\section{Introduction}
Understanding quantitatively the properties of QCD at low temperature
and high baryon density remains one of the most challenging problems in
nuclear physics.
It is increasingly important as it governs the behavior of neutron stars
including their mass-radius relations
(see Refs.~\cite{watts:2016,Raithel:2017,Ozel:2016,Nattila:2017} and
references therein),
cooling, and the emission of
gravitational waves in neutron star mergers~\cite{nstarmerger:2017}.
At low to moderate
densities it is natural to model neutron star matter via a
system of interacting nucleons, and much progress has been
made along these
lines~\cite{Gandolfi:2012,Holt:2013fwa,Gandolfi:2015,Hebeler:2015hla}
At very high densities the problem is also tractable: At
low temperatures, the ground state will be a
color-flavor-locked superfluid phase
of quark matter with large superfluid pairing
gaps~\cite{Alford:1999,Rajagopal:2001}.  At intermediate densities
(several times nuclear saturation density) the situation is much less
clear.  One expects the
dominant degrees of freedom to transition from nucleons to  quarks and gluons,
but the density at which this occurs remains very difficult to determine even qualitatively.

      Studying the neutron star mass-radius relationship and cooling of neutron stars
has been an important goal to constraining the equation of state
through astrophysical observations~\cite{Steiner:2012}.  The recent observation of two
solar mass neutron stars~\cite{Demorest:2010}, for example, has
severely constrained possible equations of state at high density.
Nevertheless these provide only some constraints on the equation of
state, and less about the relevant degrees of freedom at high density.

      In this paper we study the behavior of small numbers of baryons
at high density (or equivalently small volumes) in periodic
boundary conditions.  It may soon be possible to study
these systems in lattice QCD, as the sign problem for small baryon number
is less severe. The sign problem grows exponentially with imaginary
time and is proportional to the number of nucleons times the mass of the
nucleon minus three halves the mass of the pion~\cite{Beane:2011}.
 We evaluate these systems in both nucleonic
models and quark models, and identify specific features that
arise in the spectra as the degrees of freedom change from nucleons to quarks.
These features rely on the relatively high momenta
and short distances that arise in small periodic volumes.
The most important of these features are shell closures for small
numbers of nucleons
and pairing in open-shell systems. The latter is important even
for very small systems, in particular for the $N=4$ systems we study.

The small lattice length $L$ needed to simulate high densities with a
small number of baryons may be subject to significant corrections
from long-range physics. However, we can expect that some of the
longest-distance effects due to pions should be the same in
nuclear and QCD simulations as long as nucleon-pion interactions
are consistently included in the nuclear Hamiltonians. In any
case the smallest box considered in this study (corresponding to
$N=4$ at $\rho=0.48$ fm$^{-3}$) has a box size of $L\approx2.03$ fm.

It may also  be possible to perform lattice QCD and nuclear
simulations for unphysical heavy up- and down- quarks,
resulting in heavy pions. This would reduce the sign problem and
the corrections due to
finite volume at the cost of unphysical pion masses.
Studying the transition even for
high pion masses may be instructive as pion degrees of freedom may
not play a very important role in the high-density phase transitions.
See Ref.~\cite{halqcd:2009} for some studies of $NN$ phase
shifts at high pion mass.

Small volumes will typically result in large excitation energies, reflecting
the wider spacing in the single-particle spectra, as discussed below.
This will be less true for comparison of different pairing symmetries
that are degenerate in the free-particle limit.   However, in general,
both the quantum Monte Carlo (QMC) and lattice QCD simulations will converge
more quickly with imaginary time for small systems.

Although studies of small systems are ill-suited to capture
critical behavior and cannot precisely identify possible phase
transitions, this work is well motivated because presently we
lack even a qualitative understanding of how quark degrees
of freedom might emerge at high density. The signatures we
identify below suggest that these small systems could be quite valuable
in this regime.
Eventually one can add protons and/or hyperons to look at the
density dependence of the symmetry energy and/or the presence of
hyperons in neutron stars.  In these initial studies,
we concentrate on pure neutron matter.

\subsection{Nucleonic Models:}
       To study nucleonic matter we consider nonrelativistic nucleons
interacting via two-nucleon ($NN$) interactions:
\begin{equation}
H \ = \ -\sum_i \ \frac{\hbar^2 \nabla_i^2}{2 m} + \sum_{i<j} V_{ij},
\end{equation}
where the $NN$ interaction is taken as either the Argonne
AV18~\cite{Wiringa:1995}, the
AV$8'$~\cite{Wiringa:2002},
or the local next-to-next-to-leading order (N$^2$LO) chiral interactions
of Ref.~\cite{Gezerlis:2014}.
At low densities such interactions should be able to faithfully
reproduce the properties of QCD.
Of course the three- (and eventually four- and many-) nucleon forces
will be present, and eventually play a significant role.
The simple spectral features we identify below
will remain, though, even in a more sophisticated picture.
These features depend primarily upon the single-particle
states available to a nucleon in periodic boundary condition,
indeed some are present even for noninteracting neutrons.

In this paper we consider only cubic simulation volumes with
periodic boundary conditions.
Other geometries, such as elongated volumes or different types of boundary
conditions, might allow one to identify additional spectral
features~\cite{Bedaque:2004ax}, but since present lattice
calculations typically
use cubic symmetry, we will adopt it in this
study.
Below we show results
for different densities with various numbers of neutrons in cubes of length $L$ on each side, and we show results as a function of the density $\rho = N/L^3$.

       The nuclear interaction models described above are defined in the continuum.  To maintain periodic boundary conditions, we add the contributions from periodic images for each pair:
\begin{equation}
    V_{ij} = \sum_{i_x,i_y,i_z=-M}^{M}  V\left[ r_{ij}+ L (i_x\
\hat{x} + i_y \  \hat{y} + i_z \  \hat{z})\right] \,,
\end{equation}
where $r_{ij}$ is the minimum separation in the periodic box and
$M = 1$ to $2$ images in each direction is sufficient to obtain
periodic solutions since the $NN$ interaction is at most of pion
range.  We note that the longest-range corrections from pions ``wrapping
around'' the box are also present in lattice QCD
simulations~\cite{Colangelo:2003hf}.  For heavier pion masses
these long-range periodic images will play much less of a role.

For the larger number of neutrons ($N>4$), we restrict ourselves to solutions
fully symmetric under cubic rotations.  For the smallest systems, we also
investigate states that would correspond most closely to $p$-wave
($N=3,4$) or $d$-wave ($N=4$) solutions in the continuum.  The couplings to nonzero
angular momenta prove quite interesting in comparing neutron potential
models to quark models.

Calculations are performed using
QMC methods: Either the Green's function Monte Carlo or Auxiliary Field
Diffusion Monte Carlo methods. More details are described in~\cite{Carlson:2015}.
The simulations are fairly simple as there are only a modest
number of neutrons and the small volumes raise the energies
of the excited states allowing for a more rapid convergence.

\subsection{Quark Model:}

For comparison, we also consider very simple quark models of high-density QCD
in periodic boundary conditions.  These models are not intended to be
realistic or predictive of the behavior of QCD in this regime, but they should
illustrate possible alternative behaviors when deconfined quarks
are the dominant degrees of freedom.

We consider both free and interacting quarks in periodic boundary conditions.
In general the models can be written as:
\begin{equation}
H \ = \sum_i T_i + \sum_{i<j} V_{ij} \,.
\end{equation}
In the free case we consider for the kinetic energies $T_i$ of the  quarks
relativistic and nonrelativistic dispersion relations $T_i =  \sqrt{p_i^2 + m_i^2} $ and $T_i = m_i + p_i^2 / 2 m_i $, respectively.
For the interacting quark model, we shall assume that chiral
symmetry is not fully restored when quark degrees of freedom
first manifest in the spectrum  and use a relatively large value
of $m=300$~MeV. Under these conditions we treat
the quarks as nonrelativistic even for the small volumes that we consider.

The pair potential is chosen in order to reproduce the pairing
pattern expected from a Nambu-Jona-Lasinio-like
model~\cite{Klevansky:1992qe} where interactions are antisymmetric with
respect to color.
Furthermore, for 3 colors and 2 flavors, we look for a color
superconducting ground-state called the 2SC phase where two
species are paired in a color and flavor anti-symmetric channel
while the third does not interact with either and is effectively decoupled \cite{Alford:2007}.
In the following we will consider the blue quarks to be the decoupled species.

For a given neutron number $N$ the occupation numbers in flavor-color-spin space are chosen in order to satisfy the following constraints
\begin{description}
\item[baryon number] $3N=(N_U+N_D)$
\item[color neutrality] $N_R=N_B=N_G$
\item[charge neutrality] $N_D=2N_U$
\item[spin neutrality] $N_\uparrow=N_\downarrow$
\end{description}

When quark degrees of freedom are manifest, it is appropriate to neglect
confinement at the short-distance scales of relevance to our small volumes.
We assume that the average confining interaction per baryon is a function of
the density only independent of baryon number at fixed density.
We then compare the evolution of the energy with baryon number
at fixed density to the nucleonic models.

The shell structure can be influenced by interactions, though,
especially those that lead to pairing.
To include this we shall consider a simple local potential of the form
\begin{equation}
\label{eq_pot}
V(i,j;\vec{r}_{i},\vec{r}_{j}) = \Lambda_{ij} V(r_{ij}) \,,
\end{equation}
where $i,j$ are multi-indices containing color, flavor and spin projection.
In order to obtain the desired pairing structure expected in high
density QCD,  we choose a simple short-range interaction
\begin{equation}
V(r)=- \frac{4 \hbar^2}{\mu r_e^2} \alpha \beta^2 \frac{e^{-2\beta \frac{r}{r_e}}}{\left[1+\alpha e^{-2\beta\frac{r}{r_e}}\right]^2}.
\end{equation}
The matrix $\Lambda_{ij}$ is chosen to be anti-diagonal with entries equal
to 1,
$\mu$ is the reduced mass, and
\begin{equation}
\alpha=\sqrt{1-2\frac{r_e}{a_s}} \,, \quad\quad \beta = 1+\alpha \,,
\end{equation}
where $a_s$ and $r_e$ are the ($S$-wave) scattering-length and
effective-range respectively.
In our
calculations we choose $a_s=10$ fm
and $r_e=0.1$ fm in order to be close to the unitary limit.

It is useful to compare relevant energy scales for 4 and
14 neutrons in periodic boundary conditions.  In
Table \ref{table:scale} we compare the lowest finite energy
($k = 2 \pi /L$) free-particle modes in the box.
Since the quarks are light, it takes substantial
energy to raise them to higher momentum states,
as indicated in the table.  There are more degrees of
freedom available, however, meaning that quarks will have a
substantially lower Fermi energy for the case of
weak interactions at high density.

\begin{table}
\begin{tabular}{lcccc}
\hline
Particle   & mass (GeV) &   N  & $\rho$ ($fm^{-3}$) & E (k=1, GeV) \\
\hline
Nucleon & 0.94 & 4 & 0.16 &  0.096 \\
rel q & 0.0 & 4 & 0.16 & 0.424 \\

rel q & 0.3 & 4 & 0.16 & 0.219 \\
non-rel q & 0.3 & 4 & 0.16 & 0.299 \\
\hline
Nucleon & 0.94 & 14 & 0.16 & 0.042 \\
rel q & 0.0 & 14 & 0.16 & 0.279 \\

rel q & 0.3 & 14 & 0.16 & 0.110 \\
non-rel q & 0.3 & 14 & 0.16 & 0.130 \\
\hline
Nucleon & 0.94 & 4 & 0.32 & 0.152 \\
rel q & 0.0 & 4 & 0.32 & 0.534 \\

rel q & 0.3 & 4 & 0.32 & 0.313 \\
non-rel q & 0.3 & 4 & 0.32 & 0.476 \\
\hline
\end{tabular}
\caption{Single particle energy levels for different baryon numbers
for SU(2) quarks and nucleons at different densities.
\label{table:scale}}
\end{table}

In very simple models the strength of the quark interaction is adjusted to reproduce the $N-\Delta$ mass splitting.  The total splitting is 320 MeV, which can be
compared to the single-particle energy splittings above.
For example, for four neutrons, two nucleons have a momentum
$|k| = 1$, while with quarks one can accommodate all quarks
with $|k| = 0$ at the cost of twice the $N-\Delta$ mass
splitting.  At low densities (large volumes) the four-neutron system would
be preferred, but at high densities (small volumes)
having all the quarks at $|k| = 0$ would be preferable.

\section{Results for $\boldsymbol{N=3}$ and $\boldsymbol{4}$}

At present it is difficult to compute many-neutron states in lattice QCD because of the rapid growth in the number of correlators required and
because the signal to noise ratio for small pion mass
grows exponentially
with baryon number.
Very small systems of 4 baryons may be easiest to simulate in
lattice QCD. For these systems we calculate states with different
quantum numbers as an additional probe of hadronic versus quark degrees
of freedom.
We find distinctively different behavior even for
4 neutrons when comparing the  nuclear and quark models
for different quantum states.

Two neutrons in finite volume have been studied
in lattice QCD (see for example Refs.~\cite{Beane:2006,Ishii:2007,Berkowitz:2015})
and as two nucleons
using QMC methods in Ref.~\cite{Klos:2016}.
These results are directly tied to the phase shifts of the neutron-neutron
interaction via the L\"uscher formula. Systems of three and four neutrons
in external wells at low density have been studied in Ref.~\cite{Gandolfi:2017};
here we are interested in the behavior at high densities in periodic boundary
conditions to mimic lattice simulations.

For three neutrons we study low-lying states with the quantum numbers of
two neutrons with spin and total momentum zero, and with the extra neutron in a $|k|=1$ state. The spin of this unpaired neutron can be oriented
along or anti-aligned to the lattice equivalent of the angular momentum
giving something similar to $P_{3/2}$  or $P_{1/2}$ states.  As expected the
former are slightly lower in energy due to the spin-orbit splitting in the
neutron-neutron interaction. The total (including center-of-mass kinetic energy)
ground-state energies of three neutrons at different densities with the AV$8'$ and AV18 interactions are compared with free neutrons with the same boundary conditions in Fig. \ref{fig:3n}.

\begin{figure}[htb]
\includegraphics[width=1.0\columnwidth]{n=3.pdf}
\caption{Ground-state energies of 3 free neutrons and with
AV$8'$ and AV18 $NN$ interactions as a function of density.}
\label{fig:3n}
\end{figure}

For four neutrons we study states with two neutrons paired to total
momentum and spin zero, and then either $s$-, $p$-, or $d$-wave pairing of the remaining
two dominantly $|k|=1$ neutrons.  The  $s$- and $d$-wave states have the spins coupled to
zero while the $p$-wave states must have the spins coupled to $1$ to maintain
antisymmetry.  The $s$-wave state is the same as that considered below
for up to 14 neutrons.

\begin{figure}[htb]
\includegraphics[width=1.0\columnwidth]{n4simple.pdf}
\caption{Energy per neutron of 4 neutrons for states with different
pairing symmetries as a function of density.}
\label{fig:4n}
\end{figure}

We find the initially surprising result (see Fig. \ref{fig:4n})
that neutron-neutron interactions
favor pairing in the $d$-wave state for small box sizes.  For free neutrons
the three different ($s$-, $p$- and $d$-wave) states
would be degenerate.   The interaction between the
two neutrons in $|k|=1$ states coupled to zero total momentum dominates
the spectrum.  The relative momentum for this pair is quite large in these
small volumes, in the region of the repulsive $s$-wave neutron-neutron
interaction.  The relevant phase shifts are shown in Fig.~\ref{fig:phases-4n}.
At saturation density, two neutrons with momenta $+1$ and $-1$ are at a
total energy of nearly 200 MeV, while at twice saturation
density the center-of-mass  energy is 300 MeV.
The strong $s$-wave repulsion disfavors the $s$-wave state, and the
periodic boundary
conditions favor the $L=2$ state, as the $d$-wave pairing is symmetric
across the periodic boundaries.
That is, a pair orbiting with $L=2$ feels an attractive interaction,
while for $L=1$, the periodic images interfere as the the relative coordinates
$r$ and $L-r$ and consequently $\vec{L} \cdot\vec{S}$ are oriented in opposite directions.

\begin{figure}[htb]
\includegraphics[width=1.0\columnwidth]{phases-4n.pdf}
\caption{$S$-, $P$-, and $D$-wave phase shifts for the
different $NN$ interaction models.}
\label{fig:phases-4n}
\end{figure}

Results for the four-neutron calculations at different volumes are
shown in Fig.~\ref{fig:4n} for the AV18 and N$^2$LO $NN$ interactions.
The different states all have very similar energies at half saturation density,
while the $d$-wave state is favored in all these models
at $\rho = 0.16$ fm$^{-3}$ and above. At twice saturation
density the $s$-wave state is roughly 100 MeV higher than the $d$-wave state.

For free or paired quarks it is always advantageous to keep all the quarks
dominantly in $k=|0|$ states. The pairing energy that can be gained by promoting
some quarks to higher momentum is small compared to the energy cost
of promoting two or more quarks to $|k|=1$ states. These conclusions are unaltered by the addition of a gluon-exchange spin-interaction (which historically was invoked to explain the $N-\Delta$ mass difference) of the form \cite{Carlson:1983}
\begin{equation}
V_s(r_{ij})=\frac{2\alpha_s}{2m_im_j}\left[\frac{8\pi}{3}\vec{\sigma}_i\cdot\vec{\sigma}_j\delta^3(\vec{r}_{ij})+\frac{1}{r^3_{ij}}S^{(2)}_{ij}\right]\;,
\end{equation}
where $S^{(2)}_{ij}$ is the tensor operator. The energy difference between
the $s$-wave and $d$-wave states favors the $s$-wave as the ground state,
since the $s$-wave is lower by $\Delta E$ per baryon $\approx 27$ MeV and $45$ MeV
at 2 and 3 times nuclear saturation density, respectively.
This ordering is opposite to that seen in the neutron calculations.

\section{Results from $\boldsymbol{N=4}$ to 14 neutrons}

We also consider nuclear ground states
from $N=4$ to 14 with even numbers of neutrons paired
to spin zero with full cubic symmetry.
In the continuum, this would correspond to an $s$-wave superfluid,
which is expected to be the ground state at low
densities~\cite{Gezerlis:2008,Gandolfi:2009b}.
The special cases $N=2$ and 14 correspond to filled single-particle
nuclear shells and hence are expected to have lower
energy per particle than the remaining (open-shell) systems.
Similar behavior has been observed in calculations of neutrons
in external fields with either Woods-Saxon or harmonic
wells~\cite{gandolfi:2011,Maris:2013}.
This is confirmed by our
numerical results as shown in Fig.~\ref{fig:shell}.
The rest mass of the nucleons are not included in this figure.

\begin{figure}[htb]
\includegraphics[width=1.0\columnwidth]{evsn.pdf}
\caption{Energies per neutron versus neutron number at different densities
for free neutrons (dashed lines) or with the AV18 $NN$ interaction (points).
At lower densities the energies per neutron are quite small.  The upper
curves correspond to densities up to three times nuclear saturation
density, where the energies per particle (ignoring rest mass) are from
50-100 MeV per nucleon.  The minima at $N=14$ corresponds to a closed shell of neutrons with $|k| = 0,1$.}
\label{fig:shell}
\end{figure}

Note that the differences between different particle numbers are quite
significant at high density, of order 10 MeV per particle between adjacent $N$ and of
order 50 MeV per nucleon lower for $N=14$ compared to $N=6$. Such strong shell dependence,
absent for free quarks, is also observed in the
limit of strongly paired quark matter.

The nuclear model dependence is fairly small at modest densities, but increases substantially at the highest densities considered. This is illustrated in Fig.~\ref{fig:eofg}, where we plot the ratio of energies to Fermi gas energies
for different particle numbers and densities for both the AV$8'$ and AV18 interactions. The AV$8'$ model only fits the lower partial waves, and is therefore less reliable at moderate to high densities.
The difference between AV18 and AV$8'$ is treated perturbatively in these
calculations.
The pattern of $E/N$ versus $N$ is the same for both interactions.

\begin{figure}[htb]
\includegraphics[width=1.0\columnwidth]{eofg.pdf}
\caption{Ratio of interacting to continuum Fermi Gas energies for the AV$8'$
(open symbols)  and AV18 $NN$ (closed symbols) interactions. }
\label{fig:eofg}
\end{figure}

In Fig.~\ref{fig:quarksandneutronsvsn} we compare results of the nuclear
model with free neutrons and also free and paired quarks at twice
saturation density. The confinement energy in the quark model, which is
assumed to be constant with density,
is adjusted to match the nuclear result for 14
interacting neutrons.
Note the dramatically different behavior versus baryon number in the
nuclear and quark models, particularly for small $N$.
The same behavior is observed also at three times saturation density in
Fig.~\ref{fig:quarksandneutronsvsn2}.

\begin{figure}[htb]
\includegraphics[width=1.0\columnwidth]{quarksandneutronsvsn-pair_2rho0}
\caption{Energy per neutron for nuclear and quark models at twice saturation density.  The behavior of the quark and nuclear models versus baryon number is quite different, particularly for small numbers.}
\label{fig:quarksandneutronsvsn}
\end{figure}

For both quarks and neutrons we find that the addition of interactions does
not change the qualitative behavior apart from the cases $N=16$ and $N=6$ at high density.
The sudden increase in energy for $N=16$ compared to $N=14$, which is
caused by the filling of the $|k|=2$ momentum shell in the neutron case,
has a similar nature for the quark model: At $N=16$, it is preferable for
the (down) blue quarks to fill the $|k|=2$ shell instead of
populating the (up) blue states as this will require breaking
interactive pairs. The large energy at $N=6$ for the quark model
compared to $N=4$ is due to the fact that for the latter all of the
interacting quarks can reside in the $|k|=0$ shell while at $N=6$
two pairs of quarks have to fill the $|k|=1$ state.
This large shell effect of $\approx 310$ MeV per
baryon at $3$ times saturation density will necessarily indicate the
dominance of quarks degrees of freedom. The feature is less pronounced
at twice saturation density where the, still large, gap is $\approx 120$ MeV.
This effect seems rather
robust since even in the limit of noninteracting quarks the gap
between $N_B=6$ and $N_B=4$ is still rather large,
$\approx 170$ MeV for $\rho=0.48$ fm$^{-3}$
and  $\approx 120$ MeV for $\rho=0.32$ fm$^{-3}$.

\begin{figure}[htb]
\includegraphics[width=1.0\columnwidth]{quarksandneutronsvsn-pair_3rho0}
\caption{Same as Fig.~\ref{fig:quarksandneutronsvsn} but at three times nuclear saturation density.}
\label{fig:quarksandneutronsvsn2}
\end{figure}

\section{Conclusions}

In this work we have studied the behavior of the energy
per baryon in high density neutron matter for small volumes.
These types of simulations can be valuable in understanding the
relevant degrees of freedom at high baryon chemical potentials,
in particular to begin to identify the regime where nucleon degrees
of freedom dominate and where full QCD simulations of quarks
and gluons are required.

While these simulation volumes are small, particularly for $N=4$, and may
suffer from important finite-size effects beyond the periodic pion
potentials discussed here, further studies with larger quark masses
and the $NN$ potentials derived from them are warranted.
Eventually lattice QCD studies with larger $N$ at physical pion masses
will be important.

Even for very small systems, we find a particular
pairing pattern in neutron simulations.
For four neutrons the $d$-wave state is favored at
high density for the two neutrons in the $|k|=1$ state,
due to a combination of the $s$- and $d$-wave phase shifts
and the periodic boundary conditions. In contrast, for quark
models the $s$-wave pairing is always favored
to avoid filling the $|k|=1$ states.

A clear signature of the dominance of strongly paired quarks as
degrees of freedom can also be seen in the large energy gap
between $N_B=4$ and $N_B=6$, in particular for paired quarks
compared to nucleons at 2 (or 3) times saturation density.
For the spatially symmetric states,
we expect an increase of $\approx 120$ MeV per baryon
($\approx 310$ MeV per baryon) to be compared with a much
smaller gap of $\approx 4$ MeV per baryon
($\approx 15$ MeV per baryon) for the interacting nuclear case.

Simulations of three- and four-nucleons ($A=3$ and 4 nuclei)
have already been completed
with higher quark masses, so it is reasonable to expect that similar
comparisons of neutron simulations to full lattice QCD
could be completed in the near future. A direct comparison would require
reasonable extractions of the $NN$ scattering phase shifts for these
quark masses.  Such a direct comparison could have
dramatic implications for the behavior of QCD at low temperatures and
high baryon density and hence the mass-radius relation of neutron stars
and the gravitational waves emitted during their mergers.

\emph{Acknowledgments:}
The work of J.C. and S.G. was supported by the NUCLEI SciDAC program,
by the U.S. DOE under contract DE-AC52-06NA25396, and by the LANL LDRD
program. A.R. was supported by NSF Grant No. AST-1333607 and
by DOE Grant No. DE-AC52-06NA25396.
The work of J.L. was supported by the ERC Grant No. 307986 STRONGINT and
the BMBF under Contract No. 05P15RDFN1.
The work of S.R. was supported by DOE Grant No. DE-FG02-00ER41132.
Computational resources have been provided by Los Alamos Open
Supercomputing.
We also used resources provided by NERSC, which is supported by the US
DOE under Contract DE-AC02-05CH11231.


\section{Introduction}
The Jordan-Moore-Gibson-Thompson equation is a nonlinear Partial Differential Equation
(PDE) model which describes the acoustic velocity potential in ultrasound wave propagation;
the use of the constitutive Cattaneo law for the heat flux, in place of the Fourier law,
accounts for its being of third order in time.
The quasilinear PDE is
\begin{equation}\label{Eq:quasilineare}
\tau \psi_{ttt} + \psi_{tt}-c^2\Delta \psi - b\Delta \psi_t=
\frac{\partial}{\partial t}\Big(\frac1{c^2}\frac{B}{2A}\psi^2_t+|\nabla \psi|^2\Big)
\end{equation}
in the unknown $\psi=\psi(t,x)$, that is the acoustic velocity potential (then
$-\nabla \psi$ is the acoustic particle velocity), $A$ and $B$ being suitable constants;
{\em cf.}~Moore \& Gibson \cite{moore-gibson_1960}, Thompson \cite{thompson_1972},
Jordan \cite{jordan_2009}.
For a brief overview on nonlinear acoustics, along with a list of relevant references,
see the recent paper by Kaltenbacher \cite{kalt_2015}.
Aiming at the understanding of the nonlinear equation, a great deal of attention has
been recently devoted to its linearization---referred to in the literature as the
Moore-Gibson-Thompson (MGT) equation---whose mathematical analysis is also of independent
interest, posing already several questions and challenges.

Let $\Omega\subset \mathbb{R}^n$ be a region with smooth ($C^2$) boundary
$\Gamma:=\partial\Omega$.
(It is a natural conjecture that existence results for wave equations in non-smooth domains
({\em cf.}~\cite{Grisvard}) can be extended to wave equations with memory and to the MGT equation, by using the methods we present in this paper.
We consider the MGT equation
\begin{equation}\label{e:mgt}
\tau u_{ttt}+\alpha u_{tt} -c^2 \Delta u -b \Delta u_t =0
\qquad \text{in $(0,T)\times\Omega$}
\end{equation}
in the unknown $u=u(t,x)$, $t\ge 0$, $x\in \Omega$, representing the acoustic velocity potential or alternatively, the acoustic pressure (see \cite{kalt-las-posp_2012}
for a discussion on this issue).
The coefficients $c$, $b$, $\alpha$ are constant and positive; they represent the speed and diffusivity of sound ($c$, $b$), and, respectively, a viscosity parameter ($\alpha$).
For simplicity we set $\tau=1$ throughout the paper.

Equation \eqref{e:mgt} is supplemented with initial and boundary conditions:
\begin{align}
& u(0,\cdot)=u_0\,,\; u_t(0,\cdot)=u_1\,,\; u_{tt}(0,\cdot)=u_2(x)\,,
& \text{in $(0,T)\times\Omega$}
\label{e:IC}
\\[1mm]
& {\mathcal T} u(t,\cdot) =g(t,\cdot)
& \text{on $(0,T)\times\Gamma$};
\label{e:BC}
\end{align}
${\mathcal T}$ denotes here a boundary operator, which---for the sake of simplicity---associates to a function either the trace on $\Gamma$, or the outward normal derivative
$\frac{\partial}{\partial \nu}\big|_\Gamma$ (it would be the {\em conormal} derivative,
in the case of a more general elliptic operator than the Laplacian).

The original studies of the MGT equation with homogenous (Dirichlet or Neumann) boundary data
carried out in Kaltenbacher~{\em et al.} \cite{kalt-etal_2011} and Marchand {\em et al.} \cite{marchand-etal_2012}~establish appropriate functional settings for semigroup well-posedenss, as well as stability and spectral properties of the dynamics, depending on the
parameters values.
They obtain, in particular,
\begin{enumerate}
\item[i)]
that assuming $b>0$ the linear dynamics is governed by a strongly continuous {\em group} in
the function space $H^1_0(\Omega)\times H^1_0(\Omega)\times L^2(\Omega)$ (Dirichlet BC),
or ($H^1(\Omega)\times H^1(\Omega)\times L^2(\Omega)$ (Neumann BC);
\item[ii)]
that in the case $b=0$ the associated initial/boundary value problems
are ill-posed ({\em cf.}~Remark~\ref{r:role-of-b});
\item[iii)]
that the parameter $\gamma=\alpha - \tau c^2/b$ is a threshold of stability/instability: it must be positive, if the property of uniform stability is required.
\end{enumerate}
The critical role of $\gamma$ for a dissipative behaviour was recently pointed out
also in Dell'Oro and Pata~\cite{delloro-pata_2016}, within the framework of viscoleasticity.
(We add that linear and true nonlinear variants of the MGT equation including an {\em additional} memory term have been the object of recent investigation; see \cite{las-jee_2017} and references therein.)

\smallskip
Our interest lies in studying the regularity of the mapping
\begin{equation*}
(u_0,u_1,u_2,g)\longmapsto u
\end{equation*}
that associates to initial and boundary data---taken in appropriate spaces---the corresponding solution $u=u(t,x)$ to the initial/boundary value problem (IBVP)
\eqref{e:mgt}-\eqref{e:IC}-\eqref{e:BC}.
(We note that the time and more often the space variable $x$ will generally not be esplicit,
unless when needed for the sake of clarity.)

As it will be shown in the paper, it will be the embedding of equation \eqref{e:mgt}
in a general class of integro-differential equations (depending on a parameter)
to spark our method of proof for the regularity analysis of the associated initial/boundary value problems.
Indeed, the MGT equation is a special instance of the following wave equation with
persistent memory,
\begin{equation}\label{e:memory}
u_{tt}-b \Delta u=-b\gamma \int_0^t N(t-s) \Delta u(s)\,ds + F(t)\xi\,,
\end{equation}
which displays an affine term depending on a suitable $\xi$, and that will be
supplemented with (initial and boundary) data
\begin{equation} \label{eq:dataDIe:memory}
u(0)=u_0\,,\ u_t(0)=u_1\,, \qquad {\mathcal T} u=g\,.
\end{equation}
The assumptions on the real valued functions $N(t)$, $F(t)$ and on $\xi$ in \eqref{e:memory}
are specified later; see Theorem~\ref{t:sample}.
As it will be apparent below, the parameter $\xi$ includes the component $u_2$ of
initial data $(u_0,u_1,u_2)$ for the MGT equation, while \eqref{e:memory}-\eqref{eq:dataDIe:memory} reduces to the MGT equation (with \eqref{e:IC}-\eqref{e:BC}) when
\begin{equation*}
N(t)=F(t)=e^{-\alpha t}\,,\qquad \xi=u_2-b\Delta u_0\,.
\end{equation*}

\smallskip
The obtained regularity results will follow combining the (interior and trace) regularity
theory for wave equations with non-homogenous boundary data---the Neumann case being the most challenging (see \cite{las-trig_wave1}, and the optimal result of \cite{tataru_1998})---with the methods developed in~\cite{PandLIBRO} for equations with persistent memory.
In order to carry out a regularity analysis of the model equation with memory
\eqref{e:memory} we shall use the trick of MacCamy \cite{maccamy_1977} and the theory of Volterra equations.

For equations with memory of the form \eqref{e:memory} the reader is referred, e.g., to \cite[Chapter~2]{PandLIBRO}; see also \cite[Chapter~5]{corduneanu}.
A novelty in the equation is brought about by the presence of the (vectorial) parameter $\xi$.
A classical reference on---and thorough treatment of---evolutionary integral equations is
\cite{pruess_1993}.

\smallskip
It is important to emphasize at the very outset that the adopted perspective and approach
paves the way for establishing the (sharp) regularity of boundary traces for the MGT equation,
as well as for the solutions to a rather general family of wave equations with memory, supplemented with Dirichlet or Neumann boundary conditions.
While being a topic of recognized current interest, the only result that appears available so far is the one obtained (via energy methods) in \cite{loreti-sforza_parma},
tailored for the case of Dirichlet boundary conditions.
Our alternative proof for the model equation with memory, depending on the parameter $\xi$
(and with the same BC) is given in Theorem~\ref{t:traces-memory}, which
brings about a boundary regularity result for the MGT equation, that is
Corollary~\ref{c:traces-mgt}.
(The study of the regularity of boundary traces for wave equations with memory in
the case of Neumann boundary conditions is left to a separate, subsequent investigation.)

\subsection{Main results: synopsis}
The outcome of the {\em interior} regularity analysis carried out in this paper is stated in
Theorem~\ref{t:main_1}, pertaining to the general model equation with memory \eqref{e:memory},
and Theorem~\ref{t:main_2} for the MGT equation itself.
Beside being the former results instrumental in achieving the subsequent ones, they
are also of independent interest.
\\
Because the said results are presented by means of elaborate tables, aiming at rendering explicit the major achievements on the regularity of equations \eqref{e:memory} and
\eqref{e:mgt}---the latter linked and complementing those in our key reference~\cite{kalt-etal_2011}---we highlight them in Theorem~\ref{t:sample} below.
Theorem~\ref{t:sample} includes as well a last statement on the regularity of {\em boundary} traces, an issue which is dealt with in Section~\ref{s:traces}; see Theorem~\ref{t:traces-memory} and Corollary~\ref{c:traces-mgt}.

\medskip
For the statement and understanding of all our findings, we need to introduce appropriate
functional spaces along with the related notation.
Let $A$ be the unbounded operator defined as follows:
\begin{equation} \label{definizOPERATORE-A}
A w:=(\Delta -I) w\,, \quad
{\mathcal D}(A) =\big\{w\in H^2(\Omega)\colon {\mathcal T} w =0 \;
\text{on $\Gamma$}\big\}\,;
\end{equation}
namely, $A$ is the (so called) {\em realization} of the differential operator $\Delta -I$ in
$L^2(\Omega)$, with homogeneous boundary conditions (BC) defined by ${\mathcal T}$, in the present work
of either Dirichlet or Neumann type; of course, the domain of $A$ depends on $\mathcal{T}$.
(We might take the realization of the laplacian in the case of Dirichlet BC; translating the
differential operator allows us to deal with both significant BC at once.)
We further note that
$A$ is the infinitesimal generator of an exponentially stable analytic
semigroup and the fractional powers of $-A$ are well defined.
Thus, we are allowed to introduce the functional spaces $X_s$ definied as follows:
\begin{equation}\label{e:x_r}
X_s=
\begin{cases}
{\mathcal D}((-A)^{s/2}) & \text{if $s \ge 0$}
\\[1mm]
[{\mathcal D}((-A)^{s/2})]' & \text{if $s < 0$}\,,
\end{cases}
\end{equation}
endowed with the graph norm if $s\ge 0$, while the norm of a dual space is needed otherwise.
\\
Then, it is well known that $A\colon X_s \rightarrow  X_{s-2}$ is continuous, surjective and boundedly invertible.

\smallskip
The next theorem collects results obtained in Sections~\ref{s:main_1}, \ref{sect:RegulaMGT},
~\ref{s:traces}.
\begin{theorem}[A compendium of main results] \label{t:sample}
The following assertions hold:
\begin{itemize}
\item[i)] \label{item1THTheorem:sample}
{\rm
(Interior regularity for the equation with memory~\eqref{e:memory}) with homogeneous BC).
}
Assume $N(t)\in H^2(0,T)$ and $F(t)\in L^2(0,T)$ for every $T>0$.
Let $g\equiv 0$.
If $u_0\in X_0$ and $u_1,\xi\in X_{-1}$,
then the solution $u$ to the IBVP problem \eqref{e:memory}-\eqref{eq:dataDIe:memory}
satisfies
\begin{equation*}
u\in C([0,T];X_0)\cap C^1([0,T];X_{-1})\cap L^2(0,T;X_{-2})\,.
\end{equation*}

\item[ii)] \label{item2THTheorem:sample}
{\rm
(Interior regularity for the MGT equation \eqref{e:mgt} with homogeneous BC).
}
If $g\equiv 0$ and $(u_0,u_1,u_2)\in X_1\times X_1\times X_0$,
then the solution $u$ to the IBVP problem \eqref{e:mgt}-\eqref{e:IC}-\eqref{e:BC}
satisfies
\begin{equation} \label{e:boundary-tointerior}
u\in C([0,T];X_1)\cap C^1([0,T];X_1)\cap C^2([0,T];X_0)
\end{equation}
and the map $(u_0,u_1,u_2)\longmapsto u(t,x)$ is continuous in the specified spaces.

\item[iii)] \label{item3THTheorem:sample}
{\rm
(Boundary-to-interior regularity for equations \eqref{e:memory} and \eqref{e:mgt},
with trivial initial data).
}
Assume $u_0=u_1=\xi=0$ ($u_0=u_1=u_2=0$, respectively), and $g\in L^2(0,T;L^2(\Gamma))$.
Then there exists $\alpha_0$ such that for the solutions $u$ to the IBVP problem
\eqref{e:memory}-\eqref{eq:dataDIe:memory} (\eqref{e:mgt}-\eqref{e:IC}--\eqref{e:BC},
respectively) satisfy
\begin{equation} \label{e:boundary-to-interior}
u\in C([0,T];X_{\alpha_0})\cap C^1([0,T];X_{\alpha_0-1}\cap  L^2(0,T;X_{\alpha_0-2})\,;
\end{equation}
the value of $\alpha_0$ depends on the boundary operator $\mathcal{T}$ (and partly
on $\Omega$) and are specified in \eqref{e:alfazero} below.
\item[iv)]
{\rm
(Regularity of boundary traces for the MGT equation \eqref{e:mgt}).
}
Let $u=u(t,x)$ be a solution to the MGT equation \eqref{e:mgt} corresponding to
initial data $(u_0,u_1,u_2)$ and homogeneous boundary data.
Assume $(u_0,u_1,u_2)\in H^1_0(\Omega)\times L^2(\Omega)\times H^{-1}(\Omega)$,
along with the compatibility condition
\begin{equation*}
u_2-\Delta u_0\in L^2(\Omega)\,.
\end{equation*}
Then, for every $T>0$ there exists $M=M_T$ such that
\begin{equation*
\begin{split}
& \int_0^T\!\!\!\int_{\partial\Omega} \Big|\frac{\partial}{\partial\nu} u(x,t)\Big|^2
d\sigma\,d t
\le M\, \Big( \|u_0\|_{H^1_0(\Omega)}+\|u_1|_{L^2(\Omega)}^2 +
\\[1mm]
& \qquad\qquad\qquad \qquad\qquad\qquad \qquad
+\|u_2-\Delta u_0\|^2_{L^2(\Omega)}\Big)\,.
\end{split}
\end{equation*}

\end{itemize}

\end{theorem}

\begin{remarks}
\begin{rm}
We see from the statements in i) and ii), respectively, that while the equation with memory
\eqref{e:memory} displays a somewhat expected regularity, namely, the same as most PDE models for wave propagation, the interior regularity of solutions to the MGT equation \eqref{e:mgt} under homogeneous boundary conditions improves.
\\
Instead, the regularity result in iii)---that pertains to the case of trivial initial data ($u_0=u_1=u_2=0$) and non-homogeneous boundary data ($g\ne 0$)---is not improved by special
choices of the kernel $N(t)$, such as $N(t)=e^{-\alpha t}$.

It is worth mentioning that our analysis does not disclose that the dynamics of the MGT
equation \eqref{e:mgt} is governed by a {\em group}.
Following the studies on well-poseness performed in \cite{kalt-etal_2011} and \cite{marchand-etal_2012}, the present study focuses on the regularity analysis of a general
class of PDE systems which are governed by {\em semigroups}---not necessarily groups---and
whose solutions generally display a lower regularity than the ones of equation \eqref{e:mgt}.
The higher interior regularity for the MGT equation is obtained when we particularize
the formulas, and exploiting the smoothness of the coefficients.
\end{rm}
\end{remarks}

\smallskip
We note that the values of $\alpha_0$ which occurr in \eqref{e:boundary-to-interior}---and which correspond to appropriate Sobolev exponents---are the ones
established in the case of (linear) hyperbolic equations with $L^2(\Sigma)$ boundary data
(of either Dirichlet or Neumann type).
We record explicitly for the IBVP
\begin{equation}\label{e:ibvp-wave}
\begin{cases}
u_{tt}=\Delta u - u+f & \text{in $(0,T)\times \Omega$}
\\[1mm]
u(0,\cdot)=u_0\,, \; u_t(0,\cdot)=u_1  & \text{in $\Omega$}
\\[1mm]
{\mathcal T} u=g  & \text{on $(0,T)\times \Gamma$}
\end{cases}\,.
\end{equation}
a statement which embodies a complex of successive achivements; see the cited references.
\begin{theorem}[\cite{las-lions-trig}, \cite{las-trig_wave1}, \cite{tataru_1998}]
\label{t:tataru}
Assume that $u_0, u_1=0$, $f= 0$, and $g\in L^2(\Sigma)$.
Then, the unique solution to the initial/boundary value problem \eqref{e:ibvp-wave}
satisfies
\begin{equation*}
(u,u_t)\in C([0,T];H^{\alpha_0}(\Omega) \times H^{\alpha_0-1}(\Omega))\,,
\end{equation*}
with
\begin{equation}\label{e:alfazero}
\alpha_0 =
\begin{cases}
0 & \text{if ${\mathcal T}$ is the Dirichlet trace operator}
\\[1mm]
\frac23 & \text{if ${\mathcal T}$ is the Neumann trace operator and $\Omega$ is a
smooth domain}
\\[1mm]
\frac34 & \text{if ${\mathcal T}$ is the Neumann trace operator and $\Omega$ is a parallelepiped.}
\end{cases}
\end{equation}
\end{theorem}
For a chronological overview with historical and technical remarks see, e.g.,
\cite[Notes on Chapter~8, p.~761]{las-trig-book}.

We finally point out on the regularity of wave equations the recent progress of
\cite{triggiani_2016}, dealing with the case of boundary data $g$ that are {\em not}
`smooth in space', e.g., $g\in L^2(0,T;H^{-1/2}(\Gamma))$.
In view of the approach taken in the present work, it is clear that the results obtained
therein could be utilized as well in order to attain regularity results for equations
with memory and for the MGT equation under boundary data that are less
regular (than square integrable) in space.

\subsection{Orientation}
The plan of the paper is briefly outlined below.
For the reader's convenience and since these tools will be utilized throughout,
in Section~\ref{s:preliminaries-on-wave} we provide a minimal background and references on the
approach to linear wave equations
via cosine operator theory.

In Section~\ref{sec:memory} we perform an analysis of the equation with
memory \eqref{e:memory} that encompasses the MGT equation.
An equivalent equation---in fact easier, since the convolution term therein does not involve
differential operators at all---is derived, which in turn results
in a Volterra equation of the second kind; see Proposition~\ref{p:volterra}.
This step will play a crucial role in the proof of our first regularity result,
that is Theorem~\ref{t:main_1}, concerning the model equation with memory \eqref{e:memory}.
Section~\ref{s:main_1} is then almost entirely devoted to the proof of Theorem~\ref{t:main_1}.

In Section~\ref{sect:RegulaMGT} we return to the third order MGT equation and show how the (interior) regularity results specifically pertaining to the MGT equation, stated in
Theorem~\ref{t:main_2}, follow as a consequence of Theorem~\ref{t:main_1}.
Finally, Section~\ref{s:traces} is devoted to the regularity of boundary traces;
see Theorem~\ref{t:traces-memory} and Corollary~\ref{c:traces-mgt}.
\\
A discussion and explanation of the introduced definition of solutions to the third order
(in time) equation under investigation is postponed to Appendix~\ref{a:def-solutions}.

\section{Preliminaries on wave equations} \label{s:preliminaries-on-wave}
Consider the initial/boundary value problem for a linear wave equation
\eqref{e:ibvp-wave}.
Since the methods of proof employed in the present work rely in a crucial way on the representation of solutions to wave equations by means of cosine operators, few lines
on this approach follow.
The reader is referred to \cite{belleni} and \cite{las-trig-cos}, were a first use of cosine operators is found in order to study equations with persistent memory and in the context of boundary control theory, respectively; see also the former contribution of \cite{sova_1966}.
We adopt here the notation of \cite{belleni} and \cite{fattorini}.

We shall use the operator $A$ in~\eqref{definizOPERATORE-A}, which is the realization of the translation $\Delta -I$ of the
Laplacian in $L^2(\Omega)$, with suitable homogeneous boundary conditions, according to
a (boundary) operator ${\mathcal T}$.
(In the Dirichlet case $A$ might be simply the realization of the Laplacian.)
As noted already, $A$ is boundely invertibile, i.e. $A^{-1}$ exists and it is bounded,
in fact compact (even if ${\mathcal T}$ represents the normal derivative on $\Gamma$).
It generates an exponentially stable analytic
semigroup and the fractional powers of $-A$ are well defined and we shall use the spaces $X_s$ in \eqref{e:x_r}   ($X_s$ has the graph
norm if $s\ge 0$, and the norm as a dual space otherwise).
We recall once more that $A$: $X_s \rightarrow  X_{s -2}$ is continuous, surjective  and boundedly invertible.

Next, we introduce the Green maps $G\in {\mathcal L}(L^2(\Gamma),L^2(\Omega))$ defined
as follows:
\begin{equation}\label{e:green-map}
G\colon L^2(\Gamma)\ni \varphi\longmapsto G\varphi=:\psi \;
\Longleftrightarrow \;
\begin{cases}
\Delta \psi =\psi & \textrm{on $\Omega$}
\\[1mm]
{\mathcal T} \psi=\varphi & \textrm{on $\Gamma$}\,.
\end{cases}\,;
\end{equation}
By elliptic theory, it is known that there exists an appropriate $s >0 $ such that
$\text{im}\,G\subset X_s$ so that $A G\subset X_{s -2}$.
For instance, in the case of Dirichlet boundary conditions one has
$\text{im}\,G= H^{1/2}(\Omega) \subset X_s$, with inclusion that holds true for any
$s=1/2 -\sigma$, $0<\sigma<\frac{1}{2}$.

Thus, it is known that the solution to the IBVP \eqref{e:ibvp-wave} is given by
\begin{equation}\label{e:waves-explicit}
\begin{split}
u(t)& =R_+(t)u_0+{\mathcal A}^{-1} R_-(t)u_1-{\mathcal A}\int_0^t R_-(t-s)Gg(s)\,ds \,+
\\[1mm]
& \qquad\qquad\qquad +{\mathcal A}^{-1} \int_0^t R_-(t-s)f(s)\,ds
\end{split}
\end{equation}
where the operator ${\mathcal A}$, and the families of operators $R_+(\cdot)$, $R_{-}(\cdot)$
are defined as follows:
\begin{equation}\label{eq:defiOperRpm}
{\mathcal A}=i(-A)^{1/2}\,,\qquad
R_+(t)=\frac{e^{{\mathcal A} t}+e^{-{\mathcal A} t}}{2}\,,
\qquad R_-(t)=\frac{e^{{\mathcal A} t}-e^{-{\mathcal A} t}}{2}\,,
\end{equation}
$R_+(t)$ being the strongly continuous {\em cosine} operator generated by $A$ in
$L^2(\Omega)$; see \cite{sova_1966}, \cite{fattorini}, \cite[Vol.~II]{las-trig-book}.

\begin{remark}
\begin{rm}
The previous definitions make sense because ${\mathcal A}$ is the infinitesimal generator of a
$C_0$-{\em group} of operators;
in particular, we have as well
\begin{equation*}
X_s={\mathcal D}((i{\mathcal A})^s)={\mathcal D}({\mathcal A}^s) \qquad\qquad \text{if $s\ge 0$,}
\end{equation*}
and ${\mathcal A}$ is bounded and boundedly invertible from $X_s$ to $X_{s-1}$ for every
$s$.
\end{rm}
\end{remark}

Computing the derivatives of \eqref{e:waves-explicit} we obtain the following equalities,
valid in $H^{-1}(\Omega)$ and $H^{-2}(\Omega)$, respectively:
\begin{equation}\label{e:waves-explicit-prime}
\begin{split}
u_t(t)& ={\mathcal A} R_-(t)u_0+R_+(t)u_1-A\int_0^t R_+(t-s) Gg(s)\,ds\, +
\\[1mm]
& \qquad\qquad\qquad + \int_0^t R_+(t-s)f(s)\,ds\,,
\end{split}
\end{equation}
as well as
\begin{equation} \label{e:waves-explicit-second}
\begin{split}
u_{tt}(t)&=A R_+(t)u_0+{\mathcal A} R_-(t)u_1  -AG g(t) -A\Big({\mathcal A} \int_0^t R_-(t-s) Gg(s)\,ds\Big)+
\\[1mm]
&  + f(t) +{\mathcal A}\int_0^t R_-(t-s)f(s)\,ds =
\\[1mm]
&=Au(t)-AGg(t)+f(t)\,.
\end{split}
\end{equation}

\begin{remark}
\begin{rm}
If $f(\cdot)$ is of class $C^1([0,T])$ then it is possible to integrate by parts,
like in
\begin{equation*}
{\mathcal A}^{-1} \int_0^t R_-(t-s)f(s)\,ds=-A^{-1}\left [
f(t)-R_+(t)f(0)-\int_0^t R_+(t-s) f(s) d s
\right ]
\end{equation*}
which brings about a gain of one unity in space regularity.
The integration by parts is rigorously justified in~\cite[Lemma~5]{PandAMO}.
\end{rm}
\end{remark}

The explicit formula \eqref{e:waves-explicit}, along with \eqref{e:waves-explicit-prime}
and \eqref{e:waves-explicit-second} are among the keys for the following regularity
result.
The statement in iii) is by far the most challenging, as its proof is based on
pseudo-differential methods and microlocal analysis.
\begin{theorem}\label{teo:propertyONDE}
Let $T>0$ be given, and $s\in \mathbb{R}$.
The following statements hold true for the solutions to the initial/boundary value
problem~(\ref{e:ibvp-wave}).
\begin{enumerate}
\item[i)]
\label{teo:propertyONDE-1}
Assume $g=0$, $f=0$.
Then $(u_0,u_1)\longmapsto u(t)$ is continuous from $X_s\times X_{s-1}$
into $C([0,T],X_s)\cap C^1([0,T],X_{s-1})\cap C^2([0,T],X_{s-2})$.
\item[ii)]
\label{teo:propertyONDE-2}
Assume $u_0=0$, $u_1=0$, $g=0$.
Then the map $f \longmapsto u(t)$ is continuous from $L^2(0,T;X_s)$ into
$C([0,T],X_{s+1})\cap C^1([0,T),X_s)$
while $u_{tt}(t)-f(t)\in C([0,T],X_{s+1})$.
\item[iii)]\label{teo:propertyONDE-3}
Assume $u_0=0$, $u_1=0$, $f=0$.
Then, there exists $\alpha_0\ge 0$---depending on ${\mathcal T}$ and possibly on the geometry of
$\Omega$---such that for every $g\in L^2((0,T)\times \Gamma)$ we have
$u\in  C([0,T],X_{\alpha_0})\cap C^1([0,T];X_{\alpha_0-1})\cap C^2([0,T],X_{\alpha_0-2})$.
The mapping $g\longmapsto u$ is continuous in the indicated spaces.
\end{enumerate}

\end{theorem}

\begin{remarks}
\begin{rm}
1. With reference to the assertion $iii)$ above,
we remind the reader that the proper value of Sobolev exponent $\alpha_0$ are
given in \eqref{e:alfazero}.

\noindent
2. The properties stated in the previous Theorem justify \eqref{e:waves-explicit} as a formula for the solutions to the IBVP~\eqref{e:ibvp-wave}, since the following fact is easily checked: when $u_0, u_1\in {\mathcal D}(\Omega)$ ($C^\infty(\Omega)$ functions with compact support),
$f \in {\mathcal D}((0,T)\times \Omega)$, $g \in {\mathcal D}((0,T)\times \Gamma)$, then $u-Gg\in C([0,T],{\mathcal D}(A))\cap C^1([0,T],{\mathcal D}(A))\cap C^2([0,T];L^2(\Omega))$ and the following equality holds:
\begin{equation*}
u_{tt}(t)=A(u(t)-Gg(t))+f(t)\,,
\end{equation*}
along with $u(0)=u_0$, $u_t(0)=u_1$.
Thus, the boundary condition ${\mathcal T} u=g$ is satisfied in the sense that $u(t)-Gg(t)\in {\mathcal D}(A)$
for almost any $t$.

\end{rm}
\end{remarks}

\section{The MGT equation as an equation with memory}  \label{sec:memory}

We initially proceed formally.
Rewrite the first hand side of equation \eqref{e:mgt} as
\begin{equation}\label{eq:MGTPasso1}
\begin{split}
& u_{ttt}+\alpha u_{tt} -c^2 \Delta u -b \Delta u_t =
\\[1mm]
& \qquad\qquad\qquad
= \big(u_{tt}-b \Delta u\big)_t
+\alpha \big(u_{tt}-b\Delta u\big) - c^2\Delta u +\alpha b \Delta u=
\\[1mm]
& \qquad\qquad\qquad
= \big(u_{tt}-b \Delta u\big)_t +\alpha \big(u_{tt}-b\Delta u\big)+b \gamma \Delta u =0
\end{split}
\end{equation}
where we recall that $\gamma = \alpha -c^2/b$.
Solving the equation
\begin{equation*}
\big(u_{tt}-b \Delta u\big)_t =-\alpha \big(u_{tt}-\Delta u\big)-b \gamma \Delta u
\end{equation*}
in the `unknown' $u_{tt}-b \Delta u$ gives the following integral equation in the unknown
(and in fact not yet defined as solution) $u$:
\begin{equation}\label{eq:MGTriscritta}
u_{tt}-b \Delta u= e^{-\alpha t}\xi-b\gamma \int_0^t e^{-\alpha (t-s)}\Delta u(s) \, ds\,,
\end{equation}
with $\xi= u_2-b\Delta u_0$.

Thus, in view of the obtained equation~\eqref{eq:MGTriscritta}, we consider the following
(more general) model equation with persistent memory, depending on the parameter $\xi$:
\begin{equation} \label{eq:MEMORY}
u_{tt}-b \Delta u= -b\gamma \int_0^t N(t-s)\Delta u(s) \, ds + F(t)\xi
\end{equation}
(already appeared---as \eqref{e:memory}---in the Introduction and recorded here for the
reader's convenience; notice that both functions $N(t)$ and $F(t)$ equal $e^{-\alpha t}$
in the MGT equation).

\begin{remark}\label{r:role-of-b}
\begin{rm}
If it happens that $\gamma=0$, then \eqref{eq:MGTriscritta} is nothing but a wave equation
with affine term
$F(t)\xi$ and the regularity of the corresponding solutions
follows from Theorem~\ref{teo:propertyONDE}.
Thus, we explicitly assume $\gamma \ne 0$, and recall from the Introduction that $b>0$.
It is important to emphasize that in the case $b=0$ the problem is ill-posed,
since the semigroup generation fails, as proved in~\cite[Theorem~1.1]{kalt-etal_2011};
instead, if $b<0$ then the PDE becomes a {\em nonlocal} elliptic equation of a kind
studied by Skubacevski\v{i} in~\cite{skubacevskii}.
\end{rm}
\end{remark}

The regularity analysis of equation \eqref{eq:MEMORY} is carried out under the assumptions listed below.

\begin{hypotheses}\label{a:kernel-and-affine}
i) The  coefficient $b$ is positive.
\hskip 1mm
ii) The memory kernel $N(t)$ and the function $F(t)$
are real valued;
$N(t)\in H^2(0,T)$ while $F(t)\in L^2(0,T)$ for every $T>0$.
\end{hypotheses}

\subsection{An equivalent Volterra integral equation}

A first step in our analysis is to show that we can get rid of the (second order)
differential operator in the convolution term of \eqref{eq:MEMORY}.
To do so, let us preliminarly introduce the Volterra equation of the second kind
\begin{equation}\label{e:volterra-2-kind}
X(t)- \gamma \int_0^t N(t-s) X(s) \, ds = G(t)\,, \qquad t\in [0,T]\,.
\end{equation}
This equation has a unique solution $X(t)$ given by the following formula:
\begin{equation}\label{eq:soluFORMvolte}
X(t)=G(t)-\int_0^t R_0(t-s) G(s)\,ds\,.
\end{equation}
where $R_0(\cdot)$ is the (unique) solution to the integral equation
\begin{equation}\label{e:resolvent-kernel}
R_0(t)-\gamma \int_0^t N(t-s) R_0(s)\,ds = -\gamma N(t)\,, \qquad t\in [0,T]\,.
\end{equation}
The function $t \longmapsto R_0(t)$ is the {\em resolvent kernel} of the Volterra equation.
An important observation is that $R_0\in H^2(0,T)$ since $N\in H^2(0,T)$ and
\mbox{$R_0(0)=-\gamma N(0)$}.
We then see (either from~\eqref{eq:soluFORMvolte} or from \eqref{e:volterra-2-kind}) that if
$G(t)$ is continuous then $X(t)$ is continuous; if $G(t)$ is square integrable then
$X(t)$ is square integrable.

\smallskip
We now perform several formal computations which will lead to a definition of the solutions
to equation \eqref{eq:MEMORY} (with appropriate initial and boundary data).
Rewrite the equation \eqref{eq:MEMORY} in the following different fashion,
\begin{equation}\label{e:like-a-volterra}
\Delta u -\gamma \int_0^t N(t-s)\Delta u(s) \, ds =\frac{1}{b}\big(u_{tt}- F(t)\xi\big)\,,
\end{equation}
that is a Volterra integral equation of the second kind in the unknown $\Delta u$.
With reference to the general form \eqref{e:volterra-2-kind}, we have here
\begin{equation*}
G(t)=\frac{1}{b}\big(u_{tt}- F(t)\xi\big)\,.
\end{equation*}
The formula \eqref{eq:soluFORMvolte} gives
\begin{equation*}
b\Delta u =u_{tt}- F(t)\xi- \int_0^t R_0(t-s)\big(u_{ss}(s)- F(s)\xi\big)\,ds\,,
\end{equation*}
where $R_0(\cdot)$ is the unique solution to the integral equation \eqref{e:resolvent-kernel},
as explained above.
Since $R_0\in H^2(0,T)$ we are allowed to integrate by parts twice, thereby obtaining
\begin{equation*}
\begin{split}
b\Delta u & = u_{tt}- F(t)\xi
- \Big\{R_0(t-s)u_t(s)\big|_{s=0}^{s=t}-\int_0^t R_0'(t-s)u_s(s)\,ds\Big\} +
\\[1mm]
& \qquad\qquad\qquad + \int_0^t R_0(t-s)F(s)\xi\,ds=
\\[1mm]
& = u_{tt}- F(t)\xi-R_0(0)u_t(t) + R_0(t) u_1 - R_0'(0)u(t)-R_0'(t)u_0-
\\[1mm]
& \qquad\qquad\qquad - \int_0^t R_0''(t-s)u(s)\,ds +  \int_0^t R_0(t-s)F(s)\xi\,ds\,,
\end{split}
\end{equation*}
where the memory term does not contain differential operators.

\begin{remark}
\begin{rm}
The computations carried out so far---known as MacCamy's trick
\cite{maccamy_1977}---are purely formal, since the solutions to the equation \eqref{eq:MEMORY}
have not yet been defined.
\end{rm}
\end{remark}

The obtained equation is a wave equation perturbed by a persistent memory, namely,
\begin{equation*}
\begin{split}
u_{tt} & = b(\Delta-I) u + (R_0'(0)+b) u(t) + R_0(0)u_t(t) + \int_0^t R_0''(t-s)u(s)\,ds-
\\[1mm]
& \qquad\qquad\qquad -R_0'(t)u_0 - R_0(t) u_1 + \Big\{F(t)\xi -\int_0^t R_0(t-s)F(s)\xi\,ds\Big\}\,.
\end{split}
\end{equation*}
The introduction of the function
\begin{equation} \label{e:variable-v}
v(t) = e^{-\frac{1}{2}R_0(0)t} u(t)
\end{equation}
enables us to eliminate the term $R_0(0)u_t$, and to attain the following equation in the unknown $v$:
\begin{equation} \label{eq:MEMORY-for-v}
v_{tt}=b (\Delta v-v)+ \int_0^t K(t-s)v(s)\,ds + \beta v(t)
+ \big(h_2(t)\xi+h_1(t)u_1+ h_0(t) u_0\big)\,,
\end{equation}
with the constant $\beta$ and the functions $K(\cdot)$, $h_i(\cdot)$, $i=0,1,2$ given
by the formulas below:
\begin{equation}\label{e:various-functions}
\begin{split}
& \text{$K(t) = e^{-\frac{1}{2}R_0(0)t} R_0''(t)$ is square integrable in $(0,T)$;}
\\
& \beta=b+\frac{1}{4}R_0^2(0)+R_0'(0)\,;
\\
& h_0(t)=e^{-\frac12 R_0(0) t} R_0'(t)\in H^1(0,T)\,;
\\
& h_1(t)=e^{-\frac12 R_0(0) t} R_0(t)\in H^2(0,T)\,;
\\
& \text{$h_2(t)= e^{-\frac12 R_0(0) t}\big(F(t)-\int_0^t R_0(t-s) F(s)\,ds\big)$
is square integrable.}
\end{split}
\end{equation}
The above suggests the following Definition, which is rigorously justified in the Appendix.

\begin{definition} \label{d:def-solution}
Let ${\mathcal H}$ be a Hilbert space.
An ${\mathcal H}$-valued function $t\longmapsto u(t)$ is a solution of equation \eqref{eq:MEMORY}
supplemented with initial/boundary conditions \eqref{eq:dataDIe:memory} if the function
$t\longmapsto v(t)$ defined in \eqref{e:variable-v} is an ${\mathcal H}$-valued
continuous function which solves the Volterra integral equation \eqref{eq:MEMORY-for-v},
with $\beta$, $K(\cdot)$, $h_i(\cdot)$, $i=0,1,2$ defined by \eqref{e:various-functions}.
\end{definition}

\begin{remark}
\begin{rm}
In the case $F(t)\equiv N(t)=e^{-\alpha t}$, then the above definition yields the definition
of solutions to the MGT equation \eqref{e:mgt},
with initial/boundary conditions \eqref{e:IC}-\eqref{e:BC}.

\end{rm}
\end{remark}

On the basis of Definition~\ref{d:def-solution} we are led to study the regularity of solutions the following IBVP for the wave equation with memory \eqref{eq:MEMORY-for-v}, that is:
\begin{equation}\label{ibvp-for-v}
\begin{cases}
v_{tt}=b (\Delta v-v)+ \int_0^t K(t-s)v(s)\,ds + \beta v(t)
+ \big(h_2(t)\xi+h_1(t)u_1+ h_0(t) u_0\big)
\\[1mm]
v(0,\cdot)=v_0\,, \; v_t(0,\cdot)=v_1
\\[1mm]
{\mathcal T} v=e^{-\frac{1}{2}R_0(0)t} g\,,
\end{cases}
\end{equation}
where initial data are related to the ones of $u$ via the following relations:
\begin{equation} \label{eq:datiUdatiV}
v_0=u_0\,, \qquad v_1= u_1-\frac{1}{2}R_0(0)u_0\,.
\end{equation}

The next Proposition connects the IBVP \eqref{ibvp-for-v} to a Volterra equation of the
second kind, with suitable kernel and affine term.

\begin{proposition} \label{p:volterra}
Any solution to the initial/boundary value problem \eqref{ibvp-for-v} solves the Volterra
equation
\begin{equation} \label{eq:represent}
v(t)+\int_0^t L(t-s)v(s)\,ds=H(t)\,,
\end{equation}
where $L(\cdot)$ is a strongly continuous kernel defined by
\begin{equation} \label{eq:defiL}
L(t)v=-\frac{\beta}{\sqrt b} {\mathcal A}^{-1}R_-(\sqrt b t) v
-\frac{1}{\sqrt b}{\mathcal A}^{-1}\int_0^t R_-(\sqrt b(t-s)) K(s)v\,ds
\end{equation}
(and $K(\cdot)$ is defined explicitly in \eqref{e:various-functions}), while
the affine term $H(\cdot)$ is given by
\begin{equation} \label{eq:defiH}
\begin{split}
H(t)&=\Big[R_+(\sqrt{b}t)-\frac{R_0(0)}{2\sqrt{b}}{\mathcal A}^{-1}R_-(\sqrt{b}t)\Big]u_0
+\frac{1}{\sqrt{b}}{\mathcal A}^{-1} R_-(\sqrt{b}t) u_1-
\\[1mm]
& \qquad -\sqrt{b} {\mathcal A}\int_0^t R_-(\sqrt{b}(t-s)) G\,e^{-\frac{1}{2}R_0(0)s} g(s)\,ds+
\\[1mm]
& \qquad +\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))
\big[h_2 (s)\xi+ h_1(s)u_1+h_0(s)u_0\big]\, ds\,.
\end{split}
\end{equation}
We recall once more that $R_0(\cdot)$ is the (scalar) resolvent kernel defined---in terms of
$N(\cdot)$---by the integral equation \eqref{e:resolvent-kernel}.
\end{proposition}

\begin{proof}
The proof is straightforward, in view of formula \eqref{e:waves-explicit} for the solution
to wave equations with initial and boundary data.
We just recall here that the abstract operator $A$ is the realization of the differential operator $\Delta -I$ with boundary conditions driven by ${\mathcal T}$, while $R_+(\sqrt{b}t)$, the {\em cosine} operator generated by $bA$, and $R_-(\sqrt{b}t)$ are defined in~(\ref{eq:defiOperRpm}).
We finally note that $H(\cdot) z\in C([0,T];X_\alpha)$, for every $z\in X_\alpha$.
\end{proof}

\section{Interior regularity for the equation \eqref{e:memory}}  \label{s:main_1}
In this Section we see how the regularity results pertaining to wave equations stated
in Theorem~\ref{teo:propertyONDE} can be suitably extended to the general equation with
memory of the form \eqref{eq:MEMORY}.
This will eventually imply the {\em stronger} regularity of solutions to the third order
MGT equation \eqref{e:mgt} (see the next Section).

\smallskip
The key and starting point is the Volterra integral equation \eqref{eq:represent} in the
unknown $v$.
Its kernel $L(\cdot)$ is now operator valued and \emph{strongly continuous} from
$[0,+\infty)$ to ${\mathcal L}(X_\alpha)$ for every $\alpha$.
By using Theorem~\ref{teo:propertyONDE} we will derive the regularity properties of the
right hand side of \eqref{eq:represent}, that will be inherited by $v$ and then by the
solutions to the wave equation with memory \eqref{eq:MEMORY}.
These properties will be expressed in terms of the boundary datum $g$, as well as of
initial data $u_0$, $u_1$ and $\xi$.

It is convenient to write explicitly the solution of a Volterra integral equation in a Hilbert space ${\mathcal H}$.
We introduce the notation $*$ for the convolution,
\begin{equation*}
L * h=\int_0^t L(t-s)h(s)\,ds=\int_0^t L(s)h(t-s)\,ds\,.
\end{equation*}
Here $L(t)$ is a strongly continuous function of time, with values in ${\mathcal L}({\mathcal H})$ and
$h(t)$ is an integrable ${\mathcal H}$-valued function.
\\
Moreover, let $L^{(*n)}$ denote iterated convolutions, recursively defined by the following equalities
\begin{equation*}
L^{(*1)}=L\,,\quad  L^{(*(n+1))}*h=L*\Big( L^{(*n)}*h\Big)
\end{equation*}
(for every integrable ${\mathcal H}-$valued function $h$).
Then, the solution to the Volterra equation \eqref{eq:represent}---that is $v+L*v=H$, in
short---is
\begin{equation*}
v=H+\sum _{k=2}^{\infty} L^{(*k)}*H\,.
\end{equation*}
Uniform convergence of the series is easily proved.
In the special case of our interest, ${\mathcal H}=X_\alpha$ and $L(t)$ is given by \eqref{eq:defiL}.
The following result is well known.

\begin{lemma} \label{lemmaVOLTE}
Let $T>0$ and let the kernel $L(\cdot)$ be given by \eqref{eq:defiL}.
If $H\in C([0,T];X_\alpha)$, then the solution $v$ of the Volterra equation
$v+L*v=H$ satisfies $v\in C([0,T];X_\alpha)$.
\end{lemma}

We will repeatedly use Lemma~\ref{lemmaVOLTE} in order to pinpoint the regularity of the solutions
to the initial/boundary value problems associated with the equation \eqref{eq:MEMORY}.

\smallskip

\begin{theorem}[Regularity for equation \eqref{eq:MEMORY}]\label{t:main_1}
Consider Eq.~\eqref{eq:MEMORY} with initial data $(u_0,u_1)$ and boundary
data defined by \eqref{e:BC}.
Then, the regularity of the (linear) map $(u_0,u_1,\xi,g) \longmapsto (u,u_t,u_{tt})$
is detailed in Table~\ref{tableGENEresu}.
\begin{table}[h]
\begin{center}

\begin{tabular}{|c|c|c|c|c|}
\hline
&&&&\\
$u_0$ & $u_1$ & $\xi$ & $g $ & $u=u(t,x)$ solution of \eqref{eq:MEMORY}\\
&&&&
\\

 &&&&
\\
 \hline
&&&&
\\
$X_\lambda$ &    $0$ &    $0$ &    $0$ &
$C([0,T];X_\lambda)\cap C^1([0,T];X_{\lambda-1})\cap C^2([0,T];X_{\lambda-2})$
\\
&&&&\\
\hline
&&&&\\
$0$ &   $X_\mu$ & $0$ &   $0$ & $C([0,T];X_{\mu+1})\cap C^1([0,T];X_\mu)\cap C^2([0,T]; X_{\mu-1})$
\\
&&&&\\
\hline
&&&&\\
$0$ & $0$ & $X_\nu$ & $0$ &$\left\{\begin{array}{l}
\mbox{if $F(t)\in L^2(0,T)$   then:}\\
  C([0,T];X_{\nu+1})\cap C^1([0,T]; X_{\nu })
\cap H^2([0,T]; X_{\nu-1})\,; \\
\mbox{if $F(t)\in H^1(0,T)$ then:}\\
   C([0,T];X_{\nu+2})\cap C^1([0,T]; X_{\nu+1})
\cap H^2([0,T]; X_\nu)
\end{array}\right.$
\\
&&&&\\
  \hline
  &&&&\\
 $0$ &    $0$ &$0$ & $ L^2(\Sigma) $ &
$C([0,T];X_{\alpha_0})\cap C^1([0,T];X_{\alpha_0-1 })\cap H^2([0,T]; X_{\alpha_0-2})$
\\
&&&&\\
  \hlin
\end{tabular}
\caption{Regularity of solutions to the equation \eqref{eq:MEMORY}. The transformations are continuous between the indicated spaces.}
\label{tableGENEresu}
\end{center}
\end{table}

\end{theorem}
\begin{proof}
The proof of the several statements contained in Table~\ref{tableGENEresu}
is structured in few major steps.

\smallskip
\paragraph{\bf 0. Premise and outline.}
Consider the Volterra equation \eqref{eq:represent}, and note that the functions
$v_t(t)$ and $v_{tt}(t)$ solve the same Volterra integral equation of the second
kind in a Hilbert space, yet with different affine terms, $H_1(\cdot)$ and $H_2(\cdot)$ say,
respectively, which will be computed in the next step.
In view of Lemma~\ref{lemmaVOLTE}, the (time and space) regularity of these affine terms---depending on $u_0$, $u_1$, $\xi$ and $g$---will naturally bring about the (time and space) regularity for the triple $(v,v_t,v_{tt})$.
\\
To do so we will set to zero all data but one.
Finally, the derived regularity properties will be inherited by the triple
$(u,u_t,u_{tt})$ pertaining to the original equation with persistent memory
\eqref{eq:MEMORY}, still depending on $u_0$, $u_1$, $\xi$ and $g$.

\smallskip
\paragraph{\bf 1. The affine terms of Volterra equations.}
We rewrite \eqref{eq:represent} in the form
\begin{equation*}
v(t)+\int_0^t L(s)v(t-s) ds= H(t)
\end{equation*}
and compute the derivatives of both the sides.
Inserting the expressions \eqref{eq:defiL} and \eqref{eq:defiH} of $L(t)$ and $H(t)$, and
replacing initial data $v_0$ and $v_1$ with their respective expressions in terms of $u_0$
and $u_1$ (see \eqref{eq:datiUdatiV}), we obtain the following Volterra integral equations
in the unknowns $v_t(t)$ and $v_{tt}(t)$:
\begin{align}
v_t(t)+\int_0^t L(t-s)v_s(s)\,ds= H_1(t)\,,
\label{eq:deriPRIMA}
\\[1mm]
v_{tt}(t)+\int_0^t L(t-s)v_{ss}(s)\,ds = H_2(t)
\label{eq:deriSEconda}
\end{align}
where
\begin{align}
\nonumber & H_1(t):= L(t)v_0+H_t(t)=
\\
&=\Big[\frac{\beta}{\sqrt{b}}{\mathcal A}^{-1} R_-(\sqrt{b} t)u_0 +
  \frac{1}{\sqrt b}\int_0^t R_-(\sqrt{b}(t-s)){\mathcal A}^{-1}K(s)u_0\,ds\Big] +H_t(t)\,,
\label{e:y_1}
\\[2mm]
& H _2(t):= \Big[\frac{\beta}{\sqrt b}{\mathcal A}^{-1} R_-(\sqrt{b}t)u_1
+\frac{1}{\sqrt b} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))K(s)u_1\,ds \Big] -
\nonumber \\[1mm]
& \qquad\qquad-\frac{R_0(0)}{2\sqrt b}\Big[\beta{\mathcal A}^{-1}R_-(\sqrt{b}t)u_0
+{\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))K(s)u_0 \,ds \Big]+
\nonumber \\[1mm]
& \qquad\qquad\qquad +\beta R_+(\sqrt{b}t)u_0+\int_0^t R_+(\sqrt{b}(t-s)) K(s)u_0\,ds +H_{tt}(t)\,,
\nonumber
\end{align}
while the explicit expression \eqref{eq:defiH} of $H(t)$ is recorded here
for the reader's convenience:
\begin{equation*}
\begin{split}
H(t)&=\Big[R_+(\sqrt{b}t)-\frac{R_0(0)}{2\sqrt{b}}{\mathcal A}^{-1}R_-(\sqrt{b}t)\Big]u_0
+\frac{1}{\sqrt{b}}{\mathcal A}^{-1} R_-(\sqrt{b}t) u_1-
\\[1mm]
& \qquad -\sqrt{b} {\mathcal A}\int_0^t R_-(\sqrt{b}(t-s)) G\,e^{-\frac{1}{2}R_0(0)s} g(s)\,ds+
\\[1mm]
& \qquad +\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))
\big[h_2 (s)\xi+ h_1(s)u_1+h_0(s)u_0\big]\, ds\,.
\end{split}
\end{equation*}
As it will appear clear immediately below, we neglected to write explicitly the derivatives
of $H(t)$, just because their regularity is easily deduced invoking once more
Theorem~\ref{teo:propertyONDE}.

\smallskip
\paragraph{\bf 2a. Effects of boundary data action.}
With $u_0,\,u_1,\,\xi\equiv 0$, $g\in L^2(\Sigma)$, the affine term $H(t)$
in \eqref{eq:represent} (recorded above) reduces to
\begin{equation}
H(t)=-\sqrt{b}{\mathcal A}\int_0^t R_-(\sqrt{b}(t-s))Gg(s)\, ds\,.
\end{equation}
Therefore we know from assertion iii) \eqref{teo:propertyONDE-3} of
Theorem~\ref{teo:propertyONDE} that
\begin{equation*}
(H,H_t,H_{tt})\in C([0,T];X_{\alpha_0}\times X_{\alpha_0-1}\times X_{\alpha_0-2})\,.
\end{equation*}
Thus, Lemma~\ref{lemmaVOLTE} shows that the solutions of the Volterra
equation \eqref{eq:represent} as well as those pertaining to the former
equation with memory \eqref{eq:MEMORY}
belong to
\begin{equation*}
C([0,T];X_{\alpha_0})\cap C^1 ([0,T];X_{\alpha_0-1})\cap C^2([0,T];X_{\alpha_0-2})\,.
\end{equation*}

\smallskip
\paragraph{\bf 2b. Effects of the initial datum $u_0$.}
Assume $u_1, \,\xi=0$, $g=0$, and $u_0\in X_\lambda$.
The affine term of the equation \eqref{eq:represent} in the unknown $v$ becomes
\begin{equation*}
H(t) = \Big[R_+(\sqrt{b}t)-\frac{R_0(0)}{2\sqrt{b}}{\mathcal A}^{-1}R_-(\sqrt{b}t)\Big]u_0
+\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s)) h_0(s)u_0\, ds\,,
\end{equation*}
so that readily
\begin{equation*}
H\in C([0,T];X_\lambda)\cap C^1([0,T];X_{\lambda-1})\cap C^2([0,T];X_{\lambda-2})\,,
\end{equation*}
which immediately implies $v(t)\in C([0,T];X_\lambda)$.
Recall now   the term $H_1$ in \eqref{e:y_1} and notice that its regularity is determined
by the regularity of $H_t$.
Then, $H_1$---as well as $v_t$, in view of Lemma~\ref{lemmaVOLTE}---belongs to
$C^1([0,T];X_{\lambda-1})$, while $H_2$ and then $v_{tt}(t)$ belong to
$C^1([0,T];X_{\lambda-2})$, which establishes the first row of Table~\ref{tableGENEresu}.

\smallskip
\paragraph{\bf 2c. Effect of the initial datum $u_1$.}
Assume $u_0,\ \xi=0$, and $g=0$ while $u_1\in X_\mu$.
In this case
\begin{equation*}
H(t) = \frac{1}{\sqrt{b}}{\mathcal A}^{-1} R_-(\sqrt{b}t) u_1
+\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s)) h_1(s)u_1\, ds\,,
\end{equation*}
so that we have a slight regularization
\begin{equation*}
(H,H_t,H_{tt})\in C([0,T];X_{\mu +1}\times X_\mu\times X_{\mu-1});
\end{equation*}
the transformation $u_1\longmapsto H$ is continuous in the indicated spaces
({\em cf.}~assertion {\em iii)}
of Theorem~\ref{teo:propertyONDE}).
\\
The obtained regularity for $H$ and its derivatives holds for $H_i$, $i=1,2$, and then
is inherited by the solution $v(t)$: namely,
\begin{equation*}
v\in C([0,T];X_{\mu+1})\cap C^1([0,T];X_\mu)\cap C^2([0,T]; X_{\mu-1})\,;
\end{equation*}
in turn, the same is valid for $u$, thereby confirming the second row of
Table~\ref{tableGENEresu}.

\smallskip
\paragraph{\bf 2d. Effect of the parameter $\xi$.}
We finally discuss the dependence on $\xi$.
Assume $u_0, u_1=0$, and $g=0$ and $\xi\in X_\nu$. In this case
\begin{equation*}
H(t)=\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s)) h_2 (s)\xi \,ds
\end{equation*}
and, from \eqref{e:various-functions}, $h_2(t)\in L^2(0,T)$, just like $F(t)$.

We invoke once more item {\em ii)}
of Theorem~\ref{teo:propertyONDE}, and ascertain again a slightly regularizing property: the transformation $\xi\longmapsto v$ is continuous from $X_\nu$ to
$C([0,T];X_{\nu+1})\cap C^1([0,T];X_\nu)\cap H^2([0,T]; X_{\nu-1})$
(while if in addition $F(t)$---and consequently, $h_2(t)$---is continuous, then $v\in C^2([0,T];X_{\nu-1})$).

In the case $F\in H ^2(0,T)$ (as the case of the MGT equation) we have a stronger regularization, since we can integrate by parts as follows:
\begin{equation}
\begin{split}
H(t) &=- \frac{1}{b}\,A^{-1}\int_0^t \frac{d}{ds }R_+(\sqrt{b}(t-s)) h_2 (s)\xi \, ds=
\\[1mm]
& \qquad=- \frac{1}{b}\,A^{-1}\Big[\big(h_2(t)-R_+(\sqrt{b}t) h_2(0)\big)\xi
-\int_0^t  R_+(\sqrt{b}(t-s)) h_2'(s)\xi \,ds\Big]\,;
\end{split}
\end{equation}
a rigorous justification is found, e.g., in \cite[Lemma~5]{PandAMO}.

For a better understanding, we compute explicitly
\begin{equation*}
\begin{split}
H_t(t) &=-\frac{1}{b}\, A^{-1}\Big[\cancel{h_2'(t)\xi}-\sqrt{b}{\mathcal A} R_-(\sqrt{b}t)\xi
-\cancel{h_2'(t)\xi}-
\\[1mm]
& \qquad\qquad\qquad -\sqrt{b}{\mathcal A} \int_0^t R_-(\sqrt{b}(t-s)) h_2' (s)\xi \,ds\Big]=
\\[1mm]
&= \frac{1}{b} {\mathcal A}^{-1} R_-(\sqrt{b}t)\xi + \frac{1}{b} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s)) h_2'(s)\xi \,ds\in X_{\nu+1}\,,
\end{split}
\end{equation*}
and
\begin{equation*}
H_{tt}(t) =R_+(\sqrt{b}t)\xi + \int_0^t R_+(\sqrt{b}(t-s)) h_2'(s)\xi \,ds\in X_{\nu}\,,
\end{equation*}
that complete the membership $H\in X_{\nu+2}$.

It is important to note that the space regularity increases of one unity and we get the
result in the third row of Table~\ref{tableGENEresu}, where $H^2$ is replaced with $C^2$
if furthermore $F\in C^2(0,T)$, that is the case of the MGT equation.
\end{proof}

\begin{remark}
\begin{rm}
The noticeable outcome of the obtained regularity result is that $u_1$ and $\xi$ are regularized by one and, respectively, two unities.
Hence, when $g=0$, $u_0=0$ while $u_1$ and $\xi$ belong to $X_r$, then
$(u(t),u_t(t),u_{tt}(t))$ evolves in $X_{r+1}\times X_r\times X_{r-1}$.
\end{rm}
\end{remark}

From Table~\ref{tableGENEresu} of Theorem~\ref{t:main_1} we deduce, in particular,
the following regularity result.

\begin{corollary}\label{c:example}
Consider equation \eqref{eq:MEMORY} with initial data $(u_0,u_1)$, and homogeneous
boundary data, namely, $g\equiv 0$ in \eqref{e:BC}.
Then, if $F(\cdot)\in C^1$ and if $(u_0,u_1,\xi)\in X_r\times X_{r-1} \times X_{r-2}$,
then the corresponding weak solution satisfies
\begin{equation*}
(u,u_t,u_{tt})\in C([0,T];X_r\times X_{r-1}\times X_{r-2})\,.
\end{equation*}

\end{corollary}

\section{Interior regularity for the MGT equation}\label{sect:RegulaMGT}

In this Section we utilize the analysis performed for the general class of equations
with memory \eqref{eq:MEMORY}, in order to derive a result pertaining
to the MGT equation, that is Theorem~\ref{t:main_2} below.
This Theorem establishes, in particular, the statements of Theorem~\ref{t:sample} detailing
the regularity from the boundary to the interior for the MGT equation (i.e. item {\em iii)}),
as well as the one pertaining to the interior regularity, under homogeneous boundary data
(i.e. item {\em ii)}).
The latter result is consistent with the analyis formerly carried out in
\cite{kalt-etal_2011}, that brought about semigroup well-posedness
of the MGT equation in the space ${\mathcal D}(A^{1/2})\times {\mathcal D}(A^{1/2}) \times L^2(\Omega)$,
$A$ being the proper realization of the Laplacian in $L^2(\Omega)$;
see \cite[Theorem~1.2]{kalt-etal_2011}.
The peculiar regularity of the MGT equation is here (re)confirmed in a wealth of functional
settings.

Recall that for the special case of the MGT equation we have in particular
\begin{equation*}
N(t)=F(t) = e^{-\alpha t}\,, \qquad \xi = u_2-b\Delta u_0
\end{equation*}
in \eqref{eq:MEMORY}.
The meaning given to solutions is still the one   in Definition~\ref{d:def-solution}.

We restart from the integral equation which defines the resolvent
associated with the convolution kernel $-\gamma N(t)$ of \eqref{eq:MEMORY},
that is equation \eqref{e:resolvent-kernel} and that in the present case---with
$N(t)= e^{-\alpha t}$---reads as
\begin{equation}\label{e:resolvent-eq-mgt}
R_0(t)-\gamma \int_0^t e^{-(t-s)}\, R_0(s)\, ds= - \gamma e^{-\alpha t}\,.
\end{equation}
It is then easily verified that the solution to \eqref{e:resolvent-eq-mgt} is given by
\begin{equation}\label{e:kernel-mgt}
R_0(t)= -\gamma e^{(\gamma-\alpha)t}=-\gamma e^{-\frac{c^2}{b}t}\,,
\end{equation}
which gives $R_0(0)= -\gamma$ and hence
\begin{equation*}
\qquad v(t)= e^{\frac{\gamma}2 t}u(t)
\end{equation*}
for $v$ defined in \eqref{e:variable-v}).

\smallskip
In view of Definition~\ref{d:def-solution}, and taking into account the actual expression
of $R_0(t)$---depending on $N(t)$ and $F(t)$---in \eqref{e:kernel-mgt}, the following instance
of Definition~\ref{d:def-solution} comes into the picture.

\begin{definition}[Instance of Definition~\ref{d:def-solution}]
A function $u=u(t,x)$ is a solution of the IBVP \eqref{e:mgt}-\eqref{e:IC}-\eqref{e:BC}
for the Moore-Gibson-Thompson equationif and only if the function
$v(t,x) = e^{\frac{\gamma}{2}t}u(t,x)$ solves
\begin{equation*}
\begin{split}
v_{tt} &=b (\Delta v-v)+ \int_0^t K(t-s)v(s)\,ds + \beta v(t)
\\[1mm]
& \qquad \qquad + h_2(t)\big(u_2-b\Delta u_0\big)+h_1(t)u_1+ h_0(t) u_0\,,
\end{split}
\end{equation*}
where
\begin{equation*
\begin{split}
& K(t) = -\gamma (\gamma-\alpha)^2 e^{(\frac{3}{2}\gamma-\alpha)t}\,,
\qquad \beta=b-\gamma\Big(\frac{3}{4}\gamma -\alpha\Big)\,,
\\
& h_0(t)=-\gamma (\gamma-\alpha) e^{(\frac{3}{2}\gamma -\alpha)t}\,,
\quad
h_1(t)=-\gamma e^{(\frac{3}{2}\gamma -\alpha)t}\,,
\quad
h_2(t)= e^{(\frac{3}{2}\gamma-\alpha)t}\,.
\end{split}
\end{equation*}

\end{definition}

\medskip
Thus, on the basis of Theorem~\ref{t:main_1}, we develop a picture of the interior
regularity for the MGT equation.

\begin{theorem}[Regularity for the MGT equation]\label{t:main_2}
The regularity of the map $(u_0,u_1,u_2,g)\longmapsto u$ that associates to initial and
boundary data the solution $u=u(t,x)$
to the Moore-Gibson-Thompson equation \eqref{e:mgt} is described by the Table~\ref{tableMGTresu}.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hlin
&&&&\\
$u_0$ & $u_1$ & $u_2$ & $g $ & $u=u(t,x)$ solution of the MGT equation\\
&&&&
\\
 &&&&
\\
\hline
&&&&
\\
$X_\lambda$ &    $0$ &    $0$ &    $0$ &
$C([0,T];X_\lambda)\cap C^1([0,T];X_{\lambda })\cap C^2([0,T];X_{\lambda-1})$
\\
&&&&\\
\hline
&&&&\\
$0$ &   $X_\mu$ & $0$ &   $0$ & $C([0,T];X_{\mu+1})\cap C^1([0,T];X_\mu)\cap C^2([0,T]; X_{\mu-1})$
\\
&&&&\\
\hline
&&&&\\
$0$ & $0$ & $X_\nu$ & $0$ &  $C([0,T];X_{\nu+2})\cap C^1([0,T]; X_{\nu+1})\cap C^2([0,T]; X_{\nu })$
\\
&&&&\\
  \hline
  &&&&\\
 $0$ &    $0$ &$0$ & $ L^2(\Sigma) $ &
$C([0,T];X_{\alpha_0})\cap C^1([0,T];X_{\alpha_0-1 })\cap H^2([0,T]; X_{\alpha_0-2})$
\\
&&&&\\
  \hlin
\end{tabular}
\caption{Regularity of solutions to the equation \eqref{e:mgt}. The transformations are continuous between the indicated spaces.}
\label{tableMGTresu}
\end{center}
\end{table}

\end{theorem}
\begin{proof}
Along the lines of the first steps of the proof of Theorem~\ref{t:main_1}, we return to
the Volterra equation \eqref{eq:represent} and note that the affine term $H(t)$ in
\eqref{eq:defiH} must be rewritten taking into account that in the present case
we have $\xi=u_2-b\Delta u_0$.
Let us focus on the last summand of $H(t)$, that is
\begin{equation*}
\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))
\big[h_2 (s)\big(u_2-b\Delta u_0\big)+ h_1(s)u_1+h_0(s)u_0\big]\, ds\,,
\end{equation*}
and more specifically on the term
\begin{equation*}
T(t)=\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)[u_2-b\Delta u_0]\,ds\,.
\end{equation*}
We rewrite
\begin{equation}\label{e:t}
\begin{split}
T(t)&=\underbrace{\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)u_2\,ds}_{T_1(t)}-
\\[1mm]
& \qquad \underbrace{-\sqrt{b} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)\Delta u_0\,ds}_{T_2(t)}
\end{split}
\end{equation}
and compute
\begin{equation}\label{e:t_2}
\begin{split}
T_2(t)&=-\sqrt{b} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)\big(\Delta u_0-u_0+u_0\big)\,ds=
\\[1mm]
&=\underbrace{-\sqrt{b} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)
Au_0\,ds}_{T_{21}(t)}-
\\[1mm]
& \qquad\qquad\qquad
\underbrace{-\sqrt{b} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)u_0\,ds}_{T_{22}(t)}\,.
\end{split}
\end{equation}
Assuming $u_0\in X_\lambda$, then $Au_0\in X_{\lambda-2}$; moreover, recall that
$A={\mathcal A}^2$, and the relation between the operators $R_-(\cdot)$ and $R_+(\cdot)$.
Then proceed with the computations, integrating by parts to get
\begin{equation}\label{e:t_21}
\begin{split}
T_{21}(t)& =-\sqrt{b} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)Au_0\,ds
\\[1mm]
& = -\sqrt{b} {\mathcal A}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)u_0\,ds=
\\[1mm]
& =\int_0^t \frac{d}{ds} \big\{R_+(\sqrt{b}(t-s)) h_2 (s)u_0\Big\}\,ds
- \int_0^t R_+(\sqrt{b}(t-s))h_2'(s)u_0\,ds=
\\[1mm]
& =h_2(t)u_0-R_+(\sqrt{b}(t))h_2(0)u_0
-\int_0^t R_+(\sqrt{b}(t-s))h_2'(s)u_0\,ds\,.
\end{split}
\end{equation}

Combine \eqref{e:t_21} with \eqref{e:t_2} and \eqref{e:t}, insert the resulting expression of
$T(t)$ in $H(t)$, to obtain
\begin{equation*}
\begin{split}
H(t)& =\cancel{R_+(\sqrt{b}t)u_0}-\frac{R_0(0)}{2\sqrt{b}}{\mathcal A}^{-1} R_-(\sqrt{b}t)u_0
+\frac{1}{\sqrt{b}}{\mathcal A}^{-1} R_-(\sqrt{b}t) u_1-
\\[1mm]
& \qquad -\sqrt{b} {\mathcal A}\int_0^t R_-(\sqrt{b}(t-s)) G\,e^{-\frac{1}{2}R_0(0)s} g(s)\,ds+
\\[1mm]
& \qquad
+\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)u_2\, ds+
\\[1mm]
& \qquad
+ h_2(t) u_0\cancel{- R_+(\sqrt{b}t)u_0}
- \int_0^t R_+(\sqrt{b}(t-s))h_2'(s)u_0\, ds-
\\[1mm]
& \qquad
- \sqrt{b} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)u_0\,ds+
\\[1mm]
& \qquad
+ \frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))\big[h_1(s)u_1+h_0(s)u_0\big]\,ds\,,
\end{split}
\end{equation*}
where the term $R_+(\sqrt{b}t)u_0$ appears twice with opposite signs, and hence cancel.

Rearranging the summands and replacing $t-s$ with $s$ in the integrals we attain
\begin{equation}\label{e:h}
\begin{split}
H(t)& =\Big(h_2(t) -\frac{R_0(0)}{2\sqrt{b}}{\mathcal A}^{-1} R_-(\sqrt{b}t)\Big) u_0
- \int_0^t R_+(\sqrt{b}s)h_2'(t-s)u_0\, ds+
\\[1mm]
& \qquad
+ {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}s)
\Big(\frac{1}{\sqrt{b}} h_0(t-s)-\sqrt{b} h_2 (t-s)\Big)u_0\,ds+
\\[1mm]
& \qquad
+\frac{1}{\sqrt{b}}{\mathcal A}^{-1} R_-(\sqrt{b}t) u_1
+ \frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}s)h_1(t-s)u_1\,ds-
\\[1mm]
& \qquad
+\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}s)h_2 (t-s)u_2\, ds \\[1mm]
& \qquad -\sqrt{b} {\mathcal A}\int_0^t R_-(\sqrt{b}(t-s)) G\,e^{-\frac{1}{2}R_0(0)s} g(s)\,ds\,,
\end{split}
\end{equation}
which allows the understanding of the regularity of $H(t)$, along with the sought regularity properties of solutions to the MGT equation.

\smallskip
Notice first that in comparison with the general model equation with memory \eqref{e:memory}
the space regularity of $H(t)$ is not improved, owing to the presence of the term $h_2(t)u_0$.
Instead, the regularity of $H_t(t)$ is improved thanks to the cancellation of the
term $R_+(\sqrt{b}t)u_0$: in fact, if $g\equiv 0$, $u_1=u_2=0$, $u_0\in X_\lambda$,
then $H_t\in C([0,T];X_\lambda)$, while $H_{tt}\in C([0,T];X_{\lambda-1})$.
However, the said cancellation (of a term depending only on $u_0$) has no effect on the remaining terms: the dependence on $u_1$ and $u_2$ is subject to the smoothing effect already described in Table~\ref{tableGENEresu} (in terms of $u_1$ and $\xi$).
Thus, the results displayed in Table~\ref{tableMGTresu} follow.
(The cancellation has also another significant effect: the summand $h_2(t)u_0$ decays in time, but does not propagate in space, as explained in Remark~\ref{Rem:nonpropagazione}.
Observe that in the term
\begin{equation*}
\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)u_2\, ds
\end{equation*}
one may integrate by parts, thereby obtaining
\begin{equation*}
\begin{split}
&\frac{1}{\sqrt{b}} {\mathcal A}^{-1}\int_0^t R_-(\sqrt{b}(t-s))h_2 (s)u_2\, ds
= -\frac{1}{b} {\mathcal A}^{-2}\Big\{h_2 (t)u_2-R_+(\sqrt{b}(t)h_2 (0)u_2
\\[1mm]
&  \qquad\qquad\qquad
+ \int_0^t R_+(\sqrt{b}(t-s))h_2' (s)u_2\, ds\Big\}
\end{split}
\end{equation*}
that confirms the said smoothing effect.

Using once more that the functions $h_i(t)$, $i=0,1,2$ are twice differentiable,
it is easily seen that when $u_0\in X_\lambda$, $u_1=u_2=0$, $g=0$, then
\begin{equation}\label{e:anomaly}
H(t)\in  C([0,T];X_\lambda)\cap C^1([0,T];X_\lambda)\cap C^2([0,T];X_{\lambda-1})\,;
\end{equation}
the regularity of $v$ and the one of $u$ follow accordingly.
In conclusion, the   representation  \eqref{e:h} of  $H(t)$ shows that all the regularity results
summarized in the rows of Table~\ref{tableGENEresu} remain valid, with the exception
of those in the first row, that are improved consistently with \eqref{e:anomaly}.
\end{proof}

\begin{remark}
\begin{rm}
We note that in particular the regularity of the mapping
\begin{equation*}
g \longmapsto (u,u_t,u_{tt}) \qquad \text{(assuming $u_0=u_1=u_2=0$)}
\end{equation*}
for the Moore-Gibson-Thompson equation is the same as in the case of the wave equation
(as well as of the equation with memory \eqref{e:memory}).
Hence, the last row in Table~\ref{tableMGTresu}, explicitly stated in {\em iii)} of
Theorem~\ref{t:sample}, follow readily from Theorem~\ref{t:tataru}.
\end{rm}
\end{remark}

\begin{remark}\label{Rem:nonpropagazione}
\begin{rm}
We note that $R_-(\sqrt{b}t)u_0$ and $R_+(\sqrt{b}t)u_0$ solve the wave equation, and so
the `shape' of $u_0$ is propagated in space, as in the wave equation.
Instead, the term $h_2(t)u_0$ (which decays exponentially in time) is a stationary wave
and does not propagate in the space variables.
\\
Thanks to the formulas for the solutions of the Volterra integral equations, this stationary wave appears also in the solution of the MGT equation.
\end{rm}
\end{remark}

\section{Boundary regularity} \label{s:traces}
In this Section we establish a sharp regularity result for the normal trace
on $\Gamma=\partial \Omega$ of solutions to the the MGT equation \eqref{e:mgt},
supplemented with (homogeneous) Dirichlet boundary condition.
This result, presented as Corollary~\ref{c:traces-mgt}, follows from a boundary regularity
result pertaining to the family of wave equations with memory \eqref{e:memory}, depending
on $\xi\in L^2(\Omega)$, that is Theorem~\ref{t:traces-memory} below.
In doing so we re-obtain, in the case $\xi=0$, a result established only recently in
\cite{loreti-sforza_parma} via multiplier techniques, that is Theorem~1.1 therein.

We point out that the present approach to the analysis of wave equations with
memory enable us to pinpoint the boundary (beside the interior) regularity of solutions
in a direct and straightforward way.
The tools employed are the ones of operator and semigroup theories, along with the
view of Volterra equations; the regularity results for wave equations already available in the literature play a crucial role.
\\
Thus, our method of proof paves the way for the derivation of appropriate boundary regularity
results for the model equation with memory \eqref{e:memory}, as well as for the MGT equation
\eqref{e:mgt}, when supplemented with Neumann boundary conditions (a case which is known
to be drastically more difficult for the wave equation itself).
These are---to the authors' knowledge---both open problems (the former, even in the case $\xi=0$).

\smallskip
Let the operator ${\mathcal T}$ be the Dirichlet trace on $\Gamma = \partial \Omega$, and let
$G=G_D$ be the Green map defined by \eqref{e:green-map} accordingly.
Then, an elementary computation which utilizes the (second) Green Theorem yields,
for $\phi\in D(A)$, the following trace result:
\begin{equation}\label{e:basic-trace-d}
G^*A\phi=-\frac{\partial \phi}{\partial\nu}\Big|_\Gamma   \qquad \forall \phi\in D(A)\,;
\end{equation}
see, e.g., \cite[Vol.~I, p.~181]{las-trig-book}.

We begin by recalling the (by now well known) result
on the boundary traces of the wave equation:

\begin{theorem}[\cite{las-lions-trig}] \label{t:traces-waves}
Let $u=u(t,x)$ be a solution to the initial/boundary value problem \eqref{e:ibvp-wave}
for the wave equation with homogeneous boundary data (i.e. $g=0$).
Then, for every $T>0$ there exists $M=M_T$ such that
\begin{equation*}
\int_0^T\!\!\!\int_{\partial\Omega} \Big|\frac{\partial}{\partial\nu} u(x,t)\Big|^2
d\sigma\,d t \le
M\, \Big( \|u_0\|_{H^1_0(\Omega)}+\|u_1|_{L^2(\Omega)}^2 +\|f\|^2_{L^1(0,T;L^2(\Omega))}\Big)\,.
\end{equation*}

\end{theorem}

\smallskip
We now see that this property is inherited by the solutions to the equation with memory
\eqref{e:memory}, and next by the solutions to the MGT equation \eqref{e:mgt}, provided a suitable compatibility condition for initial data is satisfied; see \eqref{e:compatibility} below.
The first precise statement is as follows. ({\em Cf.} \cite[Theorem~1.1]{loreti-sforza_parma}
for the case $\xi=0$.)

\begin{theorem}\label{t:traces-memory}
Under the standing Hypotheses~\ref{a:kernel-and-affine} and assuming $\xi\in L^2(\Omega)$,
let $u=u(t,x)$ be a solution to the equation with memory \eqref{e:memory},
with initial data $(u_0,u_1)$ and homogeneous boundary data.
Then, for every $T>0$ there exists $M=M_T$ such that
\begin{equation}\label{e:traces-memory}
\int_0^T\!\!\!\int_{\partial\Omega} \Big|\frac{\partial}{\partial\nu} u(x,t)\Big|^2
d\sigma\,d t
\le M\, \Big(\|u_0\|_{H^1_0(\Omega)}+\|u_1|_{L^2(\Omega)}^2 +\|\xi\|^2_{L^2(\Omega)}\Big)\,.
\end{equation}
\end{theorem}

\begin{proof}
Since the equation is supplemented with Dirichlet boundary conditions, with may take
$A=\Delta$, with domain $H^2(\Omega)\cap H^1_0(\Omega)$.
The estimate \eqref{e:traces-memory} is obtained as a simple consequence of the
boundary regularity of solutions to the equation with memory in \eqref{ibvp-for-v},
whose convolution term was dispensed with differential terms.
Rewrite the equation in \eqref{ibvp-for-v} as
\begin{equation*}
v_{tt}=\Delta v+k_0v+\int_0^t K(t-s)v(s) ds+\cF(t)\,,
\end{equation*}
where we have set $b=1$ for the sake of simplicity (recall that $b$ must be positive),
while $\cF(t)$ is now
\begin{equation*}
\cF(t):=\big(h_2(t)\xi+h_1(t)u_1+ h_0(t) u_0\big)\,,
\end{equation*}
with the scalar functions $h_i(\cdot)$, $i=0,1,2$ introduced in \eqref{e:various-functions}.
It then follows that
\begin{equation*}
\begin{split}
& v(t)=\underbrace{R_+(t)v_0+{\mathcal A}^{-1}R_-(t)v_1}_{=: u(t)}+
\\
& \qquad\qquad\qquad + {\mathcal A}^{-1}\int_0^t  R_-(t-s)\Big [k_0 v(s)+\int_0^s K(s-r) v(r) dr\Big]\,ds+
\\[1mm]
& \qquad\qquad\qquad \qquad +\,\underbrace{{\mathcal A}^{-1}\int_0^t R_-(t-s) \cF(s)\,ds}_{=:T(t)}\,.
\end{split}
\end{equation*}
First of all we note that Theorem~\ref{t:traces-waves} is valid for the function $u(t)$.
Next, we observe that the integral term $T(t)$ depends on $\xi$, as the function $\cF(t)$ does.
Let us examine this dependence.
We recall the following version of Young inequality: given a Hilbert space $H$, if
$h\in L^1(0,T;\mathbb{R})$ and $X\in L^2(0,T;H)$ then the convolution $X*h$ satisfies
\begin{equation*}
\|X*h\|_{L^2(0,T;H)}\le \|X\|_{L^2(0,T;X)}\,\|h\|_{L^1(0,T;\mathbb{R})}\,.
\end{equation*}
Then, assuming $\xi\in D(A)$, we obtain
\begin{align*}
&
\frac{\partial }{\partial \nu}  {\mathcal A}^{-1}\int_0^t  R_-(t-s) h_2(s) \xi\,ds =D^*A\left [{\mathcal A}^{-1}\int_0^t  R_-(t-s) h_2(s) \xi\,ds \right ]=
\\[1mm]
& \qquad\qquad\qquad =\int_0^t h_2(t-s)X(s)\, ds\,,
\end{align*}
where we set
\begin{equation*}
X(t):= D^*A\left [{\mathcal A}^{-1} R_-(t)\xi\right ]=\frac{\partial }{\partial \nu}
\left [{\mathcal A}^{-1} R_-(t)\xi\right ]\,.
\end{equation*}
Then, the (direct) inequality pertaining to wave equation establishes
\begin{equation*}
\Big\|\frac{\partial }{\partial \nu} \left [{\mathcal A}^{-1} R_-(t)\xi\right ]\Big\|_{L^2(0,T;L^2(\Gamma))} \le M\|\xi\|_{L^2(\Omega)}\,,
\end{equation*}
which is extended by continuity to every $\xi\in L^2(\Omega)$.
Young inequality then gives
\begin{equation*}
\Big \| \int_0^\cdot h_2(\cdot-s)X(s)\, ds\Big\|\le M\|\xi\|_{L^2(\Omega)}\,.
\end{equation*}
The remaining summands within $T(t)$, depending on $u_0$ and $u_1$, are continuous
$D(A)$-valued functions, too.

Therefore, the normal trace of $v$ reads as
\begin{align*}
&\frac{\partial}{\partial \nu} v(t)\Big|_\Gamma=-G^*A v(t)=
-G^*A\Big[u(t)+{\mathcal A}^{-1}\int_0^t  R_-(t-s) \cF(s)\,ds\Big]\,ds-
\\[1mm]
& \qquad\qquad -G^*A \Big[{\mathcal A}^{-1}\int_0^t R_-(t-s)\Big(k_0 v(s)+\int_0^s K(s-r) v(r) dr\Big)\,ds\Big]
\end{align*}
and we see that there exists $M>0$ such that
\begin{align}
& \left \|-G^*A\Big[u(t)+{\mathcal A}^{-1}\int_0^t  R_-(t-s) \cF(s)\,ds\Big]\,ds\right \|^2_{L^2(0,T;L^2(\Gamma))}\le
\nonumber \\[1mm]
& \qquad\qquad\qquad \le M
\left (\|u_0\|^2_{H^1_0(\Omega)}+\|u_1\|^2_{L^2(\Omega)} +\|\xi\|^2_{L^2(\Omega)}\right )\,.
\label{e:to-be-combined}
\end{align}
A similar inequality is valid for the second summand.
In fact, we know ({\em cf.} the second statement of Theorem~\ref{t:sample}) that
$v\in C([0,T];H^1_0(\Omega))$, with continuous dependence on initial data.
Therefore, the second summand satisfies
\begin{equation*}
G^*A \Big[{\mathcal A}^{-1}\int_0^t R_-(t-s)\Big(k_0 v(s)+\int_0^s K(s-r) v(r) dr\Big)\,ds\Big]
\in C(0,T;L^2(\Gamma))\,,
\end{equation*}
which combined with \eqref{e:to-be-combined} implies the sought estimate
\eqref{e:traces-memory}
\end{proof}

For the MGT equation, one obtains readily the following result.

\begin{corollary}\label{c:traces-mgt}
Let $u=u(t,x)$ be a solution to the Moore-Gibson-Thompson equation \eqref{e:mgt},
with initial data $(u_0,u_1,u_2)$ and homogeneous boundary data.
Assume $(u_0,u_1,u_2)\in H^1_0(\Omega)\times L^2(\Omega)\times H^{-1}(\Omega)$,
along with the compatibility condition
\begin{equation}\label{e:compatibility}
\xi=u_2-\Delta u_0\in L^2(\Omega)\,.
\end{equation}
Then, for every $T>0$ there exists $M=M_T$ such that
\begin{equation*}
\int_0^T\!\!\!\int_{\partial\Omega} \Big|\frac{\partial}{\partial\nu} u(x,t)\Big|^2
d\sigma\,d t\le
M\, \Big( \|u_0\|_{H^1_0(\Omega)}+\|u_1\|_{L^2(\Omega)}^2 +\|u_2-\Delta u_0\|^2_{L^2(\Omega)}\Big)\,.
\end{equation*}
\end{corollary}