplacian. Therefore, the following identities hold:

(6.8) $\displaystyle(p\psi_{tt},\Delta^{2}\psi_{tt})_{L^{2}}=$
$\displaystyle\,(p\Delta\psi_{tt}+\psi_{tt}\Delta p+2\nabla
p\cdot\nabla\psi_{tt},\Delta\psi_{tt})_{L^{2}},$
$\displaystyle-c^{2}(\Delta\psi,\Delta^{2}\psi_{tt})_{L^{2}}=$
$\displaystyle\,c^{2}\frac{\textup{d}}{\textup{d}t}(\nabla\Delta\psi,\nabla\Delta\psi_{t})_{L^{2}}-c^{2}\|\nabla\Delta\psi_{t}\|_{L^{2}}^{2}.$

We thus have

(6.9)
$\displaystyle\begin{multlined}\frac{1}{2}\tau\frac{\textup{d}}{\textup{d}t}\|\Delta\psi_{tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\frac{\textup{d}}{\textup{d}t}\|\nabla\Delta\psi_{t}\|_{L^{2}}^{2}+\delta(\textup{D}_{t}^{2-\alpha}\nabla\Delta\psi,\nabla\Delta\psi_{tt})_{L^{2}}\end{multlined}\frac{1}{2}\tau\frac{\textup{d}}{\textup{d}t}\|\Delta\psi_{tt}\|_{L^{2}}^{2}+\frac{1}{2}\tau
c^{2}\frac{\textup{d}}{\textup{d}t}\|\nabla\Delta\psi_{t}\|_{L^{2}}^{2}+\delta(\textup{D}_{t}^{2-\alpha}\nabla\Delta\psi,\nabla\Delta\psi_{tt})_{L^{2}}$
$\displaystyle=$
$\displaystyle\,\begin{multlined}-(p\Delta\psi_{tt}+\psi_{tt}\Delta p+2\nabla
p\cdot\nabla\psi_{tt},\Delta\psi_{tt})_{L^{2}}-c^{2}\frac{\textup{d}}{\textup{d}t}(\nabla\Delta\psi,\nabla\Delta\psi_{t})_{L^{2}}\\\\[2.84526pt]
+c^{2}\|\nabla\Delta\psi_{t}\|_{L^{2}}^{2}-(\nabla\phi\cdot\nabla\psi_{t},\Delta^{2}\psi_{tt})_{L^{2}}+(\nabla
f,\nabla\Delta\psi_{tt})_{L^{2}}.\end{multlined}-(p\Delta\psi_{tt}+\psi_{tt}\Delta
p+2\nabla
p\cdot\nabla\psi_{tt},\Delta\psi_{tt})_{L^{2}}-c^{2}\frac{\textup{d}}{\textup{d}t}(\nabla\Delta\psi,\nabla\Delta\psi_{t})_{L^{2}}\\\\[2.84526pt]
+c^{2}\|\nabla\Delta\psi_{t}\|_{L^{2}}^{2}-(\nabla\phi\cdot\nabla\psi_{t},\Delta^{2}\psi_{tt})_{L^{2}}+(\nabla
f,\nabla\Delta\psi_{tt})_{L^{2}}.$

We next integrate in time and estimate the resulting terms. Note first that

$\displaystyle\int_{0}^{t}(p\Delta\psi_{tt}+\psi_{tt}\Delta p+\nabla
p\cdot\nabla\psi_{tt},\Delta\psi_{tt})_{L^{2}}\,\textup{d}s$
$\displaystyle\lesssim$
$\displaystyle\,\begin{multlined}\left\\{(1+R)\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})}+\|\psi_{tt}\|_{L^{2}_{t}(L^{\infty})}\|\Delta
p\|_{L^{\infty}_{t}(L^{2})}\right.\\\ \left.+\|\nabla
p\|_{L^{\infty}(L^{4})}\|\nabla\psi_{tt}\|_{L^{2}_{t}(L^{4})}\right\\}\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})}\end{multlined}\left\\{(1+R)\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})}+\|\psi_{tt}\|_{L^{2}_{t}(L^{\infty})}\|\Delta
p\|_{L^{\infty}_{t}(L^{2})}\right.\\\ \left.+\|\nabla
p\|_{L^{\infty}(L^{4})}\|\nabla\psi_{tt}\|_{L^{2}_{t}(L^{4})}\right\\}\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})}$
$\displaystyle\lesssim$
$\displaystyle\,\left\\{(1+R)\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})}+R\|\psi_{tt}\|_{L^{2}_{t}(L^{\infty})}+R\|\nabla\psi_{tt}\|_{L^{2}_{t}(L^{4})}\right\\}\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})},$

where we have utilized the uniform boundedness of $p=1+2\tilde{k}w_{t}$, which
follows from the fact that $w\in B_{R}$. Furthermore,

$\displaystyle-c^{2}\int_{0}^{t}\frac{\textup{d}}{\textup{d}t}(\nabla\Delta\psi,\nabla\Delta\psi_{t})_{L^{2}}\,\textup{d}s=$
$\displaystyle\,-c^{2}(\nabla\Delta\psi(t),\nabla\Delta\psi_{t}(t))_{L^{2}}+c^{2}(\nabla\Delta\psi_{0},\nabla\Delta\psi_{1})_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\,\begin{multlined}\frac{1}{2\epsilon}Tc^{4}\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}^{2}+\frac{\epsilon}{2}\|\nabla\Delta\psi_{t}(t)\|_{L^{2}}^{2}\\\
+\frac{1}{2}c^{4}\|\nabla\Delta\psi_{0}\|_{L^{2}(L^{2})}^{2}+\frac{1}{2}\|\nabla\Delta\psi_{1}\|_{L^{2}(L^{2})}^{2}.\end{multlined}\frac{1}{2\epsilon}Tc^{4}\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}^{2}+\frac{\epsilon}{2}\|\nabla\Delta\psi_{t}(t)\|_{L^{2}}^{2}\\\
+\frac{1}{2}c^{4}\|\nabla\Delta\psi_{0}\|_{L^{2}(L^{2})}^{2}+\frac{1}{2}\|\nabla\Delta\psi_{1}\|_{L^{2}(L^{2})}^{2}.$

By the semi-discrete PDE, we know that $\nabla\phi\cdot\nabla\psi_{t}=0$ on
$\partial\Omega$. Thus

$\displaystyle(\nabla\phi\cdot\nabla\psi_{t},\Delta^{2}\psi_{tt})_{L^{2}}=$
$\displaystyle\,(\Delta[\nabla\phi\cdot\nabla\psi_{t}],\Delta\psi_{tt})_{L^{2}}$
$\displaystyle=$
$\displaystyle\,(\nabla\Delta\phi\cdot\nabla\psi_{t}+2D^{2}\phi:D^{2}\psi_{t}+\nabla\phi\cdot\nabla\Delta\psi_{t},\Delta\psi_{tt})_{L^{2}}$

where $D^{2}v=(\partial_{x_{i}}\partial_{x_{j}}v)_{i,j}$ is the Hessian, which
satisfies

$\|D^{2}v\|_{L^{4}}\leq
C_{H^{1},L^{4}}(\|D^{3}v\|_{L^{2}}+\|D^{2}v\|_{L^{2}})\leq
C_{H^{1},L^{4}}C_{\textup{H}}(\|\nabla\Delta v\|_{L^{2}}+\|\Delta
v\|_{L^{2}}).$

This further implies that

$\displaystyle\int_{0}^{t}(\nabla\phi\cdot\nabla\psi_{t},\Delta^{2}\psi_{tt})_{L^{2}}\,\textup{d}s$
$\displaystyle\lesssim$
$\displaystyle\,\begin{multlined}\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})}\left\\{\vphantom{C_{H^{1},L^{4}}^{2}}\|\nabla\Delta\phi\|_{L^{\infty}(L^{2})}\|\nabla\psi_{t}\|_{L^{2}_{t}(L^{\infty})}+\|\nabla\phi\|_{L^{\infty}(L^{\infty})}\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}\right.\\\
\left.+(\|\nabla\Delta\phi\|_{L^{\infty}(L^{2})}+\|\Delta\phi\|_{L^{\infty}(L^{2})})(\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}+\|\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})})\right\\}.\end{multlined}\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})}\left\\{\vphantom{C_{H^{1},L^{4}}^{2}}\|\nabla\Delta\phi\|_{L^{\infty}(L^{2})}\|\nabla\psi_{t}\|_{L^{2}_{t}(L^{\infty})}+\|\nabla\phi\|_{L^{\infty}(L^{\infty})}\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}\right.\\\
\left.+(\|\nabla\Delta\phi\|_{L^{\infty}(L^{2})}+\|\Delta\phi\|_{L^{\infty}(L^{2})})(\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}+\|\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})})\right\\}.$

Since $w\in B_{R}$, the function $\phi$ is uniformly bounded, and so

$\displaystyle\int_{0}^{t}(\nabla\phi\cdot\nabla\psi_{t},\Delta^{2}\psi_{tt})_{L^{2}}\,\textup{d}s\lesssim$
$\displaystyle\,\begin{multlined}R\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})}\left\\{\vphantom{C_{H^{1},L^{4}}^{2}}\|\nabla\psi_{t}\|_{L^{2}_{t}(L^{\infty})}+\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}\right.\\\
\left.+\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}+\|\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}\right\\}.\end{multlined}R\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})}\left\\{\vphantom{C_{H^{1},L^{4}}^{2}}\|\nabla\psi_{t}\|_{L^{2}_{t}(L^{\infty})}+\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}\right.\\\
\left.+\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}+\|\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}\right\\}.$

Integration by parts with respect to time yields

(6.10) $\displaystyle\int_{0}^{t}(\nabla
f,\nabla\Delta\psi_{tt})_{L^{2}}\,\textup{d}s=$ $\displaystyle\,(\nabla
f,\nabla\Delta\psi_{t})_{L^{2}}\big{|}_{0}^{t}-\int_{0}^{t}(\nabla
f_{t},\nabla\Delta\psi_{t})_{L^{2}}\,\textup{d}s$ $\displaystyle\leq$
$\displaystyle\,\begin{multlined}\frac{1}{4\epsilon}\|\nabla
f(t)\|_{L^{2}}^{2}+\epsilon\|\nabla\Delta\psi_{t}(t)\|_{L^{2}}^{2}-(\nabla
f(0),\nabla\Delta\psi_{1})_{L^{2}}\\\ +\frac{1}{2}\|\nabla
f_{t}\|_{L^{2}(L^{2})}^{2}+\frac{1}{2}\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}^{2}.\end{multlined}\frac{1}{4\epsilon}\|\nabla
f(t)\|_{L^{2}}^{2}+\epsilon\|\nabla\Delta\psi_{t}(t)\|_{L^{2}}^{2}-(\nabla
f(0),\nabla\Delta\psi_{1})_{L^{2}}\\\ +\frac{1}{2}\|\nabla
f_{t}\|_{L^{2}(L^{2})}^{2}+\frac{1}{2}\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}^{2}.$

The $\delta$ fractional term can be handled by relying on estimate (4.4)
similarly to before. Fixing $\epsilon>0$ small enough and combining the
derived bounds leads to

(6.11)
$\displaystyle\begin{multlined}\frac{1}{2}\tau\|\Delta\psi_{tt}(t)\|_{L^{2}}^{2}+(\frac{1}{2}\tau
c^{2}-2\epsilon)\|\nabla\Delta\psi_{t}(t)\|_{L^{2}}^{2}+\int_{0}^{t}\|{\textup{D}}_{t}^{2-\alpha}\nabla\Delta\psi\|_{L^{2}}^{2}\,\textup{d}s\end{multlined}\frac{1}{2}\tau\|\Delta\psi_{tt}(t)\|_{L^{2}}^{2}+(\frac{1}{2}\tau
c^{2}-2\epsilon)\|\nabla\Delta\psi_{t}(t)\|_{L^{2}}^{2}+\int_{0}^{t}\|{\textup{D}}_{t}^{2-\alpha}\nabla\Delta\psi\|_{L^{2}}^{2}\,\textup{d}s$
$\displaystyle\lesssim$
$\displaystyle\,\begin{multlined}\|\Delta\psi_{2}\|_{L^{2}}^{2}+\|\nabla\Delta\psi_{1}\|_{L^{2}}^{2}+\|\nabla\Delta\psi_{0}\|_{L^{2}}^{2}\\\
+R^{2}(\|\psi_{tt}\|_{L^{2}_{t}(H^{2})}^{2}+\|\nabla\psi_{tt}\|_{L^{2}_{t}(L^{4})}^{2})+\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})}^{2}\\\
+\left.R^{2}\|\nabla\psi_{t}\|_{L^{2}_{t}(L^{\infty})}^{2}+(1+R^{2}+T)\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}^{2}\right.+R^{2}\|\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}^{2}+\|\nabla
f\|^{2}_{H^{1}(L^{2})}.\end{multlined}\|\Delta\psi_{2}\|_{L^{2}}^{2}+\|\nabla\Delta\psi_{1}\|_{L^{2}}^{2}+\|\nabla\Delta\psi_{0}\|_{L^{2}}^{2}\\\
+R^{2}(\|\psi_{tt}\|_{L^{2}_{t}(H^{2})}^{2}+\|\nabla\psi_{tt}\|_{L^{2}_{t}(L^{4})}^{2})+\|\Delta\psi_{tt}\|_{L_{t}^{2}(L^{2})}^{2}\\\
+\left.R^{2}\|\nabla\psi_{t}\|_{L^{2}_{t}(L^{\infty})}^{2}+(1+R^{2}+T)\|\nabla\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}^{2}\right.+R^{2}\|\Delta\psi_{t}\|_{L_{t}^{2}(L^{2})}^{2}+\|\nabla
f\|^{2}_{H^{1}(L^{2})}.$

Note that by elliptic regularity, we have

$\|\psi_{tt}(t)\|_{L^{2}}\leq
C_{\textup{PF}}\|\nabla\psi_{tt}(t)\|_{L^{2}}\leq\|\psi_{tt}(t)\|_{H^{2}}\lesssim\|\Delta\psi_{tt}(t)\|_{L^{2}}.$

An application of Gronwall’s inequality yields

(6.12)
$\displaystyle\sup_{t\in(0,\tilde{T})}\|\Delta\psi_{tt}(t)\|_{L^{2}}^{2}+\sup_{t\in(0,\tilde{T})}\|\nabla\Delta\psi_{t}(t)\|_{L^{2}}^{2}+\int_{0}^{t}\|{\textup{D}}_{t}^{2-\alpha}\nabla\Delta\psi\|_{L^{2}}^{2}\,\textup{d}s$
$\displaystyle\leq$ $\displaystyle\,C(T,R)(\|\nabla
f\|^{2}_{H^{1}(H^{1})}+\|\Delta\psi_{2}\|_{L^{2}}^{2}+\|\nabla\Delta\psi_{1}\|_{L^{2}}^{2}+\|\nabla\Delta\psi_{0}\|_{L^{2}}^{2}).$

The uniqueness follows by using $\psi_{tt}$ as the test function in the
homogeneous problem.

(II) It is straightforward to check now that $\mathcal{T}$ is a well-defined
self-mapping. We thus focus on proving strict contractivity. Take
$w^{(1)},w^{(2)}\in B_{R}$ and set $\psi^{(1)}=\mathcal{T}\phi^{(1)}$ and
$\psi^{(2)}=\mathcal{T}\phi^{(2)}$. Then the difference
$\overline{\psi}=\psi^{(1)}-\psi^{(2)}$ solves

(6.13)
$\displaystyle\begin{multlined}\tau\overline{\psi}_{ttt}+(1+2\tilde{k}w_{t}^{(1)})\overline{\psi}_{tt}-c^{2}\Delta\overline{\psi}-\tau
c^{2}\Delta\overline{\psi}_{t}-\delta{\textup{D}}_{t}^{2-\alpha}\Delta\overline{\psi}+2\tilde{\ell}\nabla
w^{(2)}\cdot\nabla\overline{\psi}_{t}=\tilde{f}.\end{multlined}\tau\overline{\psi}_{ttt}+(1+2\tilde{k}w_{t}^{(1)})\overline{\psi}_{tt}-c^{2}\Delta\overline{\psi}-\tau
c^{2}\Delta\overline{\psi}_{t}-\delta{\textup{D}}_{t}^{2-\alpha}\Delta\overline{\psi}+2\tilde{\ell}\nabla
w^{(2)}\cdot\nabla\overline{\psi}_{t}=\tilde{f}.$

with the right-hand side

$\tilde{f}=-2\tilde{k}\overline{w}_{t}\psi^{(2)}_{tt}-2\tilde{\ell}\nabla\overline{w}\cdot\nabla\psi^{(1)}_{t}$

and satisfies zero initial conditions. Testing with $\overline{\psi}_{tt}$
yields, after standard manipulations,

(6.14)
$\displaystyle\|\overline{\psi}_{tt}(t)\|_{L^{2}}^{2}+\|\nabla\overline{\psi}_{t}(t)\|_{L^{2}}^{2}+\|\nabla\overline{\psi}(t)\|_{L^{2}}^{2}\leq
C(\tilde{T},R)\|\tilde{f}\|_{L^{2}(L^{2})}^{2};$

see also estimate (5.5) in [30]. It remains to bound the source term. By
Hölder’s inequality, we have

(6.15) $\displaystyle\|\tilde{f}\|_{L^{2}(L^{2})}\lesssim$
$\displaystyle\,\begin{multlined}\|\psi_{tt}^{(2)}\|_{L^{\infty}(L^{4})}\|\overline{w}_{t}\|_{L^{2}(L^{4})}+\|\nabla\psi_{t}^{(1)}\|_{L^{\infty}(L^{\infty})}\|\nabla\overline{w}\|_{L^{2}(L^{2})}.\end{multlined}\|\psi_{tt}^{(2)}\|_{L^{\infty}(L^{4})}\|\overline{w}_{t}\|_{L^{2}(L^{4})}+\|\nabla\psi_{t}^{(1)}\|_{L^{\infty}(L^{\infty})}\|\nabla\overline{w}\|_{L^{2}(L^{2})}.$

The first term on the right can be further bounded as follows:

(6.16)
$\displaystyle\|\psi_{tt}^{(2)}\|_{L^{\infty}(L^{4})}\|\overline{w}_{t}\|_{L^{2}(L^{4})}\leq\,C_{H^{1},L^{4}}^{2}\|\nabla\psi_{tt}^{(2)}\|_{L^{\infty}(L^{2})}T\|\nabla\overline{w}_{t}\|_{L^{\infty}(L^{2})}.$

By noting that
$\|\nabla\overline{w}\|_{L^{2}(L^{2})}\leq\tilde{T}\|\nabla\overline{w}_{t}\|_{L^{2}(L^{2})}$,
it further follows that

(6.17) $\displaystyle\|\tilde{f}\|_{L^{2}(L^{2})}^{2}\lesssim$
$\displaystyle\,\begin{multlined}\|\nabla\psi_{tt}^{(2)}\|_{L^{\infty}(L^{2})}^{2}\tilde{T}^{2}\|\nabla\overline{w}_{t}\|_{L^{\infty}(L^{2})}^{2}+\|\nabla\psi_{t}^{(1)}\|_{L^{\infty}(L^{\infty})}^{2}\tilde{T}^{2}\|\nabla\overline{w_{t}}\|_{L^{2}(L^{2})}^{2}.\end{multlined}\|\nabla\psi_{tt}^{(2)}\|_{L^{\infty}(L^{2})}^{2}\tilde{T}^{2}\|\nabla\overline{w}_{t}\|_{L^{\infty}(L^{2})}^{2}+\|\nabla\psi_{t}^{(1)}\|_{L^{\infty}(L^{\infty})}^{2}\tilde{T}^{2}\|\nabla\overline{w_{t}}\|_{L^{2}(L^{2})}^{2}.$

Employing this bound in (6.14) and relying on Gronwall’s inequality leads to

(6.18)
$\displaystyle\sup_{t\in(0,\tilde{T})}\|\overline{\psi}_{tt}(t)\|_{L^{2}}+\sup_{t\in(0,\tilde{T})}\|\nabla\overline{\psi}_{t}(t)\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\,\begin{multlined}C(T,R)\tilde{T}(\|\nabla\psi_{tt}^{(2)}\|_{L^{\infty}(L^{2})}+\|\nabla\psi_{t}^{(1)}\|_{L^{\infty}(L^{\infty})})\sup_{t\in(0,\tilde{T})}\|\nabla\overline{w}_{t}(t)\|_{L^{2}}.\end{multlined}C(T,R)\tilde{T}(\|\nabla\psi_{tt}^{(2)}\|_{L^{\infty}(L^{2})}+\|\nabla\psi_{t}^{(1)}\|_{L^{\infty}(L^{\infty})})\sup_{t\in(0,\tilde{T})}\|\nabla\overline{w}_{t}(t)\|_{L^{2}}.$

By the energy estimate for the linear problem, we know that

$\|\nabla\psi_{tt}^{(2)}\|_{L^{\infty}(L^{2})}+\|\nabla\psi_{t}^{(1)}\|_{L^{\infty}(L^{\infty})}\leq\sqrt{\tilde{C}(\tilde{T},R)}\,r$

for some $\tilde{C}(\tilde{T},R)>0$, independent of $\alpha$. Thus we can
achieve strict contractivity of $\mathcal{T}$ with respect to the
$\|\cdot\|_{X^{\textup{low}}_{\textup{fMGT III}}}$ norm by reducing the final
time $\tilde{T}$. The claim then follows by the Banach Fixed-point theorem. ∎

We next discuss the limit of this equation 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–K III equation and let $\psi$
solve the corresponding JMGT–Kuznetsov equation obtained by setting $\alpha=1$
in (6.4). Then the difference $\overline{\psi}=\psi^{\alpha}-\psi$ solves

(6.19)
$\displaystyle\begin{multlined}\tau\overline{\psi}_{ttt}+(1+2\tilde{k}\psi_{t}^{\alpha})\overline{\psi}_{tt}-c^{2}\Delta\overline{\psi}-\tau
c^{2}\Delta\overline{\psi}_{t}-\delta\textup{D}_{t}^{1-\alpha}\Delta\overline{\psi}_{t}\\\
\hskip
56.9055pt+2\tilde{k}\psi_{tt}\overline{\psi}_{t}+2\tilde{\ell}\nabla\overline{\psi}\cdot\nabla\psi_{t}^{\alpha}+2\tilde{\ell}\nabla\psi\cdot\nabla\overline{\psi}_{t}=\,\delta(\textup{D}_{t}^{1-\alpha}\Delta\psi_{t}-\Delta\psi_{t}).\end{multlined}\tau\overline{\psi}_{ttt}+(1+2\tilde{k}\psi_{t}^{\alpha})\overline{\psi}_{tt}-c^{2}\Delta\overline{\psi}-\tau
c^{2}\Delta\overline{\psi}_{t}-\delta\textup{D}_{t}^{1-\alpha}\Delta\overline{\psi}_{t}\\\
\hskip
56.9055pt+2\tilde{k}\psi_{tt}\overline{\psi}_{t}+2\tilde{\ell}\nabla\overline{\psi}\cdot\nabla\psi_{t}^{\alpha}+2\tilde{\ell}\nabla\psi\cdot\nabla\overline{\psi}_{t}=\,\delta(\textup{D}_{t}^{1-\alpha}\Delta\psi_{t}-\Delta\psi_{t}).$

Similarly to the proof of the previous theorem, testing with
$\overline{\psi}_{tt}$ and using the uniform boundedness of
$\|\psi^{\alpha}\|_{X_{\textup{fMGT--K III}}}$ for $\alpha\in(0,1]$ leads to
the following bound:

(6.20)
$\displaystyle\|\overline{\psi}_{tt}(t)\|^{2}_{L^{2}}+\|\nabla\overline{\psi}_{t}(t)\|^{2}_{L^{2}}\lesssim\,\|\textup{D}_{t}^{1-\alpha}\nabla\psi_{t}-\nabla\psi_{t}\|_{L^{2}(L^{2})}^{2}.$

On account of Lemma 3.1, we then have the following result.

######  Proposition 6.1 (Limit of the fJMGT–K III equation).

Let the assumptions of Theorem 6.1 hold with $\psi_{1}=0$. Let
$\\{\psi^{\alpha}\\}_{\alpha\in(0,1)}$ be the family of solutions to the
fJMGT–K III equation and let $\psi$ solve the corresponding JMGT–Kuznetsov
equation. 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
$\alpha\rightarrow 1^{-}$.

##  7\. Reformulation of linear models as wave equations with memory

By neglecting the nonlinear terms in the equations given in Table 1, we arrive
at their linear counterparts, which are listed separately in Table 2. A
possible idea to facilitate the linear analysis, which we wish to explore
here, is to re-formulate these equations in terms of

$z=\tau^{\gamma}\textup{D}_{t}^{\gamma}\psi+\psi,$

where $\gamma=1$ in case of fMGT III and $\gamma=\alpha$ otherwise. These
linear models can be rewritten as second-order wave equations with memory

$z_{tt}-c^{2}\Delta z-\delta\textup{D}_{t}^{\beta}\Delta\psi=f$

where $\beta=2-\alpha$ for the fMGT III equation; otherwise it is given by
(5.2). By using the Mittag-Leffler functions $E_{\gamma,\gamma}$ and
$E_{\gamma,1}$ we can express $\psi$ as

(7.1) $\displaystyle\psi(t)=$
$\displaystyle\,{E_{\gamma,1}(-(\tfrac{t}{\tau})^{\gamma})\psi_{0}+}\tau^{-\gamma}\int_{0}^{t}(t-s)^{\gamma-1}E_{\gamma,\gamma}(-(\tfrac{t-s}{\tau})^{\gamma})z(s)\,\textup{d}s$
$\displaystyle=$
$\displaystyle\,{E_{\gamma,1}(-(\tfrac{t}{\tau})^{\gamma})\psi_{0}+}\mathfrak{k}_{\gamma}*z.$

with $\mathfrak{k}_{\gamma}$ as in (7.3) below. That is,

$\textup{D}_{t}^{\beta}\psi=\textup{D}_{t}^{\beta-\gamma}\textup{D}_{t}^{\gamma}\psi=\tau^{-\gamma}\textup{D}_{t}^{\beta-\gamma}(z-\psi)=\tau^{-\gamma}\textup{D}_{t}^{\beta-\gamma}\left(z-\mathfrak{k}_{\gamma}*z{-E_{\gamma,1}(-(\tfrac{t}{\tau})^{\gamma})\psi_{0}}\right).$

The $z$ forms of each of the linear models are also listed in Table 2, where

$\tilde{f}=f-\delta\tau^{-\gamma}\textup{D}_{t}^{\beta-\gamma}E_{\gamma,1}(-(\tfrac{t}{\tau})^{\gamma})\Delta\psi_{0}$

for the fMGT III equation. Note that for $\delta=0$ they are all the same and
their analysis can be performed as in Section 7.1. We thus focus here on the
more challenging case of $\delta>0$.

###  7.1. Analysis of the fMGT II equation with $\boldsymbol{\sigma=0}$

We carry out an analysis of the fMGT II equation

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

It can be rewritten in terms of

$z=\tau^{\alpha}\textup{D}_{t}^{\alpha}\psi+\psi$

as

(7.2) $z_{tt}-(c^{2}+\tfrac{\delta}{\tau^{\alpha}})\Delta
z+\tfrac{\delta}{\tau^{\alpha}}\,\mathfrak{k}_{\alpha}*\Delta
z=f{-\delta\tau^{-\alpha}E_{\alpha,1}(-(\tfrac{t}{\tau})^{\alpha})\Delta\psi_{0}}$

with the kernel function

(7.3)
$\mathfrak{k}_{\alpha}(t)=\tau^{-\alpha}t^{\alpha-1}E_{\alpha,\alpha}(-(\tfrac{t}{\tau})^{\alpha})=-\frac{\textup{d}}{\textup{d}t}E_{\alpha,1}(-(\tfrac{t}{\tau}^{\alpha})).$

We observe that this kernel has the following properties:

(7.4) $\mathfrak{k}_{\alpha}(t)\geq 0\,,\quad\lim_{t\to
0+}\mathfrak{k}_{\alpha}(t)=+\infty\,,\quad\int_{0}^{\infty}\mathfrak{k}_{\alpha}(t)\,\textup{d}t=1\,,\quad\mathfrak{k}_{\alpha}^{\prime}(t)\leq
0.$

fMGT |  Linear time-fractional acoustic equations
---|---
I |  $(1+\tau^{\alpha}{\textup{D}}_{t}^{\alpha})(\psi_{tt}-c^{2}\Delta\psi)-\delta\Delta\psi_{t}=f$
$z$ form |  $z_{tt}-c^{2}\Delta z-\dfrac{\delta}{\tau^{\alpha}}\textup{D}_{t}^{1-\alpha}\Delta z+\dfrac{\delta}{\tau^{\alpha}}\displaystyle\int_{0}^{t}\mathfrak{k}(t-s)\Delta z\,\textup{d}s=\tilde{f}$, $\mathfrak{k}=\frac{\textup{d}}{\textup{d}t}(\mathfrak{g}_{1-\alpha}*\mathfrak{k}_{\alpha})$
I |  $(1+\tau^{\alpha}{\textup{D}}_{t}^{\alpha})(\psi_{tt}-c^{2}\Delta\psi)-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}=f$
$z$ form |  $z_{tt}-c^{2}\Delta z-\dfrac{\delta}{\tau^{\alpha}}\textup{D}_{t}^{2-2\alpha}\Delta z+\dfrac{\delta}{\tau^{\alpha}}\displaystyle\int_{0}^{t}\mathfrak{k}(t-s)\Delta z\,\textup{d}s=\tilde{f}$, $\mathfrak{k}=\frac{\textup{d}}{\textup{d}t}(\mathfrak{g}_{2(1-\alpha)}*\mathfrak{k}_{\alpha})$
II |  $(1+\tau^{\alpha}{\textup{D}}_{t}^{\alpha})(\psi_{tt}-c^{2}\Delta\psi)-\delta{\textup{D}}_{t}^{\alpha}\Delta\psi=f$
$z$ form |  $z_{tt}-\left(c^{2}+\dfrac{\delta}{\tau^{\alpha}}\right)\Delta z+\dfrac{\delta}{\tau^{\alpha}}\displaystyle\int_{0}^{t}\mathfrak{k}(t-s)\Delta z\,\textup{d}s=f$, $\mathfrak{k}=\mathfrak{k}_{\alpha}$
III |  $(1+\tau\partial_{t})(\psi_{tt}-c^{2}\Delta\psi)-\delta{\textup{D}}_{t}^{1-\alpha}\Delta\psi_{t}=f$
$z$ form |  $z_{tt}-c^{2}\Delta z-\dfrac{\delta}{\tau}\textup{D}_{t}^{1-\alpha}\Delta z+\dfrac{\delta}{\tau}\displaystyle\int_{0}^{t}\mathfrak{k}(t-s)\Delta z\,\textup{d}s=\tilde{f}$, $\mathfrak{k}=\frac{\textup{d}}{\textup{d}t}(\mathfrak{g}_{1-\alpha}*\mathfrak{k}_{1})$
Table 2. Linear time-fractional models with
$\mathfrak{k}_{\alpha}(t)=\tau^{-\alpha}t^{\alpha-1}E_{\alpha,\alpha}(-(\tfrac{t}{\tau})^{\alpha})$,
$\mathfrak{k}_{1}(t)=\tau^{-1}\exp(-(\tfrac{t}{\tau}))$, and
$\mathfrak{g}_{\gamma}(t)=t^{-\gamma}$.

Moreover, since the function $t\mapsto
E_{\alpha,1}(-(\tfrac{t}{\tau}^{\alpha}))$ is completely monotone, by
Schoenberg’s theorem [44, Theorem 7.13] we conclude that the kernel
$\mathfrak{k}_{\alpha}$ itself and also the kernel
$t\mapsto\int_{t}^{\infty}\mathfrak{k}_{\alpha}(s)\,\textup{d}s$ is positive
definite. Therefore, the next result is a straightforward consequence of [10,
Theorem 4.5], where the regularity of $\psi$ as the unique solution of

(7.5) $\tau^{\alpha}\textup{D}_{t}^{\alpha}\psi+\psi=z$

a.e. in $(0,T)$ with $\psi(0)=\psi_{0}$ follows from the fact that
$\textup{I}^{\alpha}$ maps $L^{\infty}(0,T)$ to $C^{0,\alpha}(0,T)$; see [40,
Corollary 2, p. 56].

######  Proposition 7.1 (Well-posedness of the fMGT II equation).

Let $\alpha\in(0,1]$. Given $f\in L^{1}(0,T;L^{2}(\Omega))$, $(z_{0},z_{1})\in
H_{0}^{1}(\Omega)\times L^{2}(\Omega)$, and
$\psi_{0}\in{H_{\diamondsuit}^{2}(\Omega)}$, there exists a unique mild
solution

$z\in W^{2,\infty}(0,T;H^{-1}(\Omega))\cap W^{1,\infty}(0,T;L^{2}(\Omega))\cap
L^{\infty}(0,T;H_{0}^{1}(\Omega))$

of (7.2) with $(z,z_{t})|_{t=0}=(z_{0},z_{1})$. Correspondingly, for inital
data

$(\psi_{0},\psi_{1},\psi_{2})\in\begin{cases}{H_{\diamondsuit}^{2}(\Omega)}\times\\{0\\}\times
H^{-1}(\Omega)\hphantom{~{}}\ \mbox{ if }\alpha<1,\\\
{H_{\diamondsuit}^{2}(\Omega)}\times H_{0}^{1}(\Omega)\times L^{2}(\Omega)\
\mbox{ if }\alpha=1,\end{cases}$

there exists a unique solution

$\psi\in C^{2,\alpha}(0,T;H^{-1}(\Omega))\cap
C^{1,\alpha}(0,T;L^{2}(\Omega))\cap C^{0,\alpha}(0,T;H_{0}^{1}(\Omega))$

according to (7.1); that is, the unique solution of (7.5) with
$\psi(0)=\psi_{0}$ and thus of

(7.6)
$\displaystyle\begin{multlined}\langle\tau^{\alpha}\textup{D}_{t}^{2+\alpha}\psi+\psi_{tt},v\rangle_{H^{-1},H_{0}^{1}}\\\
+(c^{2}\nabla\psi+\tau^{\alpha}c^{2}\nabla\textup{D}_{t}^{\alpha}\psi+\delta\textup{D}_{t}^{\alpha}\nabla\psi,\nabla
v)_{L^{2}}=(f,v)_{L^{2}},\end{multlined}\langle\tau^{\alpha}\textup{D}_{t}^{2+\alpha}\psi+\psi_{tt},v\rangle_{H^{-1},H_{0}^{1}}\\\
+(c^{2}\nabla\psi+\tau^{\alpha}c^{2}\nabla\textup{D}_{t}^{\alpha}\psi+\delta\textup{D}_{t}^{\alpha}\nabla\psi,\nabla
v)_{L^{2}}=(f,v)_{L^{2}},$

a.e. in $(0,T)$, with
$(\psi,\psi_{t},\psi_{tt})|_{t=0}=(\psi_{0},\psi_{1},\psi_{2})$. Furthermore,
the solution satisfies the estimate

(7.7)
$\displaystyle\begin{multlined}\|\tau^{\alpha}D_{tt}\textup{D}_{t}^{\alpha}\psi+\psi_{tt}\|^{2}_{L^{\infty}(H^{-1})}+\|\tau^{\alpha}D_{t}\textup{D}_{t}^{\alpha}\psi+\psi_{t}\|_{L^{\infty}(L^{2})}^{2}\\\
\hskip
170.71652pt+\|\tau^{\alpha}\textup{D}_{t}^{\alpha}\nabla\psi+\nabla\psi\|_{L^{\infty}(L^{2})}^{2}\end{multlined}\|\tau^{\alpha}D_{tt}\textup{D}_{t}^{\alpha}\psi+\psi_{tt}\|^{2}_{L^{\infty}(H^{-1})}+\|\tau^{\alpha}D_{t}\textup{D}_{t}^{\alpha}\psi+\psi_{t}\|_{L^{\infty}(L^{2})}^{2}\\\
\hskip
170.71652pt+\|\tau^{\alpha}\textup{D}_{t}^{\alpha}\nabla\psi+\nabla\psi\|_{L^{\infty}(L^{2})}^{2}$
$\displaystyle\lesssim$
$\displaystyle\,\begin{cases}\|\psi_{2}\|_{H^{-1}(\Omega)}^{2}+\|\nabla\psi_{0}\|_{L^{2}(\Omega)}^{2}+\|f\|_{L^{1}(L^{2})}^{2}\hphantom{\hskip
79.6678pt}&\mbox{ for }\alpha<1,\\\
\|\tau\psi_{2}+\psi_{1}\|_{L^{2}(\Omega)}^{2}+\|\tau\nabla\psi_{1}+\nabla\psi_{0}\|_{L^{2}(\Omega)}^{2}+\|f\|_{L^{1}(L^{2})}^{2}\quad&\mbox{
for }\alpha=1.\end{cases}$

Note that the condition $\psi_{1}=0$ in case $\alpha<1$ is enforced by the
singularity at zero in the identity

$z_{t}=\textup{D}_{t}\left(\tau^{\alpha}\textup{D}_{t}^{\alpha}\psi+\psi\right)=\tau^{\alpha}\left(\textup{D}_{t}^{1+\alpha}\psi+\psi_{t}(0)\frac{t^{-\alpha}}{\Gamma(1-\alpha)}\right)+\psi_{t}\,.$

Spatially higher-order regularity can be obtained with more regular initial
data and the source term by using the multiplier $(-\Delta)^{m}z_{t}$ in place
of $z_{t}$ (which led to the energy estimate (7.7)). To study the limiting
behavior, we will make use of the following resulting estimate in case
$\alpha=1$, $m=1$:

(7.8)
$\displaystyle\|\tau\psi_{ttt}+\psi_{tt}\|^{2}_{L^{\infty}(L^{2})}+\|\tau\nabla\psi_{tt}+\nabla\psi_{t}\|^{2}_{L^{\infty}(L^{2})}+\|\tau\Delta\psi_{t}+\Delta\psi\|^{2}_{L^{\infty}(L^{2})}$
$\displaystyle\lesssim$
$\displaystyle\,\|\tau\nabla\psi_{2}+\nabla\psi_{1}\|_{L^{2}(\Omega)}^{2}+\|\tau\Delta\psi_{1}+\Delta\psi_{0}\|_{L^{2}(\Omega)}^{2}+\|\nabla
f\|_{L^{1}(L^{2})}^{2}.$

######  Remark 2 (On the $z$-form of the other linear models).

As already mentioned, a reformulation of the type (7.2) is available for the
other linear models fMGT, fMGT I, and fMGT III as well; see Table 2. However,
it is not clear whether properties (7.4) still hold for the corresponding
kernels.
Due to the term $\textup{D}_{t}^{\beta-\gamma}(z-\mathfrak{k}_{\gamma}*z)$
present in these models with

$\epsilon=\beta-\gamma>0\in\\{1-\alpha,2-2\alpha\\},$

one might consider using the adjoint of $(\textup{D}_{t}^{\epsilon})^{-1}$ in
the multiplier; that is, test with
$((\textup{D}_{t}^{\epsilon})^{-1})^{*}z_{t}$ in place of $z_{t}$. Indeed,
this leads to tractable terms

(7.9)
$\displaystyle-\tfrac{\delta}{\tau^{\alpha}}\int_{0}^{t}(\textup{D}_{t}^{\epsilon}\Delta(z-\mathfrak{k}_{\gamma}*z),((\textup{D}_{t}^{\epsilon})^{-1})^{*}z_{t})_{L^{2}}\,\textup{d}t=\tfrac{\delta}{\tau^{\alpha}}\int_{0}^{t}(\nabla(z-\mathfrak{k}_{\gamma}*z),z_{t})_{L^{2}}\,\textup{d}t,$
$\displaystyle\int_{0}^{t}(z_{tt},((\textup{D}_{t}^{\epsilon})^{-1})^{*}z_{t})_{L^{2}}\,\textup{d}t=\int_{0}^{t}((\textup{D}_{t}^{\epsilon})^{-1}\textup{D}_{t}^{\epsilon}\textup{I}^{\epsilon}z_{tt},z_{t})_{L^{2}}\,\textup{d}t=\int_{0}^{t}(\textup{D}_{t}^{2-\epsilon}z,z_{t})_{L^{2}}\,\textup{d}t\,.$

However, the $c^{2}$ term does not appear to be amenable to useful estimates
since in

$-c^{2}\int_{0}^{t}(\Delta
z,((\textup{D}_{t}^{\epsilon})^{-1})^{*}z_{t})_{L^{2}}\,\textup{d}t=c^{2}\int_{0}^{t}(\nabla\textup{I}^{\epsilon},\nabla
z_{t})_{L^{2}}\,\textup{d}t$

the difference between the time differentiation orders of the two factors is
$1-(-\epsilon)=1+\epsilon>1$, which leads to an adverse sign, while the norm
of $\nabla z_{t}$ is not controllable by any of the other left-hand side terms
resulting from (7.9).

###  7.2. Limiting behavior of the fMGT II equation

For $\alpha\in(0,1]$, we denote by $\psi^{\alpha}$ the solution according to
Proposition 7.1 under the assumptions

$(\psi_{0},\psi_{1},\psi_{2})\in{H_{\diamondsuit}^{2}(\Omega)}\times\\{0\\}\times
H_{0}^{1}(\Omega),\quad f,\nabla f\in L^{1}(0,T;L^{2}(\Omega)).$

Let $\psi$ be the solution of the corresponding MGT equation. Note that then
the corresponding functions $z^{\alpha}$ and $z$ satisfy the initial
conditions

$z^{\alpha}(0)=z(0)=\psi_{0},\quad z^{\alpha}_{t}(0)=0,\quad
z_{t}(0)=\tau\psi_{2}.$

Hence to achieve compatibility, besides $\psi_{1}=0$ (see the proof of
Proposition 7.1), we also have to assume $\psi_{2}=0$. Then the difference
$\overline{z}=z^{\alpha}-z$ solves

(7.10)
$\displaystyle\overline{z}_{tt}-(c^{2}+\tfrac{\delta}{\tau^{\alpha}})\Delta\overline{z}+\delta\,\mathfrak{k}_{\alpha}*\Delta\overline{z}$
$\displaystyle=\delta\left((\tau^{-\alpha}-\tau^{-1})\Delta
z-(\mathfrak{k}_{\alpha}-\mathfrak{k}_{1})*\Delta z\right)$

with homogeneous initial data $\overline{z}(0)=\overline{z}_{t}(0)=0$. Testing
with $\overline{z}_{t}$ leads to

(7.11)
$\|\overline{z}_{t}\|_{L^{\infty}(L^{2})}^{2}+2c^{2}\|\nabla\overline{z}\|_{L^{\infty}(L^{2})}^{2}\leq
4\delta\Bigl{(}|\tau^{-\alpha}-\tau^{-1}|+\|\mathfrak{k}_{\alpha}-\mathfrak{k}_{1}\|_{L^{1}(0,T)}\Bigr{)}\|\Delta
z\|_{L^{1}(L^{2})}^{2},$

where we can estimate $\|\Delta z\|_{L^{1}(L^{2})}^{2}$ according to (7.8) and

$\|\mathfrak{k}_{\alpha}-\mathfrak{k}_{1}\|_{L^{1}(0,T)}=\int_{0}^{T}\left|\tau^{-\alpha}t^{\alpha-1}E_{\alpha,\alpha}(-(\tfrac{t}{\tau})^{\alpha})-\tau^{-1}\exp(-(\tfrac{t}{\tau}))\right|\,\textup{d}t\to
0\ \mbox{ as }\alpha\to 1^{-},$

by Lebesgue’s Dominated Convergence theorem. Thus, with

$\displaystyle\psi^{\alpha}-\psi$
$\displaystyle=\mathfrak{k}_{\alpha}*z^{\alpha}-\mathfrak{k}_{1}*z=(\mathfrak{k}_{\alpha}-\mathfrak{k}_{1})*z+\mathfrak{k}_{\alpha}*\overline{z}$
$\displaystyle(\psi^{\alpha}-\psi)_{t}(t)$
$\displaystyle=(\mathfrak{k}_{\alpha}-\mathfrak{k}_{1})(t)\,z^{\alpha}(0)+((\mathfrak{k}_{\alpha}-\mathfrak{k}_{1})*z_{t})(t)+\mathfrak{k}_{\alpha}(t)\overline{z}(0)+(\mathfrak{k}_{\alpha}*\overline{z}_{t})(t)$

we obtain, using $\overline{z}(0)=0$, $z^{\alpha}(0)=\psi_{0}$, and Young’s
Convolution inequality,

(7.12)
$\displaystyle\|(\psi^{\alpha}-\psi)_{t}\|_{L^{\infty}(L^{2})}^{2}+\|\nabla(\psi^{\alpha}-\psi)\|_{L^{\infty}(L^{2})}^{2}$
$\displaystyle\lesssim$
$\displaystyle\,\|\mathfrak{k}_{\alpha}-\mathfrak{k}_{1}\|_{L^{p}(0,T)}^{2}\|\psi_{0}\|_{L^{2}(\Omega)}^{2}+\|\mathfrak{k}_{\alpha}-\mathfrak{k}_{1}\|_{L^{1}(0,T)}^{2}\|z_{t}\|_{L^{p}(L^{2})}^{2}+\|\mathfrak{k}_{\alpha}\|_{L^{1}(0,T)}^{2}\|\overline{z}_{t}\|_{L^{p}(L^{2})}^{2}$
$\displaystyle\quad+\|\mathfrak{k}_{\alpha}-\mathfrak{k}_{1}\|_{L^{1}(0,T)}^{2}\|\nabla
z_{t}\|_{L^{\infty}(L^{2})}^{2}+\|\mathfrak{k}_{\alpha}\|_{L^{1}(0,T)}^{2}\|\nabla\overline{z}\|_{L^{\infty}(L^{2})}^{2}\,,$

where we can use Proposition 7.1 and estimate (7.11) to further bound the
right-hand side terms. Note that since

$\|\mathfrak{k}_{\alpha}-\mathfrak{k}_{1}\|_{L^{\infty}(0,T)}=\lim_{t\to
0}|\mathfrak{k}_{\alpha}-\mathfrak{k}_{1}|(t)=+\infty,$

we only get an estimate of
$\|(\psi^{\alpha}-\psi)_{t}\|_{L^{\infty}(L^{2})}^{2}$ if we additionally
assume $\psi_{0}=0$. Furthermore,

$\displaystyle\|\mathfrak{k}_{\alpha}-\mathfrak{k}_{1}\|_{L^{p}(0,T)}\leq$
$\displaystyle|\tau^{-\alpha}-\tau^{-1}|\|\cdot^{\alpha-1}E_{\alpha,\alpha}(-(\tfrac{\cdot}{\tau})^{\alpha})\|_{L^{p}(0,T)}$
$\displaystyle+\tau^{-1}\|\cdot^{\alpha-1}-1\|_{L^{p}(0,T)}\|E_{\alpha,\alpha}(-(\tfrac{\cdot}{\tau})^{\alpha})\|_{L^{\infty}(0,T)}$
$\displaystyle+\tau^{-1}\|E_{\alpha,\alpha}(-(\tfrac{\cdot}{\tau})^{\alpha})-\exp(-(\tfrac{\cdot}{\tau}))\|_{L^{p}(0,T)}\to
0\ \ \mbox{ as }\alpha\to 1^{-}\,,$

where the only critical term is the one containing the singularity,
$\int_{0}|t^{-(1-\alpha)}-1|^{p}\,\textup{d}t$. Its convergence to zero
follows from Lebesgue’s Dominated Convergence theorem with the $L^{1}$ bound

$2^{p-1}(t^{-p(1-\alpha_{0})}+1)$

for $1\geq\alpha\geq\alpha_{0}>1-\frac{1}{p}$. Thus we arrive at the following
result.

######  Proposition 7.2.

Let $\psi_{0}\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$ and $f$, $\nabla f\in
L^{1}(0,T;L^{2}(\Omega))$. Further, let $\\{\psi^{\alpha}\\}_{\alpha\in(0,1)}$
be the family of solutions to the fMGT II equation and let $\psi$ solve the
corresponding equation with $\alpha=1$, where the initial data is in both
cases given by

$(\psi^{\alpha},\psi^{\alpha}_{t},\psi^{\alpha}_{tt})|_{t=0}=(\psi,\psi_{t},\psi_{tt})|_{t=0}=(\psi_{0},0,0).$

Then for any $p\in[1,\infty)$, $\psi^{\alpha}$ converges to $\psi$ in the
$W^{1,p}(0,T;L^{2}(\Omega))\cap L^{\infty}(0,T;H_{0}^{1}(\Omega))$ norm as
$\alpha\rightarrow 1^{-}$. If additionally $\psi_{0}=0$, then we also have
convergence in $W^{1,\infty}(0,T;L^{2}(\Omega))\cap
L^{\infty}(0,T;H_{0}^{1}(\Omega))$.

## Conclusion and Outlook

In this work, based on physical balance and constitutive laws, we have derived
four different fractional-order versions of a well-known third-order in time
model of nonlinear acoustics, the JMGT equation. The fractional-order of
differentiation $\alpha\in(0,1]$ (sometimes restricted to $\alpha\in(1/2,1]$)
appears as a parameter in each of these models. We have studied the well-
posedness of these equations and their linearizations in appropriate spaces
and justified the respective limits as $\alpha\to 1^{-}$, leading to the
(J)MGT equation.
Formally taking the limit of these equations as the relaxation time $\tau$
vanishes would lead to time-fractional second-order acoustic equations, which
are of independent interest as well. An analysis of this limit will be the
subject of future research.

## Acknowledgments

The work of the first author was supported by the Austrian Science Fund fwf
under the grants P30054 and DOC 78.

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