# Coleman-Gurtin type equations with dynamic boundary conditions

Ciprian G. Gal1 and Joseph L. Shomberg2 1Department of Mathematics, Florida
International University, Miami, FL 33199, USA,
<EMAIL_ADDRESS>2Department of Mathematics and Computer Science, Providence
College, Providence, RI 02918, USA,
<EMAIL_ADDRESS>
###### Abstract.

We present a new formulation and generalization of the classical theory of
heat conduction with or without fading memory which includes the usual heat
equation subject to a dynamic boundary condition as a special case. We
investigate the well-posedness of systems which consist of Coleman-Gurtin type
equations subject to dynamic boundary conditions, also with memory. Nonlinear
terms are defined on the interior of the domain and on the boundary and
subject to either classical dissipation assumptions, or to a nonlinear balance
condition in the sense of [11]. Additionally, we do not assume that the
interior and the boundary share the same memory kernel.

###### Key words and phrases:

Coleman-Gurtin equation, dynamic boundary conditions with memory, heat
conduction, heat equations.

###### 2000 Mathematics Subject Classification:

35B25, 35B40, 35B41, 35K57, 37L30, 45K05.

###### Contents

  1. 1 Introduction
  2. 2 Derivation of the model equations
  3. 3 Past history formulation and functional setup
  4. 4 Variational formulation and well-posedness

##  1\. Introduction

In recent years there has been an explosive growth in theoretical results
concerning dissipative infinite-dimensional systems with memory including
models arising in the theory of heat conduction in special materials and the
theory of phase-transitions. The mathematical and physical literature,
concerned primarily with qualitative/quantitative properties of solutions to
these models, is quite extensive and much of the work before 2002 is largely
referenced in the survey paper by Grasselli and Pata [19]. More recent results
and updates can be found in [7, 8, 9, 10] (cf. also [16, 17]). A basic
evolution equation considered in these references is that for an homogeneous
and isotropic heat conductor occupying a $d$-dimensional (bounded) domain
$\Omega$ with sufficiently smooth boundary $\Gamma=\partial\Omega$ and reads

$\partial_{t}u-\omega\Delta
u-\left(1-\omega\right)\int_{0}^{\infty}k_{\Omega}\left(s\right)\Delta
u\left(x,t-s\right)ds+f\left(u\right)=0,$ (1.1)

in $\Omega\times\left(0,\infty\right).$ Here $u=u\left(t\right)$ is the
(absolute) temperature distribution, $\omega>0,$
$r=-f\left(u\left(t\right)\right)$ is a temperature dependent heat supply, and
$k_{\Omega}:[0,\infty)\rightarrow\mathbb{R}$ is a continuous nonnegative
function, smooth on $(0,\infty)$ and vanishing at infinity, and summable. As
usual, (1.1) is derived by assuming the following energy balance equation

$\partial_{t}e+\text{div}\left(q\right)=r$

by considering the following relationships:

$e=e_{\infty}+c_{0}u,\text{ }q=-\omega\nabla
u-\left(1-\omega\right)\int_{0}^{\infty}k_{\Omega}\left(s\right)\nabla
u\left(x,t-s\right)ds,$ (1.2)

for some constants $e_{\infty},c_{0}>0$. Equation (1.1) is always subject to
either homogeneous Dirichlet ($u=0$) or Neumann boundary conditions
($\partial_{n}u=0$) on $\Gamma\times\left(0,\infty\right)$. The first one
asserts that the temperature is kept constant and close to a given reference
temperature at $\Gamma$ for all time $t>0$, while the second “roughly” states
that the system is thermally isolated from outside interference. This equation
is also usually supplemented by the “initial” condition
$\widetilde{u}:(-\infty,0]\rightarrow\mathbb{R}$ such that

$u_{\mid t\in(-\infty,0]}=\widetilde{u}\text{ in }\Omega.$ (1.3)

These choices of boundary conditions, although help simplify substantially the
mathematical analysis of (1.1)-(1.3), are actually debatable in practice since
in many such systems it is usually difficult, if not impossible, to keep the
temperature constant at $\Gamma$ for all positive times without exerting some
additional kind of control at $\Gamma$ for $t>0$. A matter of principle also
arises for thermally isolated systems in which, in fact, the correct physical
boundary condition for (1.1) turns out to be the following

$q\cdot
n=\omega\partial_{n}u+\left(1-\omega\right)\int_{0}^{\infty}k_{\Omega}\left(s\right)\partial_{n}u\left(x,t-s\right)ds=0\text{
on }\Gamma\times\left(0,\infty\right)\text{,}$ (1.4)

see, for instance, [5, Section 6]. Indeed, the condition $\partial_{n}u=0$ on
$\Gamma\times\left(0,\infty\right)$ implies (1.4), say when $u$ is a
sufficiently smooth solution of (1.1)-(1.3), but clearly the converse cannot
hold in general.

In the classical theory of heat conduction, it is common to model a wide range
of diffusive phenomena including heat propagation in homogeneous isotropic
conductors, but generally it is assumed, as above, that surface (i.e.,
boundary) conditions are completely static or stationary. In some important
cases this perspective neglects the contribution of boundary sources to the
total heat content of the conductor. A first step to remedy this situation was
done in Goldstein [18] for heat equations. The approach presented there
introduces dynamic boundary conditions into an _ad hoc_ fashion and lacks some
rigor in the case of reaction-diffusion equations. In the next section of the
paper we will make use of the usual physical principles and present a new
formulation and generalization of the classical theory. Our general approach
follows that of Coleman and Mizel [5] which regards the second law of
thermodynamics as included among the laws of physics and which is compatible
with the principle of equipresence in the sense of Truesdell and Toupin (see
Section 2). Thus, this new formulation is expected to give a solid foundation
to the arguments employed in derivations of the heat equation with “dynamic”
boundary conditions developed in Goldstein [18], or in models for phase
transitions developed in Gal and Grasselli [13, 14]. Accounting for the
presence of boundary sources, the new formulation naturally leads to dynamic
boundary conditions for the temperature function $u$ and that contain the
above static conditions (especially, (1.4)) as special cases (see Section 2).
In particular, we derive on $\Gamma\times\left(0,\infty\right),$ the following
boundary condition for (1.1):

$\displaystyle\partial_{t}u-\nu\Delta_{\Gamma}u+\omega\partial_{n}u+g\left(u\right)$
$\displaystyle+\left(1-\omega\right)\int_{0}^{\infty}k_{\Omega}\left(s\right)\partial_{n}u\left(x,t-s\right)ds+\left(1-\nu\right)\int_{0}^{\infty}k_{\Gamma}\left(s\right)\left(-\Delta_{\Gamma}+\beta\right)u\left(x,t-s\right)ds$
(1.5) $\displaystyle=0,$

for some $\nu\in\left(0,1\right)$ and $\beta>0$. Here
$k_{\Gamma}:[0,\infty)\rightarrow\mathbb{R}$ is also a smooth nonnegative,
summable function over $(0,\infty)$ such that $k_{\Gamma}$ is vanishing at
infinity. The last two boundary terms on the left-hand side of equation (1.5)
are due to contributions coming from a (linear) heat exchange rate between the
bulk $\Omega$ and the boundary $\Gamma$, and boundary fluxes, respectively
(cf. Section 2).

Our goal in this paper is to extend the previous well-posedness results of [7,
8, 9, 10, 19, 16, 17] and [11, 12, 15] in the following directions:

  * •

by allowing general boundary processes take place also on $\Gamma$, equation
(1.1) is now subject to boundary conditions of the form (1.5);

  * •

we consider more general functions $f,g\in C^{1}\left(\mathbb{R}\right)$
satisfying either classical dissipation assumptions, or more generally,
nonlinear balance conditions allowing for bad behavior of $f,g$ at infinity;

  * •

we develop a general framework allowing for both weak and smooth initial data
for (1.1), (1.5), and possibly _different_ memory functions
$k_{\Omega},k_{\Gamma}.$

  * •

we extend a Galerkin approximation scheme whose explicit construction is
crucial for the existence of strong solutions.

The paper is organized as follows. In Section 3, we provide the functional
setup. In Section 4, we prove theorems concerning the well-posedness of the
system, based on (1.1), (1.5), generated by the new formulation. In the
subsequent section, we present a rigorous formulation and examples in which
(1.5) naturally occurs for (1.1).

##  2\. Derivation of the model equations

To begin let us consider a bounded domain $\Omega\subset\mathbb{R}^{d}$ which
is occupied by a rigid body. The region $\Omega$ is assumed to be bounded by a
smooth boundary $\Gamma:=\partial\Omega$ which is assumed to be at least
Lipschitz continuous. As usual, a thermodynamic process taking place in
$\Omega$ is defined by five basic functions, that is, the specific internal
energy $e_{\Omega}\left(x,t\right)$, the specific entropy
$\eta_{\Omega}=\eta_{\Omega}\left(x,t\right)$, the heat flux
$q=q\left(x,t\right)$, the absolute temperature $u=u\left(x,t\right)>0$ and
the heat supply $h_{\Omega}\left(x,t\right)\,$, absorbed by the material at
$x\in\Omega,$ and possibly furnished by the external world (i.e.,
thermodynamic processes that occur outside of $\Omega$). All these quantities,
defined per unit volume and unit time, are scalars except for
$q\in\mathbb{R}^{d}$ which is a vector. The classical theory [4, 5] of heat
conduction in the body $\Omega$ ignores any heat contribution which may be
supplied from processes taking place on $\Gamma$ and, hence, this situation is
never modelled by the theory. This is the case in many applications, in
calorimetry, which go back to problems that occur as early as the mid 1950’s,
see [3, Chapter I, Section 1.9, pg. 22-24]. A typical example arises when a
given body $\Omega$ is in perfect thermal contact with a thin metal sheet,
possibly of different material $\Gamma=\partial\Omega$ completely insulating
the body $\Omega$ from contact with, say, a well-stirred hot or cold fluid.
The assumption made is that the metal sheet $\Gamma$ is sufficiently thin such
that the temperature $v\left(t\right)$ at any point on $\Gamma$ is constant
across its thickness. Since the sheet $\Gamma$ is in contact with a fluid it
will either heat or cool the body $\Omega$ in which case the heat supplied to
$\Omega$ is due to both $\Gamma$ and the body of fluid, not to mention the
fact that the temperature distribution in the sheet is also affected by heat
transfer between $\Gamma$ and the interior $\Omega$. Since the outershell
$\Gamma$ is in perfect contact with the body $\Omega$, it is reasonable to
assume by continuity that the temperature distribution $u\left(t\right)$ in
$\Omega,$ in an infinitesimal layer near $\Gamma$ is equal to
$v\left(t\right)$, for all times $t>\delta$, that is,
$u\left(t\right)_{\mid\Gamma}=v\left(t\right)$ for all $t>\delta$; they need
not, of course, be equal at $t=\delta$, where $\delta$ is the (initial)
starting time. When $\rho_{1},$ $\rho_{2}$ correspond to the densities of
$\Omega$ and $\Gamma$, respectively, and $c_{1},c_{2}$ denote the heat
capacities of $\Omega$ and $\Gamma,$ respectively, this example can be
modelled by the balance equation

$\rho_{1}c_{1}\partial_{t}u=-\text{div}\left(q\right)+h_{\Omega}\text{ in
}\Omega\times\left(\delta,\infty\right),$ (2.1)

suitably coupled with an equation for $\Gamma$, by considering the heat
balance of an element of area of the sheet $\Gamma$, which is

$\rho_{2}c_{2}\partial_{t}u=q\cdot
n-\text{div}_{\Gamma}\left(q_{\Gamma}\right)+l_{\Gamma}\text{ in
}\Gamma\times\left(\delta,\infty\right).$ (2.2)

Here $n\in\mathbb{R}^{d}$ denotes the exterior unit normal vector to $\Gamma$,
$l_{\Gamma}\left(x,t\right)$ is an external heat supply and $q_{\Gamma}$ is a
tangential heat flux on $\Gamma$ while divΓ is the surface divergence whose
definition is given below. Note that the correct term to couple the balance
equations for $\Omega$ and $\Gamma$ is given by $q\cdot n$, since this is used
to quantify a (linear) heat exchange rate across $\Gamma$ from $\Omega$ in all
directions normal to the boundary $\Gamma$. The system (2.1)-(2.2) is also
important in control problems for the heat equation, say when a specific
temperature distribution at the boundary $\Gamma$ is desired (see [21]).

As mentioned earlier, in the classical theory on heat conduction one usually
ignores boundary contributions by either prescribing the temperature on
$\Gamma$ or assuming that the flux across the surface $\Gamma$ from $\Omega$
is null, or simply, by invoking Newton’s law of cooling which states that the
flux across the surface is directly proportional to temperature differences
between the surface and the surrounding medium. In the sequel, it is our goal
to include general boundary processes into the classical theory of heat
conduction. To this end, in order to define a complete thermodynamic process
in $\overline{\Omega}=\Omega\cup\Gamma$, as in the previous example, we need
to add four more response functions, that is, the specific surface energy
$e_{\Gamma}\left(x,t\right),$ the specific surface entropy density
$\eta_{\Gamma}\left(x,t\right)$, the tangential heat flux
$q_{\Gamma}=q_{\Gamma}\left(x,t\right)\in\mathbb{R}^{d-1}$, and the external
heat supply $h_{\Gamma}\left(x,t\right),$ all defined for $x\in\Gamma,$ per
unit area and unit time. It is assumed that the absolute (local) temperature
$u\left(\cdot,t\right)$ is sufficiently smooth up to $\overline{\Omega}$ as a
function of the spatial coordinate. We now introduce the following definition.

  * •

We say that the set of nine time-dependent variables constitutes a _complete_
_thermodynamic process_ in $\overline{\Omega}$ if the following conservation
law holds, not only for $\overline{\Omega}$, but also for any subdomain
$\Omega_{0}\subset\Omega$ and any part $\Gamma_{0}\subset\Gamma$:

$\int_{\Omega}\overset{\centerdot}{e}_{\Omega}dx+\int_{\Gamma}\overset{\centerdot}{e}_{\Gamma}d\sigma=-\int_{\Omega}\text{div}\left(q\right)dx-\int_{\Gamma}\text{div}_{\Gamma}\left(q_{\Gamma}\right)d\sigma+\int_{\Omega}h_{\Omega}dx+\int_{\Gamma}h_{\Gamma}d\sigma.$
(2.3)

In (2.3), $dx$ denotes the volume element, $d\sigma$ is the element of surface
area and the superimposed dot denotes the time-derivative. Note that in
general, the external heat supply $h_{\Gamma}$ on $\Gamma$ must also depend,
possibly in a nonlinear fashion, on the heat content exchanged across $\Gamma$
from $\Omega$, i.e., $h_{\Gamma}=f\left(q\cdot n\right)+l_{\Gamma}$, for some
function $f$, where $l_{\Gamma}$ accounts either for the heat supply coming
solely from $\Gamma$ or some other source outside of $\Gamma$, see the above
example (2.1)-(2.2). In order to give a rigorous definition to
div${}_{\Gamma}\left(q_{\Gamma}\right),$ we regard $\Gamma$ as a compact
Riemanian manifold without boundary, endowed with the natural metric inherited
from $\mathbb{R}^{d}$, given in local coordinates by $\mathbf{\tau}$ and with
fundamental form $\left(\mathbf{\tau}_{ij}\right)_{i,j=1,...,d-1}$. A scalar-
valued function $w\in C^{\infty}\left(\Gamma\right)$ induces an element of the
dual space of $T_{x}\Gamma$ via the directional derivative of tangential
vectors at $x\in\Gamma$. Clearly, $T_{x}\Gamma$ is a Hilbert space when
endowed with scalar product induced from $\mathbb{R}^{d}$. For a tangential
vector field $q_{\Gamma}\in C^{\infty}\left(\Gamma\right),$ that is,
$q_{\Gamma}\left(x\right)\in T_{x}\Gamma,$ for $x\in\Gamma,$ the surface
divergence, div${}_{\Gamma}\left(q_{\Gamma}\right),$ is in the local
coordinates $\mathbf{\tau}$ for $\Gamma,$

$\text{div}_{\Gamma}q_{\Gamma}\left(\mathbf{\tau}\right)=\frac{1}{\sqrt{\left|\mathbf{\tau}\right|}}\sum_{i=1}^{d-1}\partial_{i}(\sqrt{\left|\mathbf{\tau}\right|}q_{i}\left(\mathbf{\tau}\right)),$

where $q_{i}$ are the components of $q_{\Gamma}$ with respect to the basis
$\left\\{\partial_{1}\mathbf{\tau,...,\partial}_{d-1}\mathbf{\tau}\right\\}$
of $T_{x}\Gamma$ and
$\left|\mathbf{\tau}\right|=\det\left(\mathbf{\tau}_{ij}\right)$. Moreover, we
can define the surface gradient $\nabla_{\Gamma}u$ as a unique element of
$T_{x}\Gamma$ corresponding to this dual space element via a natural
isomorphism, that is,

$\nabla_{\Gamma}u\left(\mathbf{\tau}\right)=\sum_{i,j=1}^{d-1}\mathbf{\tau}_{ij}\partial_{j}u\left(\mathbf{\tau}\right)\partial_{i}\mathbf{\tau,}$

with respect to the canonical basis
$\left\\{\partial_{1}\mathbf{\tau,...,\partial}_{d-1}\mathbf{\tau}\right\\}$
of $T_{x}\Gamma$. For a multi-index $\alpha\in\mathbb{N}_{0}^{m}$, the
operator $\nabla_{\Gamma}^{\alpha}u$ is defined by taking iteratively the
components of $\nabla_{\Gamma}u.$ It is worth emphasizing that our form of the
first law (2.3) is equivalent to

$\overset{\centerdot}{e}_{\Omega}=-\text{div}\left(q\right)+h_{\Omega}\text{
in \ }\Omega\text{, and
}\overset{\centerdot}{e}_{\Gamma}=-\text{div}_{\Gamma}\left(q_{\Gamma}\right)+h_{\Gamma}\text{
on }\Gamma,$ (2.4)

under suitable smoothness assumptions on the response functions involved in
(2.4). Equation (2.3) may be called the law of conservation of total energy or
the _extended_ First Law of Thermodynamics. For each such complete
thermodynamic process, let us define the total rate of production of entropy
in $\overline{\Omega}=\Omega\cup\Gamma$ to be

$\Upsilon:=\int_{\Omega}\overset{\centerdot}{\eta}_{\Omega}dx+\int_{\Gamma}\overset{\centerdot}{\eta}_{\Gamma}d\sigma-\int_{\Omega}\frac{h_{\Omega}}{u}dx+\int_{\Omega}\text{div}\left(\frac{q}{u}\right)dx+\int_{\Gamma}\text{div}_{\Gamma}\left(\frac{q_{\Gamma}}{u}\right)d\sigma-\int_{\Gamma}\frac{h_{\Gamma}}{u}d\sigma,$
(2.5)

where we regard $q/u$ as a vectorial flux of entropy in $\Omega$,
$h_{\Omega}/u$ as a scalar supply of entropy produced by radiation from inside
the body $\Omega$, $h_{\Gamma}/u$ is viewed as a scalar supply of entropy
produced by radiation from $\Gamma$ and $q_{\Gamma}/u$ is a tangential flux of
entropy on $\Gamma$. More precisely, we define $\Upsilon$ to be the difference
between the total rate of change in entropy of $\overline{\Omega}$ and that
rate of change which comes from the heat supplies in both $\Omega$ and
$\Gamma$, and both the inward and tangential fluxes. We postulate the
following extended version of the Second Law of Thermodynamics as follows.

  * •

For every complete thermodynamic process in $\overline{\Omega}$ the inequality

$\Upsilon\geq 0$ (2.6)

must hold for all $t$, not only in $\overline{\Omega}$, but also on all
subdomains $\Omega_{0}\subset\Omega$ and all parts $\Gamma_{0}\subset\Gamma,$
respectively111When (2.6) holds on all parts $\Omega_{0}\subset\Omega,$ it is
understood that all the boundary integrals in (2.5) drop out; in the same
fashion, when (2.6) is satisfied for all parts $\Gamma_{0}\subset\Gamma,$ the
bulk integrals are also omitted from the definition of $\Upsilon.$. For
obvious reasons, we will refer to the inequality $\Upsilon\geq 0$ as the
_extended_ Clausius-Duhem inequality. Finally, a complete thermodynamic
process is said to be _admissible_ in $\overline{\Omega}$ if it is compatible
with a set of constitutive conditions given on the response functions
introduced above, at each point of $\overline{\Omega}$ and at all times $t$.

Of course, for the postulate to hold, the various response functions must obey
some restrictions, including the usual ones which are consequences of the
_classical_ Clausius-Duhem inequality. In particular, the entropy
$\eta_{\Omega}$ at each point $x\in\Omega$ must be determined only by a
function of the specific internal energy $e_{\Omega},$ and the temperature $u$
at $x\in\Omega$ is determined only by a relation involving $e_{\Omega}$ and
$\eta_{\Omega}$. More precisely, it turns out that for the postulate to hold
on any $\Omega_{0}\subset\Omega$, both the internal energy $e_{\Omega}$ and
the entropy function $\eta_{\Omega}$ must be constitutively independent of any
higher-order stress tensors $\nabla^{\gamma}u$ for any $\gamma\geq 1$, such
that they are only functions of the local temperature, i.e., it follows that

$e_{\Omega}=e_{\Omega}\left(u\right)\text{ and
}\eta_{\Omega}=\eta_{\Omega}\left(u\right),$ (2.7)

respectively, cf. [5, Theorem 1, pg. 251]. Indeed, our postulate implies that
the local form of the second law must hold also on any subdomain $\Omega_{0}$
of $\Omega$; this implies that

$\gamma_{\Omega}:=\left(\overset{\centerdot}{\eta}_{\Omega}-\frac{h_{\Omega}}{u}+\text{div}\left(\frac{q}{u}\right)\right)\geq
0\text{ in }\Omega$ (2.8)

and

$\gamma_{\Gamma}:=\left(\overset{\centerdot}{\eta}_{\Gamma}-\frac{h_{\Gamma}}{u}+\text{div}_{\Gamma}\left(\frac{q_{\Gamma}}{u}\right)\right)\geq
0\text{ on }\Gamma.$ (2.9)

From [5], we know that $\gamma_{\Omega}\geq 0$ in the body $\Omega$ if and
only if

$q\cdot\nabla u\leq 0,$ (2.10)

for all values $u,$ $\nabla u,$…., $\nabla^{\gamma}u$, with $q=q\left(u,\nabla
u,\nabla^{2}u,...,\nabla^{\gamma}u\right)$. This inequality is called the heat
conduction inequality in $\Omega$. In fact, this inequality was established in
[20] under more general constitutive assumptions on $\eta_{\Omega},q$ and
$e_{\Omega}$, excluding memory effects, as functionals of the entropy field
over the entire body $\Omega$ at the same time.

We now find necessary and sufficient set of restrictions on the remaining
functions $\eta_{\Gamma},$ $e_{\Gamma}$, $q_{\Gamma}$. As in [5], we assume a
formulation of constitutive equations to be compatible with the Principle of
Equipresence in the sense of Truesdell and Toupin [26, pg. 293], which
basically states that “a variable present as an independent variable in one
constitutive equation should be so present in all”. In the present
formulation, the material at $x\in\Gamma$ is characterized by the response
functions $\widehat{\eta}_{\Gamma},$ $\widehat{e}_{\Gamma}$ and
$\widehat{q}_{\Gamma},$ which give the functions
$\eta_{\Gamma}\left(x,t\right),$ $e_{\Gamma}\left(x,t\right)$ and
$q_{\Gamma}\left(x,t\right)$, respectively, when the values
$\nabla_{\Gamma}^{j}u\left(x,t\right)$ are known for $j=0,1,2,...,\alpha.$
Dropping the hats for the sake of convenience and by force of this principle,
we assume that

$\displaystyle e_{\Gamma}$
$\displaystyle=e_{\Gamma}\left(u,\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right),$
(2.11) $\displaystyle\eta_{\Gamma}$
$\displaystyle=\eta_{\Gamma}\left(u,\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right),$
(2.12) $\displaystyle q_{\Gamma}$
$\displaystyle=q_{\Gamma}\left(u,\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right).$
(2.13)

Furthermore, we assume that for any fixed values of $\nabla_{\Gamma}^{j}u,$
the response function $e_{\Gamma}$ is smooth in the first variable $u$, i.e.,
we suppose $\frac{\partial e_{\Gamma}}{\partial
u}\left(u,\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right)\neq
0.$ This implies that there exist new response functions, say
$\widetilde{\eta}_{\Gamma},$ $\widetilde{e}_{\Gamma}$ and
$\widetilde{q}_{\Gamma},$ which can be used to write (2.11)-(2.13) in the
following form:

$\displaystyle u$
$\displaystyle=\widetilde{u}\left(e_{\Gamma},\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right),$
(2.14) $\displaystyle\eta_{\Gamma}$
$\displaystyle=\widetilde{\eta}_{\Gamma}\left(e_{\Gamma},\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right),$
(2.15) $\displaystyle q_{\Gamma}$
$\displaystyle=\widetilde{q}_{\Gamma}\left(e_{\Gamma},\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right).$
(2.16)

For each fixed values of the tensors $\nabla_{\Gamma}^{j}u,$
$j=0,1,2,...,\alpha$, the variable
$\widetilde{u}\left(\cdot,\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right)$
is determined through the inverse function of $e_{\Gamma},$ given by (2.11),
such that $\widetilde{\eta}_{\Gamma}$ and $\widetilde{q}_{\Gamma}$ are defined
by

$\displaystyle\widetilde{\eta}_{\Gamma}\left(e_{\Gamma},\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right)$
$\displaystyle=\eta_{\Gamma}\left(\widetilde{u}\left(e_{\Gamma},\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right),\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right),$
$\displaystyle\widetilde{q}_{\Gamma}\left(e_{\Gamma},\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right)$
$\displaystyle=q_{\Gamma}\left(\widetilde{u}\left(e_{\Gamma},\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right),\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right).$

Note that with $u\left(x,t\right)$ specified for all $x$ and $t$, equations
(2.11)-(2.13) give $\eta_{\Gamma}\left(x,t\right),$
$e_{\Gamma}\left(x,t\right)$ and $q_{\Gamma}\left(x,t\right),$ for all $x$ and
$t,$ in which case the local form of the First Law (see also (2.4)) determines
also $h_{\Gamma}$. In particular, every temperature distribution
$u\left(x,t\right)>0$ with $x$ varying over $\Gamma$, determines a unique
complete thermodynamic process in $\Gamma$. By a standard argument in [5, pg.
249], in (2.11)-(2.13) we may regard not only
$e_{\Gamma},\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u$
as independent variables, but also their time-derivatives
$\overset{\centerdot}{e}_{\Gamma}$, $\nabla_{\Gamma}\overset{\centerdot}{u},$
$\nabla_{\Gamma}^{2}\overset{\centerdot}{u},...,$
$\nabla_{\Gamma}^{\alpha}\overset{\centerdot}{u},$ to form a set of quantities
which can be chosen independently at one fixed point $x\in\Gamma$ and time.

For each complete thermodynamic process in $\overline{\Omega}$, the second
energy balance equation (2.4) allows us to write (2.9) as

$\gamma_{\Gamma}=\overset{\centerdot}{\eta}_{\Gamma}-\frac{h_{\Gamma}}{u}+\text{div}_{\Gamma}\left(\frac{q_{\Gamma}}{u}\right)=\overset{\centerdot}{\eta}_{\Gamma}-\frac{e_{\Gamma}}{u}+q_{\Gamma}\cdot\nabla_{\Gamma}\left(\frac{1}{u}\right).$
(2.17)

Since $q_{\Gamma}$ and $\eta_{\Gamma}$ must be given by (2.16) and (2.15), at
any point $\left(x,t\right),$ we have

$\overset{\centerdot}{\eta}_{\Gamma}=\frac{\partial\widetilde{\eta}_{\Gamma}}{\partial
e_{\Gamma}}\overset{\centerdot}{e}_{\Gamma}+\sum\nolimits_{j=1}^{\alpha}\left(\frac{\partial\widetilde{\eta}_{\Gamma}}{\partial
u_{,l_{1}l_{2}...l_{j}}}\right)\overset{\centerdot}{u}_{,l_{1}l_{2}...l_{j}},$

where the summation convention is used and where in local coordinates of
$\Gamma$,
$u_{,l_{1}l_{2}...l_{j}}=(\nabla_{\Gamma}^{j}u)_{l_{1}l_{2}...l_{j}}$. It
follows that

$\gamma_{\Gamma}=\left(\frac{\partial\widetilde{\eta}_{\Gamma}}{\partial
e_{\Gamma}}-\frac{1}{\widetilde{u}}\right)\overset{\centerdot}{e}_{\Gamma}+\sum\nolimits_{j=1}^{\alpha}\left(\frac{\partial\widetilde{\eta}_{\Gamma}}{\partial
u_{,l_{1}l_{2}...l_{j}}}\right)\overset{\centerdot}{u}_{,l_{1}l_{2}...l_{j}}-\frac{1}{u^{2}}\widetilde{q}_{\Gamma}\cdot\nabla_{\Gamma}u.$
(2.18)

In order for $\gamma_{\Gamma}\geq 0$ to hold on $\Gamma$ (but also on all
parts $\Gamma_{0}\subset\Gamma$), according to (2.9) and our postulate, it is
necessary and sufficient that

$\frac{\partial\widetilde{\eta}_{\Gamma}}{\partial
e_{\Gamma}}=\frac{1}{\widetilde{u}},\text{ and
}\frac{\partial\widetilde{\eta}_{\Gamma}}{\partial
u_{,l_{1}l_{2}...l_{j}}}=0,\text{ }j=1,2,...,\alpha.$ (2.19)

It follows from (2.19) that the functions $\widetilde{\eta}_{\Gamma}$ and
$\widetilde{u}_{\Gamma}$ from (2.14)-(2.15) cannot depend on
$\nabla_{\Gamma}u$, $\nabla_{\Gamma}^{2}u$, $...$,
$\nabla_{\Gamma}^{\alpha}u$, and they must reduce to functions of the scalar
variable $e_{\Gamma}$ only, i.e.,
$\eta_{\Gamma}=\widetilde{\eta}_{\Gamma}\left(e_{\Gamma}\right),$
$u=\widetilde{u}_{\Gamma}\left(e_{\Gamma}\right)$. These function must also
obey the first equation of (2.19); hence, the variables $\nabla_{\Gamma}u$,
$\nabla_{\Gamma}^{2}u$, $...$, $\nabla_{\Gamma}^{\alpha}u$ must also be
dropped out of equations (2.14) and (2.15) to get

$e_{\Gamma}=e_{\Gamma}\left(u\right)\text{ and
}\eta_{\Gamma}=\eta_{\Gamma}\left(u\right).$ (2.20)

Consequently, with this reduction we observe that (2.18) becomes

$\gamma_{\Gamma}=-\frac{1}{u^{2}}\widetilde{q}_{\Gamma}\cdot\nabla_{\Gamma}u,$

for all temperature fields $u>0$ and $q_{\Gamma}$ given by (2.16). Thus, in
order to have $\gamma_{\Gamma}\geq 0$ on $\Gamma$, it is necessary and
sufficient that $\widetilde{q}_{\Gamma}\cdot\nabla_{\Gamma}u\leq 0,$ or
equivalently,

$q_{\Gamma}\left(u,\nabla_{\Gamma}u,\nabla_{\Gamma}^{2}u,...,\nabla_{\Gamma}^{\alpha}u\right)\cdot\nabla_{\Gamma}u\leq
0,$ (2.21)

for all values $u$, $\nabla_{\Gamma}u$, $\nabla_{\Gamma}^{2}u$, $...,$
$\nabla_{\Gamma}^{\alpha}u.$ We call (2.21) the heat conduction inequality on
$\Gamma$. Therefore, we have established that a necessary and sufficient
condition for the _extended_ Clausius-Duhem inequality to hold for all
complete thermodynamic processes on $\overline{\Omega}$ is that both the
conduction inequalities (2.10)-(2.21) in $\Omega$ and $\Gamma$, respectively,
hold. An interesting consequence is that the following choices
$q=-\kappa_{\Omega}\left(u\right)\nabla u$ and
$q_{\Gamma}=-\kappa_{\Gamma}\left(u\right)\nabla_{\Gamma}u,$ where
$\kappa_{\Omega},\kappa_{\Gamma}>0$ are the thermal conductivity of $\Omega$
and $\Gamma$, respectively, are covered by this theory. Such choices were
assumed by the theories developed in [13], [14], [18] for the system
(2.1)-(2.2).

Motivated by the above result, we now wish to investigate more general
constitutive conditions for the response functions involved in (2.5), by
allowing them to depend also explicitly on histories up to time $t$ of the
temperature and/or the temperature gradients at $x$. Following the approach of
[4], using the abbreviations $g_{\Omega}:=\nabla u$,
$g_{\Gamma}:=\nabla_{\Gamma}u$, we consider a fixed point
$x\in\overline{\Omega}$, and define the functions $u^{t}$, $g_{\Omega}^{t},$
$g_{\Gamma}^{t}$ as the histories up to time $t$ of the temperature and the
temperature gradients at $x$. More precisely, we let

$\left\\{\begin{array}[]{ll}u^{t}\left(x,s\right)=u\left(x,t-s\right),&\\\
\text{ }g_{\Omega}^{t}\left(x,s\right)=g_{\Omega}\left(x,t-s\right)&\\\ \text{
}g_{\Gamma}^{t}\left(x,s\right)=g_{\Gamma}\left(x,t-s\right),&\end{array}\right.$

for all $\,s\in[0,\infty)$, on which these functions are well-defined. For a
complete thermodynamic process in $\overline{\Omega}$, we define the following
energy densities on $\Omega$ and $\Gamma$, respectively, by

$\psi_{\Omega}:=e_{\Omega}-u\eta_{\Omega},\text{
}\psi_{\Gamma}:=e_{\Gamma}-u\eta_{\Gamma}.$ (2.22)

Of course, knowledge of $e_{\Omega},e_{\Gamma}$ and
$\eta_{\Omega},\eta_{\Gamma}$ obviously determine $\psi_{\Omega}$ and
$\psi_{\Gamma}$ by these relations. We now consider a new generalization of
the constitutive equations for (2.7), (2.20) and the bulk and surface fluxes
$q,$ $q_{\Gamma},$ respectively. We shall investigate the implications that
the second law (2.6) has on these functions. We assume that the material at
$x\in\overline{\Omega}$ is characterized by three constitutive functionals
$P_{\Omega}$, $H_{\Omega}$ and $q$, in the bulk $\Omega$, and three more
constitutive functionals $P_{\Gamma},$ $H_{\Gamma}$ and $q_{\Gamma}$, on the
surface $\Gamma$, which give the present values of
$\psi_{\Omega},\psi_{\Gamma},\eta_{\Omega},\eta_{\Gamma},q$ and $q_{\Gamma}$
at any $x$, whenever the histories are specified at $x$. Note that the
restrictions of the functions $u^{t}$, $g_{\Omega}^{t},$ $g_{\Gamma}^{t}$ on
the open interval $\left(0,\infty\right)$, denoted here by $u_{r}^{t}$,
$g_{\Omega,r}^{t},$ $g_{\Gamma,r}^{t}$, are called past histories. Since a
knowledge of the histories $\left(u^{t},g_{\Omega}^{t},g_{\Gamma}^{t}\right)$
is equivalent to a knowledge of the past histories
$\left(u_{r}^{t},g_{\Omega,r}^{t},g_{\Gamma,r}^{t}\right),$ and the present
values $u^{t}\left(0\right)=u,$
$g_{\Omega}^{t}\left(0\right)=g_{\Omega}\left(t\right),$
$g_{\Gamma}^{t}\left(0\right)=g_{\Gamma}\left(t\right),$ it suffices to
consider

$\left\\{\begin{array}[]{ll}\psi_{\Omega}=P_{\Omega}\left(u^{t},g_{\Omega}^{t}\right),&\psi_{\Gamma}=P_{\Gamma}\left(u^{t},g_{\Gamma}^{t}\right),\\\
\eta_{\Omega}=H_{\Omega}\left(u^{t},g_{\Omega}^{t}\right),&\eta_{\Gamma}=H_{\Gamma}\left(u^{t},g_{\Gamma}^{t}\right),\\\
q=q\left(u^{t},g_{\Omega}^{t}\right),&q_{\Gamma}=q_{\Gamma}\left(u^{t},g_{\Gamma}^{t}\right),\end{array}\right.$
(2.23)

where the Principle of Equipresence is assumed in (2.23). We further suppose
that all the functionals in (2.23) obey the principle of fading memory as
formulated in [6] (cf. also [20, Section 5]). In particular, this assumption
means that “deformations and temperatures experienced in the distant past
should have less effect on the present values of the entropies, energies,
stresses, and heat fluxes than deformations and temperatures which occurred in
the recent past”. Such assumptions can be made precise through the so-called
“memory” functions $m_{\Omega},$ $m_{\Gamma},$ which characterize the rate at
which the memory fades both in the body $\Omega$ and on the surface $\Gamma$,
respectively. In particular, we may assume that both functions
$m_{S}\left(\cdot\right),$ $S\in\left\\{\Omega,\Gamma\right\\}$, are positive,
continuous functions on $\left(0,\infty\right)$ decaying sufficiently fast to
zero as $s\rightarrow\infty$. In this case, we let $D_{S}$ denote the common
domain for the functionals $P_{S},H_{S}$ and $q_{S}$ ($q_{\Omega}=q$), as the
set of all pairs $\left(u^{t},g_{S}^{t}\right)$ for which $u^{t}>0$ and
$\left\|\left(u^{t},g_{S}^{t}\right)\right\|<\infty$, where

$\left\|\left(u^{t},g_{S}^{t}\right)\right\|^{2}:=\left|u^{t}\left(0\right)\right|^{2}+\left|g_{S}^{t}\left(0\right)\right|^{2}+\int_{0}^{\infty}\left|u^{t}\left(s\right)\right|^{2}m_{S}\left(s\right)ds+\int_{0}^{\infty}\left(g_{S}^{t}\left(s\right)\cdot
g_{S}^{t}\left(s\right)\right)m_{S}\left(s\right)ds,$ (2.24)

and where $S\in\left\\{\Omega,\Gamma\right\\}$. Furthermore, for each
$S\in\left\\{\Omega,\Gamma\right\\}$ we assume as in [4] that $P_{S}$,
$H_{S}$, and $q_{S}$ ($q_{\Omega}=q$) are continuous over $D_{S}$ with respect
to the norm (2.24), but also that $P_{S}$ is continuously differentiable over
$D_{S}$ in the sense of Fréchet, and that the corresponding functional
derivatives are jointly continuous in their arguments.

In order to observe the set of restrictions that the postulate (2.6) puts on
the response functions, we recall (2.4) and substitute (2.22) into the local
forms (2.8), (2.9) to derive the following (local) forms of the extended
Clasius-Duhem inequality on $\overline{\Omega}$:

$\left\\{\begin{array}[]{ll}\overset{\centerdot}{\psi}_{\Omega}+\overset{\centerdot}{u}\eta_{\Omega}+\frac{1}{u}q_{\Omega}\cdot\nabla
u\leq 0&\text{in }\Omega,\\\
\overset{\centerdot}{\psi}_{\Gamma}+\overset{\centerdot}{u}\eta_{\Gamma}+\frac{1}{u}q_{\Gamma}\cdot\nabla_{\Gamma}u\leq
0&\text{on }\Gamma.\end{array}\right.$ (2.25)

We recall that a complete thermodynamic process is _admissible_ in
$\overline{\Omega}$ if it is compatible with the set of constitutive
conditions given in (2.23) at each point $x$ and at all times $t$. Since we
believe that our postulate (2.6) _should_ hold for all time-dependent
variables compatible with the extended law of balance of energy in (2.3), it
follows from [4, Theorem 6] (cf. also [6, Section 6, Theorem 1]) that the
Clausius-Duhem inequalities (2.25) imply for each
$S\in\left\\{\Omega,\Gamma\right\\}$ that

  * •

The instantaneous derivatives of $P_{S}$ and $H_{S}$ with respect to $g_{S}$
are zero; more precisely,

$D_{g_{S}}P_{S}=D_{g_{S}}H_{S}=0.$

  * •

The functional $H_{S}$ is determined by the functional $P_{S}$ through the
entropy relation:

$H_{S}=-D_{u}P_{S}.$

  * •

The modified heat conduction inequalities

$\frac{1}{u^{2}}\left(q_{S}\cdot g_{S}\right)\leq\sigma_{S},\text{
}S\in\left\\{\Omega,\Gamma\right\\},$

(with $q_{\Omega}=q$) hold for all smooth processes in $\overline{\Omega}$ and
for all $t$.

Above, $\sigma_{S}$ denotes the internal/boundary dissipation

$\sigma_{S}\left(t\right):=-\frac{1}{u\left(t\right)}\left[\delta_{u}P_{S}\left(u^{t},g_{S}^{t}\mid\overset{\centerdot}{u}_{r}^{t}\right)+\delta_{g_{S}}P_{S}\left(u^{t},g_{S}^{t}\mid\overset{\centerdot}{g}_{S,r}^{t}\right)\right],$

at time $t$, corresponding to the histories $\left(u^{t},g_{S}^{t}\right)$,
where $\overset{\centerdot}{u}$ is the present rate of change of $u$ at $x$,
$\overset{\centerdot}{u}_{r}^{t}$ is the past history of the rate of change of
$u$ at $x,$ and so on. Moreover, $D_{g_{S}}P_{S}$, $\delta_{u}P_{S}$ and
$\delta_{g_{S}}P_{S}$ denote the following linear differential operators

$\displaystyle D_{g_{S}}P_{S}\left(u^{t},g_{S}^{t}\right)\cdot l$
$\displaystyle=\left(\frac{\partial}{\partial
y}P_{S}\left(u_{r}^{t},g_{S,r}^{t},u,g_{S}+yl\right)\right)_{y=0},$
$\displaystyle\delta_{u}P_{S}\left(u^{t},g_{S}^{t}\mid k\right)$
$\displaystyle=\left(\frac{\partial}{\partial
y}P_{S}\left(u_{r}^{t}+yk,g_{S,r}^{t},u,g_{S}\right)\right)_{y=0},$
$\displaystyle\delta_{g_{S}}P_{S}\left(u^{t},g_{S}^{t}\mid\kappa\right)$
$\displaystyle=\left(\frac{\partial}{\partial
y}P_{S}\left(u_{r}^{t},g_{S,r}^{t}+y\kappa,u,g_{S}\right)\right)_{y=0},$

with identities which hold clearly for $\left(u^{t},g_{S}^{t}\right)\in
D_{S},$ $S\in\left\\{\Omega,\Gamma\right\\}$, $l\in\mathbb{R}^{\zeta_{S}}$
($\zeta_{\Omega}=d$, $\zeta_{\Gamma}=d-1$), and all $\left(k,\kappa\right)$
such that

$\int_{0}^{\infty}\left|k\left(s\right)\right|^{2}m_{S}\left(s\right)ds<\infty,\int_{0}^{\infty}\left|\kappa\left(s\right)\right|^{2}m_{S}\left(s\right)ds<\infty.$

To derive a simple model which is sufficiently general (see (2.28)-(2.29)
below), we need to consider a set of constitutive equations for $e_{S},q_{S}$,
$S\in\left\\{\Omega,\Gamma\right\\}$, which comply with the above implications
that the second law has on the response functions associated with a given
complete thermodynamic process in $\overline{\Omega}$. A fairly general
assumption is to consider small variations in the absolute temperature and
temperature gradients on both $\Omega$ and $\Gamma$, respectively, from
equilibrium reference values (cf. (2.1)-(2.2)). We take

$e_{\Omega}\left(u\right)=e_{\Omega,\infty}+\rho_{\Omega}c_{\Omega}u,\text{
}e_{\Gamma}\left(u\right)=e_{\Gamma,\infty}+\rho_{\Gamma}c_{\Gamma}u,$

where the involved positive constants $e_{S,\infty},$ $c_{S},$ $\rho_{S}$
denote the internal energies at equilibrium, the specific heat capacities and
material densities of $S\in\left\\{\Omega,\Gamma\right\\}$, respectively. In
addition, we assume that the internal and boundary fluxes satisfy the
following constitutive equations:

$\begin{array}[]{ll}q\left(t\right)=-\omega\nabla
u-\left(1-\omega\right)\int_{0}^{\infty}m_{\Omega}\left(s\right)\nabla
u^{t}\left(s\right)ds,&\\\ &\\\
q_{\Gamma}\left(t\right)=-\nu\nabla_{\Gamma}u-\left(1-\nu\right)\int_{0}^{\infty}m_{\Gamma}\left(s\right)\nabla_{\Gamma}u^{t}\left(s\right)ds,&\end{array}$
(2.26)

for some constants $\omega,\nu\in\left(0,1\right)$. Of course, when $m_{S}=0$,
$S\in\left\\{\Omega,\Gamma\right\\}$, we recover in (2.26) the usual Fourier
laws. Thus, in this context the constants $\omega,\nu$ correspond to the
instantaneous conductivities of $\Omega$ and $\Gamma$, respectively.
Furthermore, we assume in (2.4) nonlinear temperature dependent heat sources
$h_{S},$ $S\in\left\\{\Omega,\Gamma\right\\}$, namely, we take

$\begin{array}[]{ll}h_{\Omega}\left(t\right):=-f\left(u\left(t\right)\right)-\alpha\left(1-\omega\right)\int_{0}^{\infty}m_{\Omega}\left(s\right)u\left(x,t-s\right)ds,&\\\
&\\\ h_{\Gamma}\left(t\right):=-g\left(u\left(t\right)\right)-q\cdot
n-\beta\left(1-\nu\right)\int_{0}^{\infty}m_{\Gamma}\left(s\right)u\left(x,t-s\right)ds,&\end{array}$
(2.27)

for some $\beta>0,\alpha>0$, where the source on $\Gamma,$ $h_{\Gamma}$ is
also assumed to depend linearly on heat transport from inside of $\Omega$ in
directions normal to the boundary $\Gamma$. With these assumptions, (2.4)
yields the following system with memory

$\displaystyle\partial_{t}u-\omega\Delta
u-\left(1-\omega\right)\int_{0}^{\infty}m_{\Omega}\left(s\right)\Delta
u\left(x,t-s\right)ds+f\left(u\right)$ (2.28)
$\displaystyle+\alpha\left(1-\omega\right)\int_{0}^{\infty}m_{\Omega}\left(s\right)u\left(x,t-s\right)ds$
$\displaystyle=0,$

in $\Omega\times\left(0,\infty\right),$ subject to the boundary condition

$\displaystyle\partial_{t}u-\nu\Delta_{\Gamma}u+\omega\partial_{n}u+\left(1-\omega\right)\int_{0}^{\infty}m_{\Omega}\left(s\right)\partial_{n}u\left(x,t-s\right)ds$
(2.29)
$\displaystyle+\left(1-\nu\right)\int_{0}^{\infty}m_{\Gamma}\left(s\right)\left(-\Delta_{\Gamma}+\beta\right)u\left(x,t-s\right)ds+g\left(u\right)$
$\displaystyle=0,$

on $\Gamma\times\left(0,\infty\right).$

It is worth emphasizing that a different choice
$e_{\Gamma}\left(u\right)=e_{\Gamma,\infty}$ in (2.4) leads to a formulation
in which the boundary condition (2.29) is not dynamic any longer in the sense
that it does not contain the term $\partial_{t}u$ anymore. This stationary
boundary condition can be also reduced to (1.4) by a suitable choice of the
parameters $\beta,$ $\nu$ and the history $m_{\Gamma}$ involved in (2.26) and
(2.27). On the other hand, it is clear that if we (formally) choose
$m_{S}=\delta_{0}$ (the Dirac mass at zero), for each
$S\in\left\\{\Omega,\Gamma\right\\}$, equations (2.28)-(2.29) reduce into the
following system

$\left\\{\begin{array}[]{ll}\partial_{t}u-\Delta
u+\overline{f}\left(u\right)=0,&\text{in
}\Omega\times\left(0,\infty\right),\\\
\partial_{t}u-\Delta_{\Gamma}u+\partial_{n}u+\overline{g}\left(u\right)=0,&\text{on
}\Gamma\times\left(0,\infty\right),\end{array}\right.$ (2.30)

where $\overline{g}\left(x\right):=g\left(x\right)+\left(1-\nu\right)\beta x,$
$\overline{f}\left(x\right):=f\left(x\right)+\left(1-\omega\right)\alpha x$,
$x\in\mathbb{R}$. The latter has been investigated quite extensively recently
in many contexts (i.e., phase-field systems, heat conduction phenomena with
both a dissipative and non-dissipative source $\overline{g}$, Stefan problems,
and many more). We refer the reader to recent investigations pertaining the
system (2.30) in [1, 11, 12, 14, 13, 15], and the references therein.

##  3\. Past history formulation and functional setup

As in [8] (cf. also [19]), we can introduce the so-called integrated past
history of $u$, i.e., the auxiliary variable

$\eta^{t}\left(x,s\right)=\int_{0}^{s}u\left(x,t-y\right)dy,$

for $s,t>0$. Setting

$\mu_{\Omega}\left(s\right)=-\omega^{-1}\left(1-\omega\right)m_{\Omega}^{{}^{\prime}}\left(s\right),\text{
}\mu_{\Gamma}\left(s\right)=-\nu^{-1}\left(1-\nu\right)m_{\Gamma}^{{}^{\prime}}\left(s\right),$
(3.1)

assuming that $m_{S},$ $S\in\left\\{\Omega,\Gamma\right\\}$, is sufficiently
smooth and vanishing at $\infty$, formal integration by parts into
(2.28)-(2.29) yields

$\displaystyle\left(1-\omega\right)\int_{0}^{\infty}m_{\Omega}\left(s\right)\Delta
u\left(x,t-s\right)ds$
$\displaystyle=\omega\int_{0}^{\infty}\mu_{\Omega}\left(s\right)\Delta\eta^{t}\left(x,s\right)ds,$
$\displaystyle\left(1-\omega\right)\int_{0}^{\infty}m_{\Omega}\left(s\right)\partial_{n}u\left(x,t-s\right)ds$
$\displaystyle=\omega\int_{0}^{\infty}\mu_{\Omega}\left(s\right)\partial_{n}\eta^{t}\left(x,s\right)ds$

and

$\left(1-\nu\right)\int_{0}^{\infty}m_{\Gamma}\left(s\right)\left(-\Delta_{\Gamma}u\left(t-s\right)+\beta
u\left(t-s\right)\right)ds=\nu\int_{0}^{\infty}\mu_{\Gamma}\left(s\right)\left(-\Delta_{\Gamma}\eta^{t}\left(s\right)+\beta\eta^{t}\left(s\right)\right)ds.$
(3.2)

Thus, we consider the following formulation.

Problem P. Find a function $\left(u,\eta^{t}\right)$ such that

$\partial_{t}u-\omega\Delta
u-\omega\int_{0}^{\infty}\mu_{\Omega}\left(s\right)\Delta\eta^{t}\left(s\right)ds+\alpha\omega\int_{0}^{\infty}\mu_{\Omega}\left(s\right)\eta^{t}\left(x,s\right)ds+f\left(u\right)=0,$
(3.3)

in $\Omega\times\left(0,\infty\right),$

$\displaystyle\partial_{t}u-\nu\Delta_{\Gamma}u+\omega\partial_{n}u+\omega\int_{0}^{\infty}\mu_{\Omega}\left(s\right)\partial_{n}\eta^{t}\left(s\right)ds$
(3.4)
$\displaystyle+\nu\int_{0}^{\infty}\mu_{\Gamma}\left(s\right)\left(-\Delta_{\Gamma}\eta^{t}\left(s\right)+\beta\eta^{t}\left(s\right)\right)ds+g\left(u\right)$
$\displaystyle=0,$

on $\Gamma\times\left(0,\infty\right),$ and

$\partial_{t}\eta^{t}\left(s\right)+\partial_{s}\eta^{t}\left(s\right)=u\left(t\right),\text{
in }\overline{\Omega}\times\left(0,\infty\right),$ (3.5)

subject to the boundary conditions

$\eta^{t}\left(0\right)=0\text{, in
}\overline{\Omega}\times\left(0,\infty\right)$ (3.6)

and initial conditions

$u\left(0\right)=u_{0}\text{ in }\Omega,\text{ }u\left(0\right)=v_{0}\text{ on
}\Gamma,$ (3.7)

and

$\eta^{0}\left(s\right)=\eta_{0}\text{ in }\Omega\text{,
}\eta^{0}\left(s\right)=\xi_{0}\text{ on }\Gamma.$ (3.8)

Note that we do not require that the boundary traces of $u_{0}$ and $\eta_{0}$
equal to $v_{0}$ and $\xi_{0}$, respectively. Thus, we are solving a much more
general problem in which equation (3.3) is interpreted as an evolution
equation in the bulk $\Omega$ properly coupled with the equation (3.4) on the
boundary $\Gamma.$ Finally, we note that $\eta_{0},\xi_{0}$ are defined by

$\displaystyle\eta_{0}$ $\displaystyle=$
$\displaystyle\int_{0}^{s}u_{0}\left(x,-y\right)dy,\text{ in }\Omega\text{,
for }s>0,$ $\displaystyle\xi_{0}$ $\displaystyle=$
$\displaystyle\int_{0}^{s}v_{0}\left(x,-y\right)dy,\text{ on }\Gamma\text{,
for }s>0.$

However, from now on both $\eta_{0}$ and $\xi_{0}$ will be regarded as
independent of the initial data $u_{0},v_{0}.$ Indeed, below we will consider
a more general problem with respect to the original one. In order to give a
more rigorous notion of solutions for problem (3.3)-(3.8), we need to
introduce some terminology and the functional setting associated with this
system.

In the sequel, we denote by $\left\|\cdot\right\|_{L^{2}\left(\Gamma\right)}$
and $\left\|\cdot\right\|_{L^{2}\left(\Omega\right)}$ the norms on
$L^{2}\left(\Gamma\right)$ and $L^{2}\left(\Omega\right)$, whereas the inner
products in these spaces are denoted by
$\left\langle\cdot,\cdot\right\rangle_{L^{2}\left(\Gamma\right)}$ and
$\left\langle\cdot,\cdot\right\rangle_{L^{2}\left(\Omega\right)},$
respectively. Furthermore, the norms on $H^{s}\left(\Omega\right)$ and
$H^{s}\left(\Gamma\right),$ for $s>0,$ will be indicated by
$\left\|\cdot\right\|_{H^{s}}$ and
$\left\|\cdot\right\|_{H^{s}\left(\Gamma\right)}$, respectively. The symbol
$\left\langle\cdot,\cdot\right\rangle$ stands for pairing between any generic
Banach spaces $V$ and its dual $V^{\ast}$; $(u,v)^{\mathrm{tr}}$ will also
simply denote the vector-valued function $\binom{u}{v}.$ Constants below may
depend on various structural parameters such as $|\Omega|$, $|\Gamma|$,
$\ell_{1},$ $\ell_{2}$, etc, and these constants may even change from line to
line. Furthermore, we denote by $K(R)$ a generic monotonically increasing
function of $R>0,$ whose specific dependance on other parameters will be made
explicit on occurrence.

Let us now define the basic functional setup for (3.3)-(3.8). From this point
on, we assume that $\Omega$ is a bounded domain of $\mathbb{R}^{3}$ with
boundary $\Gamma$ which is of class $\mathcal{C}^{2}$. To this end, consider
the space $\mathbb{X}^{2}=L^{2}\left(\overline{\Omega},d\mu\right),$ where
$d\mu=dx_{\mid\Omega}\oplus d\sigma,$ such that $dx$ denotes the Lebesgue
measure on $\Omega$ and $d\sigma$ denotes the natural surface measure on
$\Gamma$. It is easy to see that
$\mathbb{X}^{2}=L^{2}\left(\Omega,dx\right)\oplus
L^{2}\left(\Gamma,d\sigma\right)$ may be identified under the natural norm

$\left\|u\right\|_{\mathbb{X}^{2}}^{2}=\int\limits_{\Omega}\left|u\left(x\right)\right|^{2}dx+\int\limits_{\Gamma}\left|u\left(x\right)\right|^{2}d\sigma.$

Moreover, if we identify every $u\in C\left(\overline{\Omega}\right)$ with
$U=\left(u_{\mid\Omega},u_{\mid\Gamma}\right)\in C\left(\Omega\right)\times
C\left(\Gamma\right)$, we may also define $\mathbb{X}^{2}$ to be the
completion of $C\left(\overline{\Omega}\right)$ in the norm
$\left\|\cdot\right\|_{\mathbb{X}^{2}}$. In general, any function
$u\in\mathbb{X}^{2}$ will be of the form $u=\binom{u_{1}}{u_{2}}$ with
$u_{1}\in L^{2}\left(\Omega,dx\right)$ and $u_{2}\in
L^{2}\left(\Gamma,d\sigma\right),$ and there need not be any connection
between $u_{1}$ and $u_{2}$. From now on, the inner product in the Hilbert
space $\mathbb{X}^{2}$ will be denoted by
$\left\langle\cdot,\cdot\right\rangle_{\mathbb{X}^{2}}.$ Next, recall that the
Dirichlet trace map
${\mathrm{tr_{D}}}:C^{\infty}\left(\overline{\Omega}\right)\rightarrow
C^{\infty}\left(\Gamma\right),$ defined by
${\mathrm{tr_{D}}}\left(u\right)=u_{\mid\Gamma}$ extends to a linear
continuous operator ${\mathrm{tr_{D}}}:H^{r}\left(\Omega\right)\rightarrow
H^{r-1/2}\left(\Gamma\right),$ for all $r>1/2$, which is onto for $1/2<r<3/2.$
This map also possesses a bounded right inverse
${\mathrm{tr_{D}}}^{-1}:H^{r-1/2}\left(\Gamma\right)\rightarrow
H^{r}\left(\Omega\right)$ such that
${\mathrm{tr_{D}}}\left({\mathrm{tr_{D}}}^{-1}\psi\right)=\psi,$ for any
$\psi\in H^{r-1/2}\left(\Gamma\right)$. We can thus introduce the subspaces of
$H^{r}\left(\Omega\right)\times H^{r-1/2}\left(\Gamma\right)$ and
$H^{r}\left(\Omega\right)\times H^{r}\left(\Gamma\right)$, respectively, by

$\displaystyle\mathbb{V}_{0}^{r}$ $\displaystyle:=\\{U=\left(u,\psi\right)\in
H^{r}\left(\Omega\right)\times
H^{r-1/2}\left(\Gamma\right):{\mathrm{tr_{D}}}\left(u\right)=\psi\\},$ (3.9)
$\displaystyle\mathbb{V}^{r}$
$\displaystyle:=\\{U=\left(u,\psi\right)\in\mathbb{V}_{0}^{r}:{\mathrm{tr_{D}}}\left(u\right)=\psi\in
H^{r}\left(\Gamma\right)\\},$

for every $r>1/2,$ and note that $\mathbb{V}_{0}^{r},$ $\mathbb{V}^{r}$ are
not product spaces. However, we have the following dense and compact
embeddings $\mathbb{V}_{0}^{r_{1}}\subset\mathbb{V}_{0}^{r_{2}},$ for any
$r_{1}>r_{2}>1/2$ (by definition, this also true for the sequence of spaces
$\mathbb{V}^{r_{1}}\subset\mathbb{V}^{r_{2}}$). Naturally, the norm on the
spaces $\mathbb{V}_{0}^{r},$ $\mathbb{V}^{r}$ are defined by

$\|U\|_{\mathbb{V}_{0}^{r}}^{2}:=\|u\|_{H^{r}}^{2}+\|\psi\|_{H^{r-1/2}(\Gamma)}^{2},\text{
}\|U\|_{\mathbb{V}^{r}}^{2}:=\|u\|_{H^{r}}^{2}+\|\psi\|_{H^{r}(\Gamma)}^{2}.$
(3.10)

In particular, the norm in the spaces $\mathbb{V}^{1},$ $\mathbb{V}_{0}^{1}$
can be defined as in terms of the following equivalent norms:

$\displaystyle\|U\|_{\mathbb{V}^{1}}$ $\displaystyle:$
$\displaystyle=\left(\omega\|\nabla
u\|_{L^{2}\left(\Omega\right)}^{2}+\nu\|\nabla_{\Gamma}\psi\|_{L^{2}(\Gamma)}^{2}+\beta\nu\left\|\psi\right\|_{L^{2}\left(\Gamma\right)}^{2}\right)^{1/2},\text{
}\nu>0,$ $\displaystyle\|U\|_{\mathbb{V}_{0}^{1}}$ $\displaystyle:$
$\displaystyle=\left(\omega\|\nabla
u\|_{L^{2}\left(\Omega\right)}^{2}+\alpha\omega\left\|u\right\|_{L^{2}\left(\Omega\right)}^{2}\right)^{1/2}.$

Now we introduce the spaces for the memory vector-valued function
$\left(\eta,\xi\right)$. For a given nonnegative, not identically equal to
zero, and measurable function $\theta_{S},$
$S\in\left\\{\Omega,\Gamma\right\\}$, defined on $\mathbb{R}_{+},$ and a real
Hilbert space $W$ (with inner product denoted by
$\left\langle\cdot,\cdot\right\rangle_{\mathrm{W}}$), let
$L_{\theta_{S}}^{2}\left(\mathbb{R}_{+};W\right)$ be the Hilbert space of
$W$-valued functions on $\mathbb{R}_{+}$, endowed with the following inner
product

$\left\langle\phi_{1},\phi_{2}\right\rangle_{L_{\theta_{S}}^{2}\left(\mathbb{R}_{+};W\right)}=\int_{0}^{\infty}\theta_{S}(s)\left\langle\phi_{1}\left(s\right),\phi_{2}\left(s\right)\right\rangle_{\mathrm{W}}ds.$
(3.11)

Moreover, for each $r>1/2$ we define

$L_{\theta_{\Omega}\oplus\theta_{\Gamma}}^{2}\left(\mathbb{R}_{+};\mathbb{V}^{r}\right)\simeq
L_{\theta_{\Omega}}^{2}\left(\mathbb{R}_{+};\mathbb{V}_{0}^{r}\right)\oplus
L_{\theta_{\Gamma}}^{2}\left(\mathbb{R}_{+};H^{r}\left(\Gamma\right)\right)$

as the Hilbert space of $\mathbb{V}^{r}$-valued functions
$\left(\eta,\xi\right)^{\mathrm{tr}}$ on $\mathbb{R}_{+}$ endowed with the
inner product

$\left\langle\binom{\eta_{1}}{\xi_{1}},\binom{\eta_{2}}{\xi_{2}}\right\rangle_{L_{\theta_{\Omega}\oplus\theta_{\Gamma}}^{2}\left(\mathbb{R}_{+};\mathbb{V}^{r}\right)}=\int_{0}^{\infty}\left(\theta_{\Omega}(s)\left\langle\eta_{1}\left(s\right),\eta_{2}\left(s\right)\right\rangle_{H^{r}}+\theta_{\Gamma}(s)\left\langle\xi_{1}\left(s\right),\xi_{2}\left(s\right)\right\rangle_{H^{r}\left(\Gamma\right)}\right)ds.$

Consequently, for $r>1/2$ we set

$\mathcal{M}_{\Omega}^{0}:=L_{\mu_{\Omega}}^{2}\left(\mathbb{R}_{+};L^{2}\left(\Omega\right)\right)\text{,
}\mathcal{M}_{\Omega}^{r}:=L_{\mu_{\Omega}}^{2}(\mathbb{R}_{+};\mathbb{V}_{0}^{r})\text{,
}\mathcal{M}_{\Gamma}^{r}:=L_{\mu_{\Gamma}}^{2}(\mathbb{R}_{+};H^{r}\left(\Gamma\right))$

and

$\mathcal{M}_{\Omega,\Gamma}^{0}:=L_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{2}\left(\mathbb{R}_{+};\mathbb{X}^{2}\right),\text{
}\mathcal{M}_{\Omega,\Gamma}^{r}:=L_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{2}\left(\mathbb{R}_{+};\mathbb{V}^{r}\right).$

Clearly, because of the topological identification
$H^{r}\left(\Omega\right)\simeq\mathbb{V}_{0}^{r}$, one has the inclusion
$\mathcal{M}_{\Omega,\Gamma}^{r}\subset\mathcal{M}_{\Omega}^{r}$ for each
$r>1/2$. In the sequel, we will also consider Hilbert spaces of the form
$W_{\mu_{\Omega}}^{k,2}\left(\mathbb{R}_{+};\mathbb{V}_{0}^{r}\right)$ for
$k\in\mathbb{N}$. When it is convenient, we will also use the notation

$\mathcal{H}_{\Omega,\Gamma}^{0,1}:=\mathbb{X}^{2}\times\mathcal{M}_{\Omega,\Gamma}^{1}\text{,
}\mathcal{H}_{\Omega,\Gamma}^{s,r}:=\mathbb{V}^{s}\times\mathcal{M}_{\Omega,\Gamma}^{r}\text{
for }s,r\geq 1.$

For matter of convenience, we will also set the inner product in
$\mathcal{M}_{\Omega,\Gamma}^{1},$ as follows:

$\displaystyle\left\langle\binom{\eta_{1}}{\xi_{1}},\binom{\eta_{2}}{\xi_{2}}\right\rangle_{L_{\theta_{\Omega}\oplus\theta_{\Gamma}}^{2}\left(\mathbb{R}_{+};\mathbb{V}^{1}\right)}$
$\displaystyle=\omega\int_{0}^{\infty}\theta_{\Omega}(s)\left(\left\langle\nabla\eta_{1}\left(s\right),\nabla\eta_{2}\left(s\right)\right\rangle_{L^{2}\left(\Omega\right)}+\alpha\left\langle\eta_{1}\left(s\right),\eta_{2}\left(s\right)\right\rangle_{L^{2}\left(\Omega\right)}\right)ds$
$\displaystyle+\nu\int_{0}^{\infty}\theta_{\Gamma}(s)\left(\left\langle\nabla_{\Gamma}\xi_{1}\left(s\right),\nabla_{\Gamma}\xi_{2}\left(s\right)\right\rangle_{L^{2}\left(\Gamma\right)}+\beta\left\langle\xi_{1}\left(s\right),\xi_{2}\left(s\right)\right\rangle_{L^{2}\left(\Gamma\right)}\right)ds.$

The following basic elliptic estimate is taken from [13, Lemma 2.2].

######  Lemma 3.1.

Consider the linear boundary value problem,

$\left\\{\begin{array}[]{rl}-\Delta u&=p_{1}~{}~{}\text{in}~{}\Omega,\\\
-\Delta_{\Gamma}u+\partial_{n}u+\beta
u&=p_{2}~{}~{}\text{on}~{}\Gamma.\end{array}\right.$ (3.12)

If $(p_{1},p_{2})^{{\mathrm{tr}}}\in H^{s}(\Omega)\times H^{s}(\Gamma)$, for
$s\geq 0$ and $s+\frac{1}{2}\not\in\mathbb{N}$, then the following estimate
holds for some constant $C>0$,

$\|u\|_{H^{s+2}}+\|u\|_{H^{s+2}(\Gamma)}\leq
C\left(\|p_{1}\|_{H^{s}}+\|p_{2}\|_{H^{s}(\Gamma)}\right).$ (3.13)

We also recall the following basic inequality from [11, Lemma A.2].

######  Lemma 3.2.

Let $s>1$ and $u\in H^{1}(\Omega)$. Then, for every $\varepsilon>0$, there
exists a positive constant $C_{\varepsilon}\sim\varepsilon^{-1}$ such that,

$\|u\|_{L^{s}(\Gamma)}^{s}\leq\varepsilon\|\nabla
u\|_{L^{2}(\Omega)}^{2}+C_{\varepsilon}\left(\|u\|_{L^{\gamma}(\Omega)}^{\gamma}+1\right),$
(3.14)

where $\gamma=\max\\{s,2(s-1)\\}$.

Next, we consider the linear (self-adjoint, positive) operator
$\mathrm{C}\psi:=\mathrm{C}_{\beta}\psi=-\Delta_{\Gamma}\psi+\beta\psi$ acting
on $D\left(\mathrm{C}\right)=H^{2}\left(\Gamma\right)$. The basic (linear)
operator, associated with problem (3.3)-(3.5), is the so-called “Wentzell”
Laplace operator. Recall that $\omega\in\left(0,1\right)$. We let

$\displaystyle\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}}\binom{u_{1}}{u_{2}}$
$\displaystyle:=\left(\begin{array}[]{cc}-\omega\Delta+\alpha\omega I&0\\\
\omega\partial_{n}\left(\cdot\right)&\nu\mathrm{C}\end{array}\right)\left(\begin{array}[]{c}u_{1}\\\
u_{2}\end{array}\right)$ (3.19)
$\displaystyle=\mathrm{A_{W}^{\alpha,0,0,\omega}}\binom{u_{1}}{u_{2}}+\binom{0}{\nu\mathrm{C}u_{2}},$

with

$D\left(\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}}\right):=\left\\{U=\binom{u_{1}}{u_{2}}\in\mathbb{Y}:-\Delta
u_{1}\in L^{2}\left(\Omega\right),\text{
}\omega\partial_{n}u_{1}-\nu\mathrm{C}u_{2}\in
L^{2}\left(\Gamma\right)\right\\},$ (3.20)

where $\mathbb{Y}:=\mathbb{V}_{0}^{1}$ if $\nu=0,$ and
$\mathbb{Y}:=\mathbb{V}^{1}$ if $\nu>0.$ It is well-known that
$(\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}},D(\mathrm{A_{W}^{\alpha,\beta,\nu,\omega})})$
is self-adjoint and nonnegative operator on $\mathbb{X}^{2}$ whenever
$\alpha,\beta,\nu\geq 0,$ and $\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}}>0$ if
either $\alpha>0$ or $\beta>0$. Moreover, the resolvent operator
$(I+\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}})^{-1}\in\mathcal{L}\left(\mathbb{X}^{2}\right)$
is compact. Moreover, since $\Gamma$ is of class $\mathcal{C}^{2},$ then
$D(\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}})=\mathbb{V}^{2}$ if $\nu>0$.
Indeed, for any $\alpha,\beta\geq 0$ with
$\left(\alpha,\beta\right)\neq\left(0,0\right),$ the map
$\Psi:U\mapsto\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}}U,$ when viewed as a map
from $\mathbb{V}_{2}$ into $\mathbb{X}^{2}=L^{2}\left(\Omega\right)\times
L^{2}\left(\Gamma\right),$ is an isomorphism and there exists a positive
constant $C_{\ast}$, independent of $U=\left(u,\psi\right)^{\mathrm{tr}}$,
such that

$C_{\ast}^{-1}\left\|U\right\|_{\mathbb{V}^{2}}\leq\left\|\Psi\left(U\right)\right\|_{\mathbb{X}^{2}}\leq
C_{\ast}\left\|U\right\|_{\mathbb{V}^{2}},$ (3.21)

for all $U\in\mathbb{V}^{2}$ (cf. Lemma 3.1). Whenever $\nu=0$, by elliptic
regularity theory and $U\in D(\mathrm{A_{W}^{\alpha,\beta,0,\omega}})$ one has
$u\in H^{3/2}\left(\Omega\right)$ and $\psi={\mathrm{tr_{D}}}\left(u\right)\in
H^{1}\left(\Gamma\right)$, since the Dirichlet-to-Neumann map is bounded from
$H^{1}\left(\Gamma\right)$ to $L^{2}\left(\Gamma\right)$; hence
$D(\mathrm{A_{W}^{\alpha,\beta,0,\omega}})=\mathbb{W}$, where $\mathbb{W}$ is
the Hilbert space equipped with the following (equivalent) norm

$\left\|U\right\|_{\mathbb{W}}^{2}:=\left\|U\right\|_{\mathbb{V}_{0}^{3/2}}^{2}+\left\|\Delta
u\right\|_{L^{2}\left(\Omega\right)}^{2}+\left\|\partial_{n}u\right\|_{L^{2}\left(\Gamma\right)}^{2}.$

We refer the reader to more details to e.g., [1], [15], [2] and the references
therein. We now have all the necessary ingredients to introduce a rigorous
formulation of problem P in the next section.

##  4\. Variational formulation and well-posedness

We need the following hypotheses for problem P. For the function $\mu_{S},$
$S\in\left\\{\Omega,\Gamma\right\\}$, given by (3.1), we consider the
following assumptions (cf., e.g. [8], [16] and [17]). Assume

$\displaystyle\mu_{S}\in C^{1}(\mathbb{R}_{+})\cap L^{1}(\mathbb{R}_{+}),$
(4.1) $\displaystyle\mu_{S}(s)\geq 0~{}~{}\text{for all}~{}~{}s\geq 0,$ (4.2)
$\displaystyle\mu_{S}^{\prime}(s)\leq 0~{}~{}\text{for all}~{}~{}s\geq 0.$
(4.3)

These assumptions are equivalent to assuming that $m_{S}(s),$
$S\in\left\\{\Omega,\Gamma\right\\}$, is a bounded, positive, nonincreasing,
convex function of class $C^{2}$. These conditions are commonly used in the
literature (see, for example, [8], [16] and [19]) to establish existence and
uniqueness of continuous global weak solutions for Coleman-Gurtin type
equations subject to Dirichlet boundary conditions.

As far as natural conditions for the nonlinear terms are concerned, we assume
$f$, $g\in C^{1}(\mathbb{R})$ satisfy the sign conditions

$f^{\prime}(s)\geq-M_{f},\text{ }g^{\prime}(s)\geq-M_{g},\text{ for all
}s\in\mathbb{R}\text{,}$ (4.4)

for some $M_{f},M_{g}>0$ and the growth assumptions, for all $s\in\mathbb{R}$,

$|f(s)|\leq\ell_{1}(1+|s|^{r_{1}-1}),\text{
}|g(s)|\leq\ell_{2}(1+|s|^{r_{2}-1}),$ (4.5)

for some positive constants $\ell_{1}$ and $\ell_{2}$, and where
$r_{1},r_{2}\geq 2$. Let now

$\widetilde{g}\left(s\right):=g\left(s\right)-\nu\beta s,\text{ for
}s\in\mathbb{R}\text{.}$ (4.6)

In addition, we assume there exists $\varepsilon\in(0,\omega)$ so that the
following balance condition

$\liminf_{|s|\rightarrow\infty}\frac{f(s)s+\frac{|\Gamma|}{|\Omega|}\widetilde{g}(s)s-\frac{C_{\Omega}^{2}|\Gamma|^{2}}{4\varepsilon|\Omega|^{2}}|\widetilde{g}^{\prime}(s)s+\widetilde{g}(s)|^{2}}{\left|s\right|^{r_{1}}}>0$
(4.7)

holds for $r_{1}\geq\max\\{r_{2},2(r_{2}-1)\\}$. The number $C_{\Omega}>0$ is
the best Sobolev constant in the following Sobolev-Poincaré inequality

$\left\|u-\left\langle
u\right\rangle_{\Gamma}\right\|_{L^{2}\left(\Omega\right)}\leq
C_{\Omega}\left\|\nabla u\right\|_{L^{2}(\Omega)},\text{ }\left\langle
u\right\rangle_{\Gamma}:=\frac{1}{\left|\Gamma\right|}\int\limits_{\Gamma}\mathrm{tr_{D}}\left(u\right)d\sigma,$
(4.8)

for all $u\in H^{1}\left(\Omega\right)$, see [25, Lemma 3.1].

The assumption (4.7) deserves some additional comments. Suppose that that for
$\left|y\right|\rightarrow\infty,$ both the internal and boundary functions
behave accordingly to the following laws:

$\lim_{\left|y\right|\rightarrow\infty}\frac{f^{{}^{\prime}}\left(y\right)}{\left|y\right|^{r_{1}-2}}=\left(r_{1}-1\right)c_{f}\text{,
}\lim_{\left|y\right|\rightarrow\infty}\frac{\widetilde{g}^{{}^{\prime}}\left(y\right)}{\left|y\right|^{r_{2}-2}}=\left(r_{2}-1\right)c_{\widetilde{g}},$
(4.9)

for some $c_{f},c_{\widetilde{g}}\in\mathbb{R}\backslash\left\\{0\right\\}$.
In particular, it holds

$f\left(y\right)y\sim c_{f}\left|y\right|^{r_{1}},\text{
}\widetilde{g}\left(y\right)y\sim
c_{\widetilde{g}}\left|y\right|^{r_{2}}\text{ as
}\left|y\right|\rightarrow\infty.$

For the case of bulk dissipation (i.e., $c_{f}>0$) and anti-dissipative
behavior at the boundary $\Gamma$ (i.e., $c_{\widetilde{g}}<0$), assumption
(4.7) is automatically satisfied provided that
$r_{1}>\max\\{r_{2},2(r_{2}-1)\\}$. Furthermore, if
$2<r_{2}<2\left(r_{2}-1\right)=r_{1}$ and

$c_{f}>\frac{1}{4\varepsilon}\left(\frac{C_{\Omega}\left|\Gamma\right|c_{\widetilde{g}}r_{2}}{\left|\Omega\right|}\right)^{2},$
(4.10)

for some $\varepsilon\in(0,\omega)$, then once again (4.7) is satisfied. In
the case when $f$ and $g$ are sublinear (i.e., $r_{1}=r_{2}=2$ in (4.5)), the
condition (4.7) is also automatically satisfied provided that

$\left(c_{f}+\frac{|\Gamma|}{|\Omega|}c_{\widetilde{g}}\right)>\frac{1}{\varepsilon}\left(\frac{C_{\Omega}\left|\Gamma\right|c_{\widetilde{g}}}{\left|\Omega\right|}\right)^{2}$
(4.11)

for some $\varepsilon\in\left(0,\omega\right)$. Of course, when both the bulk
and boundary nonlinearities are dissipative, i.e., there exist two constants
$C_{f}>0,C_{g}>0$ such that, additionally to (4.5),

$\left\\{\begin{array}[]{l}f\left(s\right)s\geq
C_{f}\left|s\right|^{r_{1}},\\\ \widetilde{g}\left(s\right)s\geq
C_{g}\left|s\right|^{r_{2}},\end{array}\right.$ (4.12)

for all $\left|s\right|\geq s_{0}$, for some sufficiently large $s_{0}>0,$
condition (4.7) can be dropped and is no longer required (see [11]).

In order to introduce a rigorous formulation for problem P, we define

$D(\mathrm{T}):=\left\\{\Phi=\binom{\eta^{t}}{\xi^{t}}\in\mathcal{M}_{\Omega,\Gamma}^{1}:\partial_{s}\Phi\in\mathcal{M}_{\Omega,\Gamma}^{1},\text{
}\Phi(0)=0\right\\}$ (4.13)

and consider the linear (unbounded) operator
$\mathrm{T}:D(\mathrm{T})\rightarrow\mathcal{M}_{\Omega,\Gamma}^{1}$ by

$\mathrm{T}\Phi=-\binom{\frac{d\eta}{ds}}{\frac{d\xi}{ds}},\text{
}\Phi=\binom{\eta^{t}}{\xi^{t}}\in D(\mathrm{T}).$

The follow result can be proven following [19, Theorem 3.1].

######  Proposition 4.1.

The operator $\mathrm{T}$ with domain $D(\mathrm{T})$ is an infinitesimal
generator of a strongly continuous semigroup of contractions on
$\mathcal{M}_{\Omega,\Gamma}^{1}$, denoted $e^{\mathrm{T}t}$.

As a consequence, we also have (cf., e.g. [23, Corollary IV.2.2]).

######  Corollary 4.2.

Let $T>0$ and assume $U=\binom{u}{v}\in L^{1}(0,T;\mathbb{V}^{1})$. Then, for
every $\Phi_{0}\in\mathcal{M}_{\Omega,\Gamma}^{1}$, the Cauchy problem for
$\Phi^{t}=\binom{\eta^{t}}{\xi^{t}},$

$\left\\{\begin{array}[]{ll}\partial_{t}\Phi^{t}=\mathrm{T}\Phi^{t}+U(t),&\text{for}~{}~{}t>0,\\\
\Phi^{0}=\Phi_{0},&\end{array}\right.$ (4.14)

has a unique (mild) solution $\Phi\in
C([0,T];\mathcal{M}_{\Omega,\Gamma}^{1})$ which can be explicitly given as

$\Phi^{t}(s)=\left\\{\begin{array}[]{ll}\displaystyle\int_{0}^{s}U(t-y)dy,&\text{for}~{}~{}0<s\leq
t,\\\ \displaystyle\Phi_{0}(s-t)+\int_{s}^{t}U(t-y)dy,&\text{for
}~{}~{}s>t,\end{array}\right.$ (4.15)

cf. also [8, Section 3.2] and [19, Section 3].

######  Remark 4.3.

(i) Note that, from assumption (4.3), the following inequality

$\left\langle\mathrm{T}\Phi,\Phi\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}\leq
0$ (4.16)

holds for all $\Phi\in D(\mathrm{T})$.

(ii) If $\Phi_{0}\in D(\mathrm{T})$ and $\partial_{t}U\in
L^{1}\left(0,T;\mathbb{V}^{1}\right)$, the function $\Phi^{t}$ given by (4.15)
satisfies (4.14) in the strong sense a.e. on $\left(0,T\right),$ for any
$T>0.$

We are now ready to introduce the rigorous (variational) formulation of
problem P.

######  Definition 4.4.

Let $\alpha,\beta>0$, $\omega,\nu\in(0,1)$ and $T>0$. Given
$\binom{u_{0}}{v_{0}}\in\mathbb{X}^{2}$,
$\binom{\eta_{0}}{\xi_{0}}\in\mathcal{M}_{\Omega,\Gamma}^{1}$, we seek to find
functions $U\left(t\right)=\binom{u\left(t\right)}{v\left(t\right)},$
$\Phi^{t}=\binom{\eta^{t}}{\xi^{t}}$ with the following properties:

$\displaystyle U$ $\displaystyle\in
L^{\infty}\left(0,T;\mathbb{X}^{2}\right)\cap L^{2}(0,T;\mathbb{V}^{1}),\text{
}\Phi\in L^{\infty}\left(0,T;\mathcal{M}_{\Omega,\Gamma}^{1}\right),$ (4.17)
$\displaystyle u$ $\displaystyle\in
L^{r_{1}}(\Omega\times\left(0,T\right)),\text{ }v\in
L^{r_{2}}(\Gamma\times(0,T)),$ (4.18) $\displaystyle\partial_{t}U$
$\displaystyle\in
L^{2}\left(0,T;(\mathbb{V}^{1})^{\ast}\right)\oplus\left(L^{r_{1}^{\prime}}(\Omega\times(0,T))\times
L^{r_{2}^{\prime}}(\Gamma\times(0,T))\right),$ (4.19)
$\displaystyle\partial_{t}\Phi$ $\displaystyle\in
L^{2}\left(0,T;W_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{-1,2}(\mathbb{R}_{+};\mathbb{V}^{1})\right).$
(4.20)

$\left(U,\Phi^{t}\right)$ is said to be a weak solution to problem P if
$v\left(t\right)={\mathrm{tr_{D}}}\left(u\left(t\right)\right)$ and
$\xi^{t}={\mathrm{tr_{D}}}\left(\eta^{t}\right)$ for almost all $t\in(0,T]$,
and $\left(U\left(t\right),\Phi^{t}\right)$ satisfies, for almost all
$t\in(0,T]$,

$\begin{array}[]{ll}\left\langle\partial_{t}U(t),\Xi\right\rangle_{\mathbb{X}^{2}}+\left\langle\mathrm{A_{W}^{0,\beta,\nu,\omega}}U(t),\Xi\right\rangle_{\mathbb{X}^{2}}+\int_{0}^{\infty}\mu_{\Omega}(s)\left\langle\mathrm{A_{W}^{\alpha,0,0,\omega}}\Phi^{t}\left(s\right),\Xi\right\rangle_{\mathbb{X}^{2}}ds&\\\
+\nu\int_{0}^{\infty}\mu_{\Gamma}(s)\left\langle\mathrm{C}\xi^{t}\left(s\right),\varsigma_{\mid\Gamma}\right\rangle_{L^{2}\left(\Gamma\right)}ds+\left\langle
F\left(U(t)\right),\Xi\right\rangle_{\mathbb{X}^{2}}=0,&\\\
\left\langle\partial_{t}\eta^{t},\rho\right\rangle_{\mathcal{M}_{\Omega}^{1}}=\left\langle-\frac{d}{ds}\eta^{t},\rho\right\rangle_{\mathcal{M}_{\Omega}^{1}}+\left\langle
u(t),\rho\right\rangle_{\mathcal{M}_{\Omega}^{1}},&\\\
\left\langle\partial_{t}\xi^{t},\rho_{\mid\Gamma}\right\rangle_{\mathcal{M}_{\Gamma}^{1}}=\left\langle-\frac{d}{ds}\xi^{t},\rho_{\mid\Gamma}\right\rangle_{\mathcal{M}_{\Gamma}^{1}}+\left\langle
v(t),\rho_{\mid\Gamma}\right\rangle_{\mathcal{M}_{\Gamma}^{1}},&\end{array}$
(4.21)

for all
$\Xi=\binom{\varsigma}{\varsigma_{\mid\Gamma}}\in\mathbb{V}^{1}\oplus\left(L^{r_{1}}(\Omega)\times
L^{r_{2}}(\Gamma)\right)$, all
$\Pi=\binom{\rho}{\rho_{\mid\Gamma}}\in\mathcal{M}_{\Omega,\Gamma}^{1}$ and

$U\left(0\right)=U_{0}=\left(u_{0},v_{0}\right)^{{\mathrm{tr}}},\text{
}\Phi^{0}=\Phi_{0}=\left(\eta_{0},\xi_{0}\right)^{{\mathrm{tr}}}.$ (4.22)

Above, we have set $F:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},$

$F\left(U\right):=\binom{f\left(u\right)}{\widetilde{g}\left(v\right)},$

with $\widetilde{g}$ defined as in (4.6). The function $[0,T]\ni
t\mapsto(U(t),\Phi^{t})$ is called a global weak solution if it is a weak
solution for every $T>0$.

In the sequel, if the initial datum $\left(U_{0},\Phi_{0}\right)$ is more
smooth, the following notion of strong solution will also become important.

######  Definition 4.5.

Let $\alpha,\beta>0$, $\omega,\nu\in(0,1)$ and $T>0$. Given
$\binom{u_{0}}{v_{0}}\in\mathbb{V}^{1}$,
$\binom{\eta_{0}}{\xi_{0}}\in\mathcal{M}_{\Omega,\Gamma}^{2}$, the pair of
functions $U\left(t\right)=\binom{u\left(t\right)}{v\left(t\right)},$
$\Phi^{t}=\binom{\eta^{t}}{\xi^{t}}$ satisfying

$\displaystyle U$ $\displaystyle\in
L^{\infty}\left(0,T;\mathbb{V}^{1}\right)\cap L^{2}(0,T;\mathbb{V}^{2}),\text{
}$ (4.23) $\displaystyle\Phi$ $\displaystyle\in
L^{\infty}(0,T;\mathcal{M}_{\Omega,\Gamma}^{2}),$ $\displaystyle\partial_{t}U$
$\displaystyle\in L^{\infty}\left(0,T;(\mathbb{V}^{1})^{\ast}\right)\cap
L^{2}\left(0,T;\mathbb{X}^{2}\right),$ $\displaystyle\partial_{t}\Phi$
$\displaystyle\in
L^{2}\left(0,T;L_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{2}\left(\mathbb{R}_{+};\mathbb{X}^{2}\right)\right),$

is called a strong solution to problem P if
$v\left(t\right)={\mathrm{tr_{D}}}\left(u\left(t\right)\right)$ and
$\xi^{t}={\mathrm{tr_{D}}}\left(\eta^{t}\right)$ for almost all $t\in(0,T]$,
and additionally, $\left(U\left(t\right),\Phi^{t}\right)$ satisfies (4.21),
a.e. for $t\in(0,T]$, for all $\Xi\in\mathbb{V}^{1}$,
$\Pi\in\mathcal{M}_{\Omega,\Gamma}^{1}$, and

$U\left(0\right)=U_{0}=\left(u_{0},v_{0}\right)^{{\mathrm{tr}}},\text{
}\Phi^{0}=\Phi_{0}=\left(\eta_{0},\xi_{0}\right)^{{\mathrm{tr}}}.$ (4.24)

The function $[0,T]\ni t\mapsto(U(t),\Phi^{t})$ is called a global strong
solution if it is a strong solution for every $T>0$.

######  Remark 4.6.

Note that a strong solution is incidently more smooth than a weak solution in
the sense of Definition 4.4. Moreover, on account of standard embedding
theorems the regularity $U\in L^{\infty}\left(0,T;\mathbb{V}^{1}\right)\cap
L^{2}(0,T;\mathbb{V}^{2})$ implies that

$u\in L^{\infty}\left(0,T;L^{6}\left(\Omega\right)\right)\cap
L^{q}\left(0,T;L^{p}\left(\Omega\right)\right)$

for any $p\in\left(6,\infty\right)$ , $1\leq q\leq 4p/\left(p-6\right)$, and
${\mathrm{tr_{D}}}\left(u\right)\in
L^{\infty}\left(0,T;L^{s}\left(\Omega\right)\right)$, for any
$s\in\left(1,\infty\right)$.

Another notion of strong solution to problem P, although weaker than the
notion in Definition 4.5, can be introduced as follows.

######  Definition 4.7.

The pair $U=\binom{u}{v}$ and $\Phi=\binom{\eta}{\xi}$ is called a quasi-
strong solution of problem P on $[0,T)$ if $(U(t),\Phi^{t})$ satisfies the
equations (4.21)-(4.22) for all $\Xi\in\mathbb{V}^{1}$,
$\Pi\in\mathcal{M}_{\Omega,\Gamma}^{1}$, almost everywhere on
$\left(0,T\right)$ and if it has the regularity properties:

$\displaystyle U$ $\displaystyle\in$ $\displaystyle
L^{\infty}(0,T;\mathbb{V}^{1})\cap W^{1,2}(0,T;\mathbb{V}^{1}),$ (4.25)
$\displaystyle\Phi$ $\displaystyle\in$ $\displaystyle
L^{\infty}(0,T;D\left(\mathrm{T}\right)),$ (4.26) $\displaystyle\partial_{t}U$
$\displaystyle\in$ $\displaystyle L^{\infty}\left(0,T;\mathbb{X}^{2}\right),$
(4.27) $\displaystyle\partial_{t}\Phi$ $\displaystyle\in$ $\displaystyle
L^{\infty}\left(0,T;\mathcal{M}_{\Omega,\Gamma}^{1}\right).$ (4.28)

As before, the function $[0,T]\ni t\mapsto(U(t),\Phi^{t})$ is called a global
quasi-strong solution if it is a quasi-strong solution for every $T>0$.

Our first result in this section is contained in the following theorem. It
allows us to obtain generalized solutions in the sense of Definition 4.4.

######  Theorem 4.8.

Assume (4.1)-(4.3) and (4.5)-(4.7) hold. For each $\alpha,\beta>0$,
$\omega,\nu\in(0,1)$ and $T>0$, and for any
$U_{0}=(u_{0},v_{0})^{{\mathrm{tr}}}\in\mathbb{X}^{2}$,
$\Phi_{0}=(\eta_{0},\xi_{0})^{{\mathrm{tr}}}\in\mathcal{M}_{\Omega,\Gamma}^{1},$
there exists at least one (global) weak solution $\left(U,\Phi\right)\in
C(\left[0,T\right];\mathcal{H}_{\Omega,\Gamma}^{0,1})$ to problem P.

###### Proof.

The proof is divided into several steps. Much of the motivation for the above
theorem comes from [11]. Indeed, the dissipativity induced by the balance
condition (4.7) will be exploited to obtain an _apriori_ bound. Of course,
several modifications need to be made in order to incorporate the dynamic
boundary conditions with memory into the framework.

Step 1. (An _apriori_ bound) To begin, we derive an _apriori_ energy estimate
for any (sufficiently) smooth solution $(U,\Phi)$ of problem P. Under the
assumptions of the theorem, we claim that the following estimate holds:

$\displaystyle\|U(t)\|_{\mathbb{X}^{2}}^{2}+\left\|\Phi^{t}\right\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}^{2}-2\left\langle\mathrm{T}\Phi^{t},\Phi^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+2\int_{0}^{t}\left(\|U(\tau)\|_{\mathbb{V}^{1}}^{2}+\|u(\tau)\|_{L^{r_{1}}(\Omega)}^{r_{1}}\right)d\tau$
(4.29) $\displaystyle\leq
C_{T}\left(1+\|U(0)\|_{\mathbb{X}^{2}}^{2}+\left\|\Phi^{0}\right\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}^{2}\right),$

for all $t\in[0,T]$, for some constant $C>0$, independent of $(U,\Phi)$ and
$t$.

We now show (4.29). In Definition 4.4 we are allowed to take, for almost all
$t\in[0,T]$,

$\Xi=U(t)=\left(u(t),u(t)_{\mid\Gamma}\right)^{{\mathrm{tr}}}\in\mathbb{V}^{1}\cap\left(L^{r_{1}}(\Omega)\times
L^{r_{2}}(\Gamma)\right)$

and

$\Pi=\Phi^{t}=\left(\eta^{t},\xi^{t}\right)^{{\mathrm{tr}}}\in\mathcal{M}_{\Omega,\Gamma}^{1}.$

Then we obtain the differential identities

$\frac{1}{2}\frac{d}{dt}\|U\|_{\mathbb{X}^{2}}^{2}+\left\langle\mathrm{A_{W}^{0,\beta,\nu,\omega}}U,U\right\rangle_{\mathbb{X}^{2}}+\left\langle\Phi^{t},U\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left\langle
F(U),U\right\rangle_{\mathbb{X}^{2}}=0,$ (4.30)

where

$\displaystyle\left\langle\Phi^{t},U\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}$
$\displaystyle=\omega\int_{0}^{\infty}\mu_{\Omega}(s)\left(\left\langle\nabla\eta^{t}\left(s\right),\nabla
u\right\rangle_{L^{2}\left(\Omega\right)}+\alpha\left\langle\eta^{t}\left(s\right),u\right\rangle_{L^{2}\left(\Omega\right)}\right)ds$
(4.31)
$\displaystyle+\nu\int_{0}^{\infty}\mu_{\Gamma}(s)\left(\left\langle\nabla_{\Gamma}\xi^{t}\left(s\right),\nabla_{\Gamma}u\right\rangle_{L^{2}\left(\Gamma\right)}+\beta\left\langle\xi^{t}\left(s\right),u\right\rangle_{L^{2}\left(\Gamma\right)}\right)ds$
$\displaystyle=\int_{0}^{\infty}\mu_{\Omega}(s)\left\langle\mathrm{A_{W}^{\alpha,0,0,\omega}}\Phi^{t}\left(s\right),U\right\rangle_{\mathbb{X}^{2}}ds+\nu\int_{0}^{\infty}\mu_{\Gamma}(s)\left\langle\mathrm{C}\xi^{t}\left(s\right),u\right\rangle_{L^{2}\left(\Gamma\right)}ds,$

and

$\
\frac{1}{2}\frac{d}{dt}\|\Phi^{t}\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}^{2}=\left\langle\mathrm{T}\Phi^{t},\Phi^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left\langle
U,\Phi^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}},$ (4.32)

which hold for almost all $t\in[0,T]$. Adding these identities together and
recalling (4.16), we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\left(\|U\|_{\mathbb{X}^{2}}^{2}+\|\Phi^{t}\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}^{2}\right)-\left\langle\mathrm{T}\Phi^{t},\Phi^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left(\omega\|\nabla
u\|_{L^{2}\left(\Omega\right)}^{2}+\nu\|\nabla_{\Gamma}u\|_{L^{2}(\Gamma)}^{2}+\beta\left\|u\right\|_{L^{2}\left(\Gamma\right)}^{2}\right)$
(4.33) $\displaystyle\leq-\left\langle
f(u),u\right\rangle_{L^{2}\left(\Omega\right)}-\left\langle\widetilde{g}\left(u\right),u\right\rangle_{L^{2}\left(\Gamma\right)}.$

Following [11, (2.22)] and [25, (3.11)], we estimate the product with $F$ on
the right-hand side of (4.33), as follows:

$\displaystyle\left\langle F(U),U\right\rangle_{\mathbb{X}^{2}}$
$\displaystyle=\left\langle
f(u),u\right\rangle_{L^{2}\left(\Omega\right)}+\left\langle\widetilde{g}\left(u\right),u\right\rangle_{L^{2}\left(\Gamma\right)}$
(4.34)
$\displaystyle=\int_{\Omega}\left(f(u)u+\frac{|\Gamma|}{|\Omega|}\widetilde{g}(u)u\right)dx-\frac{|\Gamma|}{|\Omega|}\int_{\Omega}\left(\widetilde{g}(u)u-\frac{1}{|\Gamma|}\int_{\Gamma}\widetilde{g}(u)u\mathrm{d}\sigma\right)dx.$

Exploiting Poincaré inequality (4.8) and Young’s inequality, we see that for
all $\varepsilon\in(0,\omega)$,

$\displaystyle\frac{|\Gamma|}{|\Omega|}\int_{\Omega}\left(\widetilde{g}(u)u-\frac{1}{|\Gamma|}\int_{\Gamma}\widetilde{g}(u)ud\sigma\right)dx$
$\displaystyle\leq
C_{\Omega}\frac{|\Gamma|}{|\Omega|}\int_{\Omega}|\nabla(\widetilde{g}(u)u)|dx$
(4.35) $\displaystyle=C_{\Omega}\frac{|\Gamma|}{|\Omega|}\int_{\Omega}|\nabla
u(\widetilde{g}^{\prime}(u)u+\widetilde{g}(u))|dx$
$\displaystyle\leq\varepsilon\|\nabla
u\|_{L^{2}(\Omega)}^{2}+\frac{C_{\Omega}^{2}|\Gamma|^{2}}{4\varepsilon|\Omega|^{2}}\int_{\Omega}|\widetilde{g}^{\prime}(u)u+\widetilde{g}(u)|^{2}dx.$

Combining (4.34)-(4.35) and applying assumption (4.7) yields

$\left\langle
F(U),U\right\rangle_{\mathbb{X}^{2}}\geq\delta\|u\|_{L^{r_{1}}(\Omega)}^{r_{1}}-\varepsilon\|\nabla
u\|_{L^{2}(\Omega)}^{2}-C_{\delta},$ (4.36)

for some positive constants $\delta$ and $C_{\delta}$ that are independent of
$U$, $t$ and $\varepsilon$. Plugging (4.36) into (4.33) gives, for almost all
$t\in[0,T]$,

$\displaystyle\frac{1}{2}\frac{d}{dt}\left(\|U\|_{\mathbb{X}^{2}}^{2}+\|\Phi^{t}\|_{\mathcal{M}_{\varepsilon}^{0}}^{2}\right)-\left\langle\mathrm{T}\Phi^{t},\Phi^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left(\omega-\varepsilon\right)\|\nabla
u\|_{L^{2}\left(\Omega\right)}^{2}$ (4.37)
$\displaystyle+\left(\nu\|\nabla_{\Gamma}u\|_{L^{2}(\Gamma)}^{2}+\beta\left\|u\right\|_{L^{2}\left(\Gamma\right)}^{2}\right)+\delta\|u\|_{L^{r_{1}}(\Omega)}^{r_{1}}$
$\displaystyle\leq C.$

Integrating (4.37 over the interval $\left(0,t\right)$ yields the desired
estimate (4.29). Additionally, from the above _apriori_ estimate (4.29), we
immediately see that

$\displaystyle U$ $\displaystyle\in
L^{\infty}\left(0,T;\mathbb{X}^{2}\right)\cap
L^{2}\left(0,T;\mathbb{V}^{1}\right),$ (4.38) $\displaystyle\Phi$
$\displaystyle\in L^{\infty}\left(0,T;\mathcal{M}_{\Omega,\Gamma}^{1}\right),$
(4.39) $\displaystyle u$ $\displaystyle\in
L^{r_{1}}\left(\Omega\times(0,T)\right).$ (4.40)

Applying Lemma 3.2, in view of of (4.38) and (4.40), we also get

$u\in L^{r_{2}}(\Gamma\times(0,T)).$ (4.41)

Thus, we indeed recover the bounds (4.17)-(4.18) through estimate (4.29).
Moreover, we have from (4.40) and (4.41) that $f\left(u\right)\in
L^{r_{1}^{\prime}}(\Omega\times\left(0,T\right))$,
$\widetilde{g}\left(v\right)\in
L^{r_{2}^{\prime}}(\Gamma\times\left(0,T\right))$; hence,

$F(U)\in L^{r_{1}^{\prime}}(\Omega\times\left(0,T\right))\times
L^{r_{2}^{\prime}}(\Gamma\times\left(0,T\right)).$ (4.42)

Clearly, since $U\in L^{2}(0,T;\mathbb{V}^{1})$ and $\Phi\in
L^{\infty}(0,T;\mathcal{M}_{\Omega,\Gamma}^{1})$ we also have
$\mathrm{A_{W}^{0,\beta,\nu,\omega}}\Phi(s)\in
L^{2}(0,T;(\mathbb{V}^{1})^{\ast})$ for almost all $s\in\mathbb{R}_{+},$ and
$\mathrm{A_{W}^{0,\beta,\nu,\omega}}U\in L^{2}(0,T;(\mathbb{V}^{1})^{\ast}),$
respectively. Therefore, after comparing terms in the first equation of
(4.21), we see that

$\partial_{t}U\in
L^{2}\left(0,T;(\mathbb{V}^{1})^{\ast}\right)\oplus\left(L^{r_{1}^{\prime}}(\Omega\times\left(0,T\right))\times
L^{r_{2}^{\prime}}(\Gamma\times\left(0,T\right))\right).$ (4.43)

Hence, this justifies our choice of test function for the first of (4.21).
Concerning the second equation of (4.21), in view of (4.38) and the
representation formula (4.15) we have

$\mathrm{T}\Phi^{t}(s)=-\partial_{s}\Phi^{t}(s)=\left\\{\begin{array}[]{ll}-U(t-s)&\text{for}~{}0<s\leq
t,\\\ -\partial_{s}\Phi_{0}(s-t)+U(t-s)&\text{for}~{}s>t.\end{array}\right.$

Then, with a given $\Phi_{0}\in\mathcal{M}_{\Omega,\Gamma}^{1},$
$\partial_{s}\Phi_{0}(\cdot)\in
W_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{-1,2}\left(\mathbb{R}_{+};\mathbb{V}^{1}\right),$
we conclude

$\partial_{t}\Phi\in
L^{2}\left(0,T;W_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{-1,2}\left(\mathbb{R}_{+};\mathbb{V}^{1}\right)\right).$
(4.44)

This concludes Step 1.

Step 2. (A Galerkin basis) First, for any $\alpha,\beta\geq$ $0$ we recall
that
($\mathrm{A_{W}^{\alpha,\beta,\nu,\omega})}^{-1}\in\mathcal{L}\left(\mathbb{X}^{2}\right)$
is compact provided that either $\beta>0$ or $\alpha>0$. This means that, for
$i\in\mathbb{N}$, there is a complete system of eigenfunctions
$\Psi_{i}^{\alpha,\beta,\nu,\omega}=(\vartheta_{i}^{\alpha,\beta,\nu,\omega},\vartheta_{i\mid\Gamma}^{\alpha,\beta,\nu,\omega})^{\mathrm{tr}}$
of the eigenvalue problem

$\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}}\Psi_{i}^{\alpha,\beta,\nu,\omega}=\lambda_{i}\Psi_{i}^{\alpha,\beta,\nu,\omega}\text{
in }\mathbb{X}^{2}$

with

$\Psi_{i}^{\alpha,\beta,\nu,\omega}\in
D\left(\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}}\right)\cap\left(C^{2}({\overline{\Omega}})\times
C^{2}\left(\Gamma\right)\right),$

see [12, Appendix]. The eigenvalues
$\lambda_{i}=\lambda_{i}^{\alpha,\beta,\nu,\omega}\in(0,\infty)$ may be put
into increasing order and counted according to their multiplicity to form a
divergent sequence going to infinity. In addition, also due to standard
spectral theory, the related eigenfunctions form an orthogonal basis in
$\mathbb{V}^{1}$ that is orthonormal in $\mathbb{X}^{2}$. Note that for each
$i\in\mathbb{N}$, the pair
$\left(\lambda_{i},\vartheta_{i}\right)\in\mathbb{R}_{+}\times
C^{2}\left(\overline{\Omega}\right),$
$\vartheta_{i}=\vartheta_{i}^{\alpha,\beta,\nu,\omega},$ is a classical
solution of the elliptic problem

$\left\\{\begin{array}[]{ll}-\omega\Delta\vartheta_{i}+\alpha\omega\vartheta_{i}=\lambda_{i}\vartheta_{i},&\text{in
}\Omega,\\\
-\nu\Delta_{\Gamma}\left(\vartheta_{i\mid\Gamma}\right)+\omega\partial_{n}\vartheta_{i}+\beta\nu\vartheta_{i\mid\Gamma}=\lambda_{i}\vartheta_{i\mid\Gamma},&\text{on
}\Gamma.\end{array}\right.$ (4.45)

It remains to select an orthonormal basis $\\{\zeta_{i}\\}_{i=1}^{\infty}$ of
$\mathcal{M}_{\Omega,\Gamma}^{1}=L_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{2}(\mathbb{R}_{+};\mathbb{V}^{1})$
that also belongs to $D(\mathrm{T})\cap
W_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{1,2}(\mathbb{R}_{+};\mathbb{V}^{1})$. We
can choose vectors
$\zeta_{i}=\varkappa_{i}\Psi_{i}^{\alpha,\beta,\nu,\omega}$, with eigenvectors
$\Psi_{i}^{\alpha,\beta,\nu,\omega}\in
D(\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}})$ satisfying (4.45) above, such
that $\\{\varkappa_{i}\\}_{i=1}^{\infty}\in C_{c}^{\infty}(\mathbb{R}_{+})$ is
an orthonormal basis for
$L_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{2}(\mathbb{R}_{+})$. This choice will be
crucial for the derivation of strong solutions in the section later.

Let $T>0$ be fixed. For $n\in\mathbb{N}$, set the spaces

$X_{n}=\mathrm{span}\left\\{\Psi_{1}^{\alpha,\beta,\nu,\omega},\dots,\Psi_{n}^{\alpha,\beta,\nu,\omega}\right\\}\subset\mathbb{X}^{2},~{}~{}X_{\infty}=\bigcup_{n=1}^{\infty}X_{n},$

and

$M_{n}=\mathrm{span}\left\\{\zeta_{1},\zeta_{2},\dots,\zeta_{n}\right\\}\subset\mathcal{M}_{\Omega,\Gamma}^{1},~{}~{}M_{\infty}=\bigcup_{n=1}^{\infty}M_{n}.$

Obviously, $X_{\infty}$ is a dense subspace of $\mathbb{V}^{1}$. For each
$n\in\mathbb{N}$, let $P_{n}:\mathbb{X}^{2}\rightarrow X_{n}$ denote the
orthogonal projection of $\mathbb{X}^{2}$ onto $X_{n}$ and let
$Q_{n}:\mathcal{M}_{\Omega,\Gamma}^{1}\rightarrow M_{n}$ denote the orthogonal
projection of $\mathcal{M}_{\Omega,\Gamma}^{1}$ onto $M_{n}$. Thus, we seek
functions of the form

$U_{n}(t)=\sum_{i=1}^{n}a_{i}(t)\Psi_{i}^{\alpha,\beta,\nu,\omega}~{}~{}\text{and}~{}~{}\Phi_{n}^{t}(s)=\sum_{i=1}^{n}b_{i}(t)\zeta_{i}(s)=\sum_{i=1}^{n}b_{i}(t)\varkappa_{i}\left(s\right)\Psi_{i}^{\alpha,\beta,\nu,\omega}$
(4.46)

that will satisfy the associated discretized problem Pn described below. The
functions $a_{i}$ and $b_{i}$ are assumed to be (at least) $C^{2}(0,T)$ for
$i=1,\dots,n$. By definition, note that

$u_{n}(t)=\sum_{i=1}^{n}a_{i}(t)\vartheta_{i}^{\alpha,\beta,\nu,\omega}~{}~{}\text{and}~{}~{}u_{n}(t)_{\mid\Gamma}=\sum_{i=1}^{n}a_{i}(t)\vartheta_{i\mid\Gamma}^{\alpha,\beta,\nu,\omega},$
(4.47)

also

$\eta_{n}^{t}(s)=\sum_{i=1}^{n}b_{i}(t)\zeta_{i}(s)~{}~{}\text{and}~{}~{}\xi_{n}^{t}(s)=\sum_{i=1}^{n}b_{i}(t)\zeta_{i}(s)_{\mid\Gamma}.$
(4.48)

As usual, to approximate the given initial data $U_{0}\in\mathbb{X}^{2}$ and
$\Phi_{0}\in\mathcal{M}_{\Omega,\Gamma}^{1}$, we take
$U_{n0}\in\mathbb{V}^{1}$ such that $U_{n0}\rightarrow U_{0}~{}~{}$(in
$\mathbb{X}^{2}$), since $\mathbb{V}^{1}$ is dense in $\mathbb{X}^{2}$, and
$\Phi_{n0}\rightarrow\Phi_{0}~{}~{}$(in $\mathcal{M}_{\Omega,\Gamma}^{1}$).

For $T>0$ and for each integer $n\geq 1$, the weak formulation of the
approximate problem Pn is the following: find $(U_{n},\Phi_{n})$, given by
(4.46) such that, for all ${\overline{U}}=(\bar{u},\bar{v})^{\mathrm{tr}}\in
X_{n}$ and ${\overline{\Phi}}=(\bar{\eta},\bar{\xi})^{\mathrm{tr}}\in M_{n}$,
the equations

$\left\langle\partial_{t}U_{n},{\overline{U}}\right\rangle_{\mathbb{X}^{2}}+\left\langle\mathrm{A_{W}^{0,\beta,\nu,\omega}}U_{n},{\overline{U}}\right\rangle_{\mathbb{X}^{2}}+\left\langle\Phi_{n}^{t},{\overline{U}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left\langle
P_{n}F(U_{n}),{\overline{U}}\right\rangle_{\mathbb{X}^{2}}=0$ (4.49)

and

$\left\langle\partial_{t}\Phi_{n}^{t},{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}=\left\langle\mathrm{T}\Phi_{n}^{t},{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left\langle
U_{n},{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}$ (4.50)

hold for almost all $t\in\left(0,T\right)$, subject to the initial conditions

$\left\langle
U_{n}(0),{\overline{U}}\right\rangle_{\mathbb{X}^{2}}=\left\langle
U_{n0},{\overline{U}}\right\rangle_{\mathbb{X}^{2}}~{}~{}\text{and}~{}~{}\left\langle\Phi_{n}^{0},{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}=\left\langle\Phi_{n0},{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}.$
(4.51)

To show the existence of at least one solution to (4.49)-(4.51), we now
suppose that $n$ is fixed and we take ${\overline{U}}=\Psi_{k}$ and
${\overline{\Phi}}=\zeta_{k}$ for some $1\leq k\leq n$. Then substituting the
discretized functions (4.46) into (4.49)-(4.51), we easily arrive at a system
of ordinary differential equations in the unknowns $a_{k}=a_{k}(t)$ and
$b_{k}=b_{k}(t)$ on $X_{n}$ and $M_{n},$ respectively. We need to recall that

$\langle P_{n}F(U_{n}),U_{k}\rangle=\langle F(U_{n}),P_{n}U_{k}\rangle=\langle
F(U_{n}),U_{k}\rangle.$

Since $f,$ $g\in C^{1}(\mathbb{R})$, we may apply Cauchy’s theorem for ODEs to
find that there is $T_{n}\in(0,T)$ such that $a_{k},b_{k}\in C^{2}(0,T_{n})$,
for $1\leq k\leq n$ and both (4.49) and (4.50) hold in the classical sense for
all $t\in[0,T_{n}]$. This argument shows the existence of at least one local
solution to problem Pn and ends Step 2.

Step 3. (Boundedness and continuation of approximate maximal solutions) Now we
apply the (uniform) _apriori_ estimate (4.29) which also holds for any
approximate solution $(U_{n},\Phi_{n})$ of problem Pn on the interval
$[0,T_{n})$, where $T_{n}<T$. Owing to the boundedness of the projectors
$P_{n}$ and $Q_{n}$ on the corresponding spaces, we infer

$\displaystyle\|U_{n}(t)\|_{\mathbb{X}^{2}}^{2}+\left\|\Phi_{n}^{t}\right\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}^{2}-2\left\langle\mathrm{T}\Phi_{n}^{t},\Phi_{n}^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+2\int_{0}^{t}\left(\|U_{n}(\tau)\|_{\mathbb{V}^{1}}^{2}+\|u_{n}(\tau)\|_{L^{r_{1}}(\Omega)}^{r_{1}}\right)d\tau$
(4.52) $\displaystyle\leq
C_{T}\left(1+\|U(0)\|_{\mathbb{X}^{2}}^{2}+\left\|\Phi^{0}\right\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}^{2}\right),$

for some constant $C_{T}>0$ independent of $n$ and $t$. Hence, every
approximate solution may be extended to the whole interval $[0,T]$, and
because $T>0$ is arbitrary, any approximate solution is a global one. As in
Step 1, we also obtain the uniform bounds (4.38)-(4.44) for each approximate
solution $(U_{n},\Phi_{n})$. Thus,

$\displaystyle U_{n}$ $\displaystyle~{}\text{is uniformly bounded
in}~{}L^{\infty}\left(0,T;\mathbb{X}^{2}\right),$ (4.53) $\displaystyle U_{n}$
$\displaystyle~{}\text{is uniformly bounded
in}~{}L^{2}\left(0,T;\mathbb{V}^{1}\right),$ (4.54) $\displaystyle u_{n}$
$\displaystyle~{}\text{is uniformly bounded
in}~{}L^{r_{1}}(\Omega\times\left(0,T\right)),$ (4.55) $\displaystyle u_{n}$
$\displaystyle~{}\text{is uniformly bounded
in}~{}L^{r_{2}}(\Gamma\times\left(0,T\right)),$ (4.56) $\displaystyle\Phi_{n}$
$\displaystyle~{}\text{is uniformly bounded
in}~{}L^{\infty}\left(0,T;\mathcal{M}_{\Omega,\Gamma}^{1}\right),$ (4.57)
$\displaystyle F(U_{n})$ $\displaystyle~{}\text{is uniformly bounded
in}~{}L^{r_{1}^{\prime}}(\Omega\times\left(0,T\right))\times
L^{r_{2}^{\prime}}(\Gamma\times\left(0,T\right)),$ (4.58)
$\displaystyle\partial_{t}U_{n}$ $\displaystyle~{}\text{is uniformly bounded
in}~{}L^{2}\left(0,T;\left(\mathbb{V}^{1}\right)^{\ast}\right)\oplus\left(L^{r_{1}^{\prime}}(\Omega\times\left(0,T\right))\times
L^{r_{2}^{\prime}}(\Gamma\times\left(0,T\right))\right),$ (4.59)
$\displaystyle\partial_{t}\Phi_{n}$ $\displaystyle~{}\text{is uniformly
bounded
in}~{}L^{2}\left(0,T;W_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{-1,2}\left(\mathbb{R}_{+};\mathbb{V}^{1}\right)\right).$
(4.60)

This concludes Step 3.

Step 4. (Convergence of approximate solutions) By Alaoglu’s theorem (cf. e.g.
[24, Theorem 6.64]) and the uniform bounds (4.53)-(4.58), there is a
subsequence of $(U_{n},\Phi_{n})$, generally not relabelled, and functions $U$
and $\Phi$, obeying (4.38)-(4.44), such that as $n\rightarrow\infty$,

$\begin{array}[]{ll}U_{n}\rightharpoonup U&\text{weakly-* in
}L^{\infty}\left(0,T;\mathbb{X}^{2}\right),\\\ U_{n}\rightharpoonup
U&\text{weakly in }L^{2}\left(0,T;\mathbb{V}^{1}\right),\\\
u_{n}\rightharpoonup u&\text{weakly in
}L^{r_{1}}(\Omega\times\left(0,T\right)),\\\ u_{n}\rightharpoonup
u&\text{weakly in }L^{r_{2}}(\Gamma\times\left(0,T\right)),\\\
\Phi_{n}\rightharpoonup\Phi&\text{weakly-* in
}L^{\infty}\left(0,T;\mathcal{M}_{\Omega,\Gamma}^{1}\right).\end{array}$
(4.61)

Moreover, setting $k_{S}:=(-\mu_{S}^{{}^{\prime}})^{1/2}\geq 0$,
$S\in\left\\{\Omega,\Gamma\right\\}$ we have

$\partial_{t}U_{n}\rightharpoonup\partial_{t}U~{}\text{weakly in
}L^{2}\left(0,T;\left(\mathbb{V}^{1}\right)^{\ast}\right)\oplus\left(L^{r_{1}^{\prime}}(\Omega\times\left(0,T\right))\times
L^{r_{2}^{\prime}}(\Gamma\times\left(0,T\right))\right),$ (4.62)
$\Phi_{n}\rightharpoonup\Phi\text{ weakly in
}L^{2}\left(0,T;L_{k_{\Omega}\oplus
k_{\Gamma}}^{2}\left(\mathbb{R}_{+};\mathbb{V}^{1}\right)\right),$ (4.63)

owing to the bound on
$\left\langle\mathrm{T}\Phi_{n},\Phi_{n}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}$
from (4.52) and

$\partial_{t}\Phi_{n}\rightarrow\partial_{t}\Phi~{}\text{weakly
in}~{}L^{2}\left(0,T;W_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{-1,2}\left(\mathbb{R}_{+};\mathbb{V}^{1}\right)\right).$
(4.64)

Indeed, we observe that the last of (4.61) and integration by parts yield, for
any $\zeta\in
C_{0}^{\infty}\left(J;C_{0}^{\infty}\left(\mathbb{R}_{+};\mathbb{V}^{1}\right)\right),$

$\int_{0}^{T}\left\langle\partial_{t}\Phi_{n}^{y},\zeta\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}dy=-\int_{0}^{T}\left\langle\Phi_{n}^{y},\partial_{t}\zeta\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}dy\;\rightarrow\;-\int_{0}^{T}\left\langle\Phi^{y},\partial_{t}\zeta\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}dy,$

and that $\Phi^{t}\in
C(0,T;W_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{-1,2}(\mathbb{R}_{+};\mathbb{V}^{1}))$.
We can exploit the second of (4.61) and (4.62) to deduce

$U_{n}\rightarrow U~{}\text{strongly
in}~{}L^{2}\left(0,T;\mathbb{X}^{2}\right),$ (4.65)

by application of the Agmon-Lions compactness criterion since $\mathbb{V}^{1}$
is compactly embedded in $\mathbb{X}^{2}$. This last strong convergence
property is enough to pass to the limit in the nonlinear terms since $f$,
$g\in C^{1}$ (see, e.g., [11, 15]). Indeed, on account of standard arguments
(cf. also [1]) we have

$P_{n}F(U_{n})\rightharpoonup F\left(U\right)~{}\text{weakly
in}~{}L^{2}\left(0,T;\mathbb{X}^{2}\right).$ (4.66)

The convergence properties (4.61)-(4.65) allow us to pass to the limit as
$n\rightarrow\infty$ in equation (4.49) in order to recover (4.21), using
standard density arguments. Indeed, in order to pass to the limit in the
equations for memory, we use (4.63) and the following distributional equality

$\displaystyle-\int_{0}^{T}\left\langle\Phi^{y},\partial_{t}\zeta\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}dy-\int_{0}^{T}\mu_{\Omega}^{\prime}\left(s\right)\left\langle\eta^{y},\zeta\right\rangle_{\mathcal{M}_{\Omega}^{1}}dy-\int_{0}^{T}\mu_{\Gamma}^{{}^{\prime}}\left(s\right)\left\langle\xi^{y},\partial_{t}\zeta\right\rangle_{\mathcal{M}_{\Gamma}^{1}}dy$
$\displaystyle=\int_{0}^{t}\left\langle\partial_{t}\Phi^{t}-\mathrm{T}\Phi^{y},\zeta\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}dy.$

Thus, we also get the last two equations of (4.21) by virtue of the last of
(4.61).

Step 5. (Continuity of the solution) According to the description for problem
P, see (4.21), we have

$\begin{array}[]{ll}\partial_{t}U\in
L^{2}\left(0,T;\left(\mathbb{V}^{1}\right)^{\ast}\right)\oplus\left(L^{r_{1}^{\prime}}(\Omega\times\left(0,T\right))\times
L^{r_{2}^{\prime}}(\Gamma\times\left(0,T\right))\right),&\\\
\partial_{t}\Phi\in
L^{2}\left(0,T;W_{\mu_{\Omega}\oplus\mu_{\Gamma}}^{-1,2}\left(\mathbb{R}_{+};\mathbb{V}^{1}\right)\right).&\end{array}$
(4.67)

Since the spaces $L^{2}\left(0,T;(\mathbb{V}^{1})^{\ast}\right),$
$L^{r_{1}^{\prime}}(\Omega\times\left(0,T\right))\times
L^{r_{2}^{\prime}}(\Gamma\times\left(0,T\right))$ are the dual of
$L^{2}\left(0,T;\mathbb{V}^{1}\right)$ and
$L^{r_{1}}(\Omega\times\left(0,T\right))\times
L^{r_{2}}(\Gamma\times\left(0,T\right))$, respectively, recalling (4.61), we
can argue exactly as in the proof of [11, Proposition 2.5] to deduce that
$U\in C\left(\left[0,T\right];\mathbb{X}^{2}\right)$. Finally, owing to $U\in$
$L^{2}(0,T;\mathbb{V}^{1})$ and Corollary 4.2, it follows that $\Phi\in
C\left(\left[0,T\right];\mathcal{M}_{\Omega,\Gamma}^{1}\right)$. Thus, both
$U\left(0\right)$ and $\Phi\left(0\right)$ make sense and the equalities
$U\left(0\right)=U_{0}$ and $\Phi^{0}=\Phi_{0}$ hold in the usual sense due to
the strong convergence of $U_{0n}\rightarrow U_{0}$ in $\mathbb{X}^{2}$, and
$\Phi_{0n}\rightarrow\Phi_{0}$ in $\mathcal{M}_{\Omega,\Gamma}^{1}$,
respectively. The proof of the theorem is finished. ∎

When both the bulk and boundary nonlinearities are dissipative (i.e., (4.12)
holds in place of the balance (4.7)), we also have the following.

######  Theorem 4.9.

Assume (4.1)-(4.3) and (4.5), (4.12) hold. For each $\alpha,\beta>0$,
$\omega,\nu\in(0,1)$ and $T>0$, and for any
$U_{0}=(u_{0},v_{0})^{{\mathrm{tr}}}\in\mathbb{X}^{2}$,
$\Phi_{0}=(\eta_{0},\xi_{0})^{{\mathrm{tr}}}\in\mathcal{M}_{\Omega,\Gamma}^{1},$
there exists at least one (global) weak solution $\left(U,\Phi\right)\in
C(\left[0,T\right];\mathcal{H}_{\Omega,\Gamma}^{0,1})$ to problem P in the
sense of Definition 4.4.

###### Proof.

The proof is essentially the same as the proof of Theorem 4.8 with the
exception that one employs the estimate

$f\left(u\right)u\geq C_{f}\left|u\right|^{r_{1}}-C_{1},\text{
}\widetilde{g}\left(u\right)u\geq C_{g}\left|u\right|^{r_{2}}-C_{2},\text{
}\forall s\in\mathbb{R},$

in place of (4.36), owing to (4.12). This implies the same apriori estimate
(4.29). ∎

Finally, we also have uniqueness of the weak solution in some cases.

######  Proposition 4.10.

Let $\left(U_{i},\Phi_{i}\right)$ be any two weak solutions of problem P in
the sense of Definition 4.4, for $i=1,2.$ Assume (4.4). Then the following
estimate holds:

$\left\|U_{1}(t)-U_{2}\left(t\right)\right\|_{\mathbb{X}^{2}}+\left\|\Phi_{1}^{t}-\Phi_{2}^{t}\right\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}\leq\left(\left\|U_{1}(0)-U_{2}\left(0\right)\right\|_{\mathbb{X}^{2}}+\left\|\Phi_{1}^{0}-\Phi_{2}^{0}\right\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}\right)e^{Ct},$
(4.68)

for some constant $C>0$ independent of time, $U_{i}$ and $\Phi_{i}.$

###### Proof.

Set ${\widetilde{U}}=U_{1}-U_{2}$, ${\widetilde{\Phi}}=\Phi_{1}-\Phi_{2}$. The
function $({\widetilde{U}},{\widetilde{\Phi}})$ satisfies the equations:

$\displaystyle\left\langle\partial_{t}\widetilde{U}(t),V\right\rangle_{\mathbb{X}^{2}}+\left\langle\mathrm{A_{W}^{0,\beta,\nu,\omega}}\widetilde{U}(t),V\right\rangle_{\mathbb{X}^{2}}+\left\langle
F(U_{1})-F(U_{2}),V\right\rangle_{\mathbb{X}^{2}}$ (4.69)
$\displaystyle+\int_{0}^{\infty}\mu_{\Omega}\left(s\right)\left\langle\mathrm{A_{W}^{\alpha,0,0,\omega}}\widetilde{\Phi}^{t}\left(s\right),V\right\rangle_{\mathbb{X}^{2}}ds+\nu\int_{0}^{\infty}\mu_{\Gamma}(s)\left\langle\mathrm{C}\widetilde{\xi}^{t}\left(s\right),v\right\rangle_{L^{2}\left(\Gamma\right)}ds$
$\displaystyle=0$

and

$\left\langle\partial_{t}\widetilde{\Phi}^{t}\left(s\right)-\mathrm{T}\widetilde{\Phi}^{t}(s)-\widetilde{U}\left(t\right),\Pi\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}=0,$
(4.70)

for all
$\left(V,\Pi\right)\in\left(\mathbb{V}^{1}\oplus\left(L^{r_{1}}(\Omega)\times
L^{r_{2}}(\Gamma)\right)\right)\times\mathcal{M}_{\Omega,\Gamma}^{1}$, subject
to the associated initial conditions

$\widetilde{U}(0)=U_{1}\left(0\right)-U_{2}\left(0\right)~{}\text{and}~{}\widetilde{\Phi}^{0}=\Phi_{1}^{0}-\Phi_{2}^{0}.$

Multiplication of (4.69) by $V=\widetilde{U}(t)$ in $\mathbb{X}^{2}$ and
multiplication of (4.70) by $\Pi=\widetilde{\Phi}^{t}$ in
$\mathcal{M}_{\Omega,\Gamma}^{1}$, followed by summing the resulting
identities, leads us to the differential inequality

$\displaystyle\frac{d}{dt}\left(\left\|U_{1}-U_{2}\right\|_{\mathbb{X}^{2}}^{2}+\left\|\Phi_{1}-\Phi_{2}\right\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}^{2}\right)$
(4.71) $\displaystyle\leq-2\left\langle
F(U_{1})-F(U_{2}),\widetilde{U}\right\rangle_{\mathbb{X}^{2}}$
$\displaystyle=-2\left\langle
f(u_{1})-f(u_{2}),u_{1}-u_{2}\right\rangle_{L^{2}\left(\Omega\right)}-2\left\langle\widetilde{g}(u_{1})-\widetilde{g}(u_{2}),u_{1}-u_{2}\right\rangle_{L^{2}\left(\Gamma\right)}.$

Employing assumption (4.4) on the nonlinear terms, we easily find that

$\frac{d}{dt}\left(\left\|U_{1}-U_{2}\right\|_{\mathbb{X}^{2}}^{2}+\left\|\Phi_{1}-\Phi_{2}\right\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}^{2}\right)\leq
C\left\|U_{1}-U_{2}\right\|_{\mathbb{X}^{2}}^{2},$ (4.72)

for some $C=C\left(M_{f},M_{g},\beta\right)>0$. Application of the standard
Gronwall lemma to (4.72) yields the desired claim (4.68). ∎

In the final part of this section, we turn our attention to the existence of
global strong solutions for problem P. First, assuming that the interior and
boundary share the same memory kernel, we can derive the existence of strong
solutions in the case when the bulk and boundary nonlinearities have
supercritical polynomial growth of order at most $7/2$. Let $\overline{f}$,
$\overline{g}$ denote the primitives of $f$ and $\widetilde{g}$, respectively,
such that $\overline{f}\left(0\right)=\overline{g}\left(0\right)=0.$

######  Theorem 4.11.

Let (4.1)-(4.3) be satisfied for $\mu_{\Omega}\equiv\mu_{\Gamma}$, and assume
that $f,$ $g\in C^{1}\left(\mathbb{R}\right)$ satisfy the following
assumptions:

(i) $|f^{{}^{\prime}}\left(s\right)|$
$\leq\ell_{1}\left(1+\left|s\right|^{r_{1}}\right),$ for all $s\in\mathbb{R}$,
for some (arbitrary) $1\leq r_{1}<\frac{5}{2}.$

(ii) $|g^{{}^{\prime}}\left(s\right)|$ $\leq\ell_{2}(1+|s|^{r_{2}}),$ for all
$s\in\mathbb{R}$, for some (arbitrary) $1\leq r_{2}<\frac{5}{2}.$

(iii) (4.4) holds and there exist constants $C_{i}>0,$ $i=1,...,4,$ such that

$f\left(s\right)s\geq-C_{1}\left|s\right|^{2}-C_{2},\text{
}g\left(s\right)s\geq-C_{3}\left|s\right|^{2}-C_{4},\text{ }\forall
s\in\mathbb{R}\text{.}$ (4.73)

Given $\alpha,\beta>0$, $\omega,\nu\in(0,1)$,
$\left(U_{0},\Phi_{0}\right)\in\mathcal{H}_{\Omega,\Gamma}^{1,2}$, there
exists a unique global strong solution $\left(U,\Phi\right)$ to problem P in
the sense of Definition 4.5.

###### Proof.

Step 1 (The existence argument). By Remark 4.6 it suffices to deduce
additional regularity for $\left(U,\Phi\right)$. In order to get the crucial
estimate we rely once again on various dissipative estimates. First, we notice
that using the condition of (4.73), we obtain

$\left\langle F\left(U_{n}\right),U_{n}\right\rangle_{\mathbb{X}^{2}}\geq-
C_{F}\left(\left\|U_{n}\right\|_{\mathbb{X}^{2}}^{2}+1\right),$

for some $C_{F}>0$. Thus, arguing in the same fashion as in getting (4.33), in
view of Gronwall’s lemma we obtain

$\displaystyle\|U_{n}(t)\|_{\mathbb{X}^{2}}^{2}+\left\|\Phi_{n}^{t}\right\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}^{2}-2\left\langle\mathrm{T}\Phi_{n}^{t},\Phi_{n}^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+C\int_{0}^{t}\|U_{n}(\tau)\|_{\mathbb{V}^{1}}^{2}d\tau$
(4.74) $\displaystyle\leq
C_{T}\left(1+\|U(0)\|_{\mathbb{X}^{2}}^{2}+\left\|\Phi^{0}\right\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}^{2}\right),$

where $C_{T}\sim e^{CT},$ for some $C>0$ which is independent of $T,$ $n,$
$t.$

Next, we derive an estimate for $U_{n}\in L^{\infty}(0,T;\mathbb{V}^{1})$ and
$\Phi_{n}\in L^{\infty}(0,T;\mathcal{M}_{\Omega,\Gamma}^{2})$. We use again
the scheme (4.49)-(4.51) in which we test equation (4.49) with the function

$\overline{U}=Z_{n}:=\binom{z_{n}}{z_{n\mid\Gamma}},\text{
}z_{n}:=\sum_{i=1}^{n}a_{i}(t)\lambda_{i}\theta_{i}^{\alpha,\beta,\nu,\omega}\in
C^{2}\left(\left(0,T\right)\times\overline{\Omega}\right).$

We get

$\left\langle\partial_{t}U_{n},{Z}_{n}\right\rangle_{\mathbb{X}^{2}}+\left\langle\mathrm{A_{W}^{0,\beta,\nu,\omega}}U_{n},{Z}_{n}\right\rangle_{\mathbb{X}^{2}}+\left\langle\Phi_{n}^{t}\left(s\right),Z_{n}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left\langle
F(U_{n}),{Z}_{n}\right\rangle_{\mathbb{X}^{2}}=0.$ (4.75)

Moreover, testing (4.50) with

$\overline{\Phi}=\Xi_{n}^{t}:=\binom{\varphi_{n}^{t}}{\varphi_{n\mid\Gamma}^{t}},\text{
}\varphi_{n}^{t}:=\sum_{i=1}^{n}b_{i}(t)\varkappa_{i}\left(s\right)\lambda_{i}\theta_{i}^{\alpha,\beta,\nu,\omega}=\sum_{i=1}^{n}b_{i}(t)\lambda_{i}\zeta_{i}\left(s\right)$

we find

$\left\langle\partial_{t}\Phi_{n}^{t},\Xi_{n}^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}=\left\langle\mathrm{T}\Phi_{n}^{t},\Xi_{n}^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left\langle
U_{n},\Xi_{n}^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}.$ (4.76)

Indeed, $\left(Z_{n},\Xi_{n}^{t}\right)\in X_{n}\times M_{n}$ is admissible as
a test function in (4.49)-(4.50). Recalling (4.46), we further notice that
$Z_{n}=\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}}U_{n}$ and
$\Xi_{n}^{t}=\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}}\Phi_{n}^{t},$
respectively, due to the fact that the eigenpair
$(\lambda_{i},\theta_{i}^{\alpha,\beta,\nu,\omega})$ solves (4.45). Owing to
these identities and (4.31), we have

$\displaystyle\left\langle\Phi_{n}^{t}\left(s\right),Z_{n}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}$
$\displaystyle=\int_{0}^{\infty}\mu_{\Omega}(s)\left\langle\mathrm{A_{W}^{\alpha,0,0,\omega}}\Phi_{n}^{t}\left(s\right),Z_{n}\right\rangle_{\mathbb{X}^{2}}ds+\nu\int_{0}^{\infty}\mu_{\Gamma}(s)\left\langle\mathrm{C}\xi_{n}^{t}\left(s\right),z_{n}\right\rangle_{L^{2}\left(\Gamma\right)}ds$
(4.77)
$\displaystyle\overset{\mu_{\Omega}\equiv\mu_{\Gamma}}{=}\int_{0}^{\infty}\mu_{\Omega}(s)\left\langle\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}}\Phi_{n}^{t}\left(s\right),\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}}U_{n}\right\rangle_{\mathbb{X}^{2}}ds$
$\displaystyle=\left\langle
U_{n},\Xi_{n}^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}.$

Adding relations (4.75)-(4.76) together, and using (4.77) we further deduce

$\displaystyle\frac{1}{2}\frac{d}{dt}\left(\left\|U_{n}\right\|_{\mathbb{V}^{1}}^{2}+\left\|\Xi_{n}^{t}\right\|_{L_{\mu_{\Omega}}^{2}(\mathbb{R}_{+};\mathbb{X}^{2})}^{2}\right)-\left\langle\mathrm{T}\Phi_{n}^{t},\Xi_{n}^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left\|Z_{n}\right\|_{\mathbb{X}^{2}}^{2}$
(4.78) $\displaystyle=\alpha\omega\left\langle
u_{n},z_{n}\right\rangle_{L^{2}\left(\Omega\right)}-\left\langle
F(U_{n}),{Z}_{n}\right\rangle_{\mathbb{X}^{2}},$

and

$\left\langle\mathrm{T}\Phi_{n}^{t},\Xi_{n}^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}=\int_{0}^{\infty}\mu_{\Omega}(s)\left\langle\mathrm{A_{W}^{\alpha,\beta,\nu,\omega}T}\Phi_{n}^{t},\Xi_{n}^{t}\right\rangle_{\mathbb{X}^{2}}ds=\frac{1}{2}\int_{0}^{\infty}\mu_{\Omega}^{{}^{\prime}}(s)\left\|\Xi_{n}^{t}\left(s\right)\right\|_{\mathbb{X}^{2}}^{2}ds,$
(4.79)

thanks to the fact that $\mu_{\Omega}\equiv\mu_{\Gamma}$. We begin estimating
both terms on the right-hand side of (4.78). The first one is easy,

$\alpha\omega\left\langle
u_{n},z_{n}\right\rangle_{L^{2}\left(\Omega\right)}\leq\delta\left\|z_{n}\right\|_{L^{2}\left(\Omega\right)}^{2}+C_{\delta}\left\|u_{n}\right\|_{L^{2}\left(\Omega\right)}^{2},$
(4.80)

for any $\delta\in(0,1]$. To bound the last term we integrate by parts in the
following way:

$\displaystyle\left\langle F(U_{n}),{Z}_{n}\right\rangle_{\mathbb{X}^{2}}$
$\displaystyle=\int_{\Omega}f\left(u_{n}\right)\left(-\omega\Delta
u_{n}+\alpha\omega
u_{n}\right)dx+\int_{\Gamma}\widetilde{g}\left(u_{n}\right)\left(-\nu\Delta_{\Gamma}u_{n}+\omega\partial_{n}u_{n}+\nu\beta
u_{n}\right)d\sigma$ (4.81)
$\displaystyle=\omega\int_{\Omega}f^{{}^{\prime}}\left(u_{n}\right)\left|\nabla
u_{n}\right|^{2}dx+\nu\int_{\Gamma}\widetilde{g}^{{}^{\prime}}\left(u_{n}\right)\left|\nabla_{\Gamma}u_{n}\right|^{2}d\sigma$
$\displaystyle+\alpha\omega\int_{\Omega}f\left(u_{n}\right)u_{n}dx+\nu\beta\int_{\Gamma}\widetilde{g}\left(u_{n}\right)u_{n}d\sigma$
$\displaystyle+\omega\int_{\Gamma}\left(\widetilde{g}\left(u_{n}\right)-f\left(u_{n}\right)\right)\partial_{n}u_{n}d\sigma.$

By assumptions (4.4) and (4.73), we can easily find a positive constant $C$
independent of $t,T$ and $n$ such that

$\omega\int_{\Omega}f^{{}^{\prime}}\left(u_{n}\right)\left|\nabla
u_{n}\right|^{2}dx+\nu\int_{\Gamma}\widetilde{g}^{{}^{\prime}}\left(u_{n}\right)\left|\nabla_{\Gamma}u_{n}\right|^{2}d\sigma\geq-
M_{f}\omega\left\|\nabla
u_{n}\right\|_{L^{2}\left(\Omega\right)}^{2}-M_{g}\nu\left\|\nabla_{\Gamma}u_{n}\right\|_{L^{2}\left(\Gamma\right)}^{2}$
(4.82)

and

$\alpha\omega\int_{\Omega}f\left(u_{n}\right)u_{n}dx+\nu\beta\int_{\Gamma}\widetilde{g}\left(u_{n}\right)u_{n}d\sigma\geq-C\left(\left\|U_{n}\right\|_{\mathbb{X}^{2}}^{2}+1\right).$
(4.83)

In order to estimate the last boundary integral on the right-hand side of
(4.81), we observe that due to assumptions (i)-(ii) it suffices to estimate
boundary integrals of the form

$I:=\int_{\Gamma}u_{n}^{r+1}\partial_{n}u_{n}d\sigma,\text{ for some }r<5/2.$

Indeed, due to classical trace regularity and embedding results, for every
$\delta\in(0,1]$ we have

$I\leq\left\|\partial_{n}u_{n}\right\|_{H^{1/2}\left(\Gamma\right)}\left\|u_{n}^{r+1}\right\|_{H^{-1/2}\left(\Gamma\right)}\leq\delta\left\|u_{n}\right\|_{H^{2}\left(\Omega\right)}^{2}+C_{\delta}\left\|u_{n}^{r+1}\right\|_{H^{-1/2}\left(\Gamma\right)}^{2}.$
(4.84)

It remains to estimate the last term in (4.84). To this end, we employ the
basic Sobolev embeddings $H^{1/2}\left(\Gamma\right)\subset
L^{4}\left(\Gamma\right)$ and $H^{1}\left(\Gamma\right)\subset
L^{s}\left(\Gamma\right),$ for any $s\in(\frac{4}{3},\infty)$, respectively.
Owing to elementary Holder inequalities, we deduce that

$\displaystyle\left\|u_{n}^{r+1}\right\|_{H^{-1/2}\left(\Gamma\right)}^{2}$
$\displaystyle=\sup_{\psi\in
H^{1/2}\left(\Gamma\right):\left\|\psi\right\|_{H^{1/2}\left(\Gamma\right)}=1}\left|\left\langle
u_{n}^{r+1},\psi\right\rangle\right|^{2}$ (4.85)
$\displaystyle\leq\left\|u_{n}\right\|_{L^{s}\left(\Gamma\right)}^{2}\left\|u_{n}\right\|_{L^{\overline{s}r}\left(\Gamma\right)}^{2r}$
$\displaystyle\leq
C\left\|u_{n}\right\|_{H^{1}\left(\Gamma\right)}^{2}\left\|u_{n}\right\|_{L^{\overline{s}r}\left(\Gamma\right)}^{2r},$

for some positive constant $C$ independent of $u,n,t,T$, for sufficiently
large $s\in(\frac{4}{3},\infty)$, where
$\overline{s}:=4s/\left(3s-4\right)>4/3$. Exploiting now the interpolation
inequality

$\left\|u\right\|_{L^{\overline{s}r}\left(\Gamma\right)}\leq
C\left\|u\right\|_{H^{2}\left(\Gamma\right)}^{1/\left(2r\right)}\left\|u\right\|_{L^{2}\left(\Gamma\right)}^{1-1/\left(2r\right)},$

provided that $r=1+2/\overline{s}<5/2$, we further infer from (4.85) that

$\displaystyle\left\|u_{n}^{r+1}\right\|_{H^{-1/2}\left(\Gamma\right)}^{2}$
$\displaystyle\leq
C\left\|u_{n}\right\|_{H^{1}\left(\Gamma\right)}^{2}\left\|u_{n}\right\|_{H^{2}\left(\Gamma\right)}\left\|u_{n}\right\|_{L^{2}\left(\Gamma\right)}^{2r-1}$
(4.86)
$\displaystyle\leq\eta\left\|u_{n}\right\|_{H^{2}\left(\Gamma\right)}^{2}+C_{\eta}\left\|u_{n}\right\|_{H^{1}\left(\Gamma\right)}^{2}\left(\left\|u_{n}\right\|_{H^{1}\left(\Gamma\right)}^{2}\left\|u_{n}\right\|_{L^{2}\left(\Gamma\right)}^{2\left(2r-1\right)}\right),$

for any $\eta\in(0,1]$. Inserting (4.86) into (4.84) and choosing a
sufficiently small $\eta=\delta/C_{\delta}$, by virtue of (3.21), we easily
deduce

$I\leq\delta\left\|Z_{n}\right\|_{\mathbb{X}^{2}}^{2}+C_{\delta}\left\|u_{n}\right\|_{H^{1}\left(\Gamma\right)}^{2}\left(\left\|u_{n}\right\|_{H^{1}\left(\Gamma\right)}^{2}\left\|u_{n}\right\|_{L^{2}\left(\Gamma\right)}^{2\left(2r-1\right)}\right).$
(4.87)

Thus, setting

$\displaystyle\Xi\left(t\right)$
$\displaystyle:=\left\|U_{n}\left(t\right)\right\|_{\mathbb{V}^{1}}^{2}+\left\|\Xi_{n}^{t}\right\|_{L_{\mu_{\Omega}}^{2}(\mathbb{R}_{+};\mathbb{X}^{2})}^{2},\text{
}$ $\displaystyle\Lambda\left(t\right)$
$\displaystyle:=C_{\delta}\left(1+\left\|u_{n}\right\|_{H^{1}\left(\Gamma\right)}^{2}\left\|u_{n}\right\|_{L^{2}\left(\Gamma\right)}^{2\left(2r-1\right)}\right),$

it follows from (4.78), (4.80)-(4.83) and (4.87) that

$\frac{d}{dt}\Xi\left(t\right)-2\left\langle\mathrm{T}\Phi_{n}^{t},\Xi_{n}^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left(2-\delta\right)\left\|Z_{n}\right\|_{\mathbb{X}^{2}}^{2}\leq\Xi\left(t\right)\Lambda\left(t\right),$
(4.88)

for a sufficiently small $\delta\in(0,1]$. Gronwall’s inequality together with
(4.74) yields

$\displaystyle\left\|U_{n}\left(t\right)\right\|_{\mathbb{V}^{1}}^{2}+\left\|\Xi_{n}^{t}\right\|_{L_{\mu_{\Omega}}^{2}(\mathbb{R}_{+};\mathbb{X}^{2})}^{2}+\int_{0}^{t}\left(\|Z_{n}(\tau)\|_{\mathbb{X}^{2}}^{2}-2\left\langle\mathrm{T}\Phi_{n}^{\tau},\Xi_{n}^{\tau}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}\right)d\tau$
(4.89) $\displaystyle\leq
C_{T}\left(\left\|U\left(0\right)\right\|_{\mathbb{V}^{1}}^{2}+\left\|\Xi^{0}\right\|_{L_{\mu_{\Omega}}^{2}(\mathbb{R}_{+};\mathbb{X}^{2})}^{2}\right),$

owing to the boundedness of the (orthogonal) projectors
$P_{n}:\mathbb{X}^{2}\rightarrow X_{n}$ and
$Q_{n}:\mathcal{M}_{\Omega,\Gamma}^{1}\rightarrow M_{n}$, and the fact that
$\Lambda\in L^{1}\left(0,T\right),$ for any $T>0.$

From (4.89), recalling (3.21) we obtain the following uniform (in $n$) bounds
for each approximate solution $(U_{n},\Phi_{n})$:

$\displaystyle U_{n}$ $\displaystyle~{}\text{is uniformly bounded
in}~{}L^{\infty}\left(0,T;\mathbb{V}^{1}\right),$ (4.90) $\displaystyle U_{n}$
$\displaystyle~{}\text{is uniformly bounded
in}~{}L^{2}\left(0,T;\mathbb{V}^{2}\right),$ (4.91) $\displaystyle\Phi_{n}$
$\displaystyle~{}\text{is uniformly bounded
in}~{}L^{\infty}\left(0,T;\mathcal{M}_{\Omega,\Gamma}^{2}\right),$ (4.92)
$\displaystyle\Phi_{n}$ $\displaystyle~{}\text{is uniformly bounded
in}~{}L^{2}\left(0,T;L_{k_{\Omega}}^{2}\left(\mathbb{R}_{+};\mathbb{V}^{2}\right)\right).$
(4.93)

Observe now that by (4.49)-(4.50), we also have

$\displaystyle\left\langle\partial_{t}U_{n},{\overline{U}}\right\rangle_{\mathbb{X}^{2}}$
$\displaystyle=\left\langle\partial_{t}U_{n},P_{n}{\overline{U}}\right\rangle_{\mathbb{X}^{2}}$
(4.94)
$\displaystyle=-\left\langle\mathrm{A_{W}^{0,\beta,\nu,\omega}}U_{n},P_{n}{\overline{U}}\right\rangle_{\mathbb{X}^{2}}-\left\langle\Phi_{n}^{t},P_{n}{\overline{U}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}-\left\langle
F(U_{n}),P_{n}{\overline{U}}\right\rangle_{\mathbb{X}^{2}}$

and

$\displaystyle\left\langle\partial_{t}\Phi_{n}^{t},{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}$
$\displaystyle=\left\langle\partial_{t}\Phi_{n}^{t},Q_{n}{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}$
(4.95)
$\displaystyle=\left\langle\mathrm{T}\Phi_{n}^{t},Q_{n}{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left\langle
U_{n},Q_{n}{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}},$

respectively. Thus, from the uniform bounds (4.90)-(4.93), we deduce by
comparison in equations (4.94)-(4.95) that

$\displaystyle\partial_{t}U_{n}\text{ is uniformly bounded in
}L^{\infty}\left(0,T;\left(\mathbb{V}^{1}\right)^{\ast}\right)\cap
L^{2}\left(0,T;\mathbb{X}^{2}\right),$ (4.96)
$\displaystyle\partial_{t}\Phi_{n}^{t}\text{ is uniformly bounded in
}L^{2}\left(0,T;L_{\mu_{\Omega}}^{2}\left(\mathbb{R}_{+};\mathbb{X}^{2}\right)\right)\cap
L^{\infty}\left(0,T;L_{\mu_{\Omega}}^{2}\left(\mathbb{R}_{+};\left(\mathbb{V}^{1}\right)^{\ast}\right)\right).$
(4.97)

We are now ready to pass to the limit as $n$ goes to infinity. On account of
the above uniform inequalities, we can find $U$ and $\Phi$ such that, up to
subsequences,

$\displaystyle U_{n}$ $\displaystyle\rightarrow U~{}\text{weakly * in
}L^{\infty}\left(0,T;\mathbb{V}^{1}\right),$ (4.98) $\displaystyle U_{n}$
$\displaystyle\rightarrow U\text{ weakly in
}L^{2}\left(0,T;\mathbb{V}^{2}\right),$ (4.99) $\displaystyle\Phi_{n}$
$\displaystyle\rightarrow\Phi\text{ weakly * in
}L^{\infty}\left(0,T;\mathcal{M}_{\Omega,\Gamma}^{2}\right),$ (4.100)
$\displaystyle\Phi_{n}$ $\displaystyle\rightarrow\Phi\text{ weakly in
}L^{2}\left(0,T;L_{k_{\Omega}}^{2}\left(\mathbb{R}_{+};\mathbb{V}^{2}\right)\right),$
(4.101) $\displaystyle\partial_{t}U_{n}$
$\displaystyle\rightarrow\partial_{t}U\text{ in
}L_{w^{\ast}}^{\infty}\left(0,T;\left(\mathbb{V}^{1}\right)^{\ast}\right)\cap
L_{w}^{2}\left(0,T;\mathbb{X}^{2}\right),$ (4.102)
$\displaystyle\partial_{t}\Phi_{n}^{t}$
$\displaystyle\rightarrow\partial_{t}\Phi^{t}\text{ in
}L_{w}^{2}\left(0,T;L_{\mu_{\Omega}}^{2}\left(\mathbb{R}_{+};\mathbb{X}^{2}\right)\right).$
(4.103)

Due to (4.98) and (4.102) and the classical Agmon-Lions compactness theorem,
we also have

$U_{n}\rightarrow U~{}\text{strongly in }C(\left[0,T\right];\mathbb{X}^{2}).$
(4.104)

Thanks to (4.98)-(4.103) and (4.104), we can easily control the nonlinear
terms in (4.49)-(4.50). By means of the above convergence properties, we can
pass to the limit in these equations and show that $\left(U,\Phi\right)$
solves (4.21) in the sense of Definition 4.5.

Finally, uniqueness follows from Proposition 4.10 owing to assumption (4.4).
The proof of the theorem is finished. ∎

######  Remark 4.12.

Observe that the assumption $\mu_{\Omega}\equiv\mu_{\Gamma}$ in Theorem 4.11
is crucial for the identity (4.77) to hold. Without it, cancellation in (4.78)
does not generally occur and (4.79) does not hold.

We now let

$h_{f}\left(s\right)=\int_{0}^{s}f^{{}^{\prime}}\left(\tau\right)\tau
d\tau\text{ and
}h_{g}\left(s\right)=\int_{0}^{s}\widetilde{g}^{{}^{\prime}}\left(\tau\right)\tau
d\tau.$

The next result states that there exist strong solutions, albeit in a much
weaker sense than in Theorem 4.11, even when the interior and boundary memory
kernels $\mu_{S}\left(\cdot\right):\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ do
_not_ coincide but both decay exponentially fast as $s$ goes to infinity.

######  Theorem 4.13.

Let (4.1)-(4.3) be satisfied and assume that $f,$ $g\in
C^{1}\left(\mathbb{R}\right)$ satisfy the following conditions:

(i) $|f^{{}^{\prime}}\left(s\right)|$
$\leq\ell_{1}\left(1+\left|s\right|^{2}\right),$ for all $s\in\mathbb{R}$.

(ii) $|g^{{}^{\prime}}\left(s\right)|$ $\leq\ell_{2}(1+|s|^{r_{2}}),$ for all
$s\in\mathbb{R}$, for some (arbitrary) $r_{2}>2.$

(iii) (4.4) holds and there exist $C_{i}>0,$ $i=1,\dots,8,$ such that

$\left\\{\begin{array}[]{ll}f\left(s\right)s\geq-
C_{1}\left|s\right|^{2}-C_{2},\text{ }g\left(s\right)s\geq-
C_{3}\left|s\right|^{2}-C_{4},&\forall s\in\mathbb{R}\\\
h_{f}\left(s\right)\geq-C_{5}\left|s\right|^{2}-C_{6},\text{
}h_{g}\left(s\right)\geq-C_{7}\left|s\right|^{2}-C_{8},&\forall
s\in\mathbb{R}\text{.}\end{array}\right.$ (4.105)

In addition, assume there exist constants $\delta_{S}>0$ such that

$\mu_{S}^{{}^{\prime}}\left(s\right)+\delta_{S}\mu_{S}\left(s\right)\leq
0\text{, for all }s\in\mathbb{R}_{+}\text{,
}S\in\left\\{\Omega,\Gamma\right\\}.$ (4.106)

Given $\alpha,\beta>0$, $\omega,\nu\in(0,1)$,
$\left(U_{0},\Phi_{0}\right)\in\mathbb{V}^{2}\times\left(\mathcal{M}_{\Omega,\Gamma}^{2}\cap
D\left(\mathrm{T}\right)\right)$, there exists a unique global quasi-strong
solution $\left(U,\Phi\right)$ to problem P in the sense of Definition 4.7.

###### Proof.

It suffices to provide bounds for $(U,\Phi^{t})$ in the (more regular) spaces
in (4.25)-(4.28). With reference to problem P${}_{n},$ we consider the
approximate problem of finding $\left(U_{n},\Phi_{n}\right)$ of the form
(4.46) such that, $\left(U_{n},\Phi_{n}\right)$ already satisfies
(4.49)-(4.50), and

$\displaystyle\left\langle\partial_{tt}U_{n},{\overline{U}}\right\rangle_{\mathbb{X}^{2}}+\left\langle\mathrm{A_{W}^{0,\beta,\nu,\omega}}\partial_{t}U_{n},{\overline{U}}\right\rangle_{\mathbb{X}^{2}}+\left\langle\partial_{t}\Phi_{n}^{t},{\overline{U}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}$
(4.107) $\displaystyle=-\left\langle
f^{{}^{\prime}}\left(u_{n}\right)\partial_{t}u_{n},\bar{u}\right\rangle_{L^{2}\left(\Omega\right)}-\left\langle\widetilde{g}^{{}^{\prime}}\left(u_{n}\right)\partial_{t}u_{n},\bar{v}\right\rangle_{L^{2}\left(\Gamma\right)}$

and

$\left\langle\partial_{tt}\Phi_{n}^{t},{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}=\left\langle\mathrm{T}\partial_{t}\Phi_{n}^{t},{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}+\left\langle\partial_{t}U_{n},{\overline{\Phi}}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}$
(4.108)

hold for almost all $t\in\left(0,T\right)$, for all
${\overline{U}}=(\bar{u},\bar{v})^{\mathrm{tr}}\in X_{n}$ and
${\overline{\Phi}}=(\bar{\eta},\bar{\xi})^{\mathrm{tr}}\in M_{n}$; moreover,
the function $\left(U_{n},\Phi_{n}\right)$ fulfils the conditions
$U_{n}\left(0\right)=P_{n}U_{0},$ $\Phi_{n}^{0}=Q_{n}\Phi^{0}$ and

$\partial_{t}U_{n}\left(0\right)=P_{n}\widehat{U}_{0},\text{
}\partial_{t}\Phi_{n}^{0}=Q_{n}\widehat{\Phi}^{0},$ (4.109)

where we have set

$\displaystyle\widehat{U}_{0}$
$\displaystyle:=-\mathrm{A_{W}^{0,\beta,\nu,\omega}}U_{0}-\int_{0}^{\infty}\mu_{\Omega}(s)\mathrm{A_{W}^{\alpha,0,0,\omega}}\Phi_{0}(s)ds-\nu\int_{0}^{\infty}\mu_{\Gamma}(s)\binom{0}{\mathrm{C}\xi_{0}(s)}ds-F(U_{0})\text{,
}$ $\displaystyle\widehat{\Phi}^{0}$
$\displaystyle:=\mathrm{T}\Phi_{0}(s)+U_{0}.$

Note that, if $U_{0}\in\mathbb{V}^{2}$ and $\Phi^{0}\in
D\left(\mathrm{T}\right)\cap\mathcal{M}_{\Omega,\Gamma}^{2}$, then
$(\widehat{U}_{0},\widehat{\Phi}^{0})\in\mathbb{X}^{2}\times\mathcal{M}_{\Omega,\Gamma}^{1}=\mathcal{H}_{\Omega,\Gamma}^{0,1}$,
owing to the continuous embeddings $H^{2}\left(\Omega\right)\subset
L^{\infty}\left(\Omega\right),$ $H^{2}\left(\Gamma\right)\subset
L^{\infty}\left(\Gamma\right)$. In particular, owing to the boundedness of the
projectors $P_{n}$ and $Q_{n}$ on the corresponding subspaces, we have

$\left\|\left(\partial_{t}U_{n}\left(0\right),\partial_{t}\Phi_{n}^{0}\right)\right\|_{\mathcal{H}_{\Omega,\Gamma}^{0,1}}\leq
K\left(R\right),$ (4.110)

for all
$\left(U_{0},\Phi^{0}\right)\in\mathbb{V}^{2}\times\left(D\left(\mathrm{T}\right)\cap\mathcal{M}_{\Omega,\Gamma}^{2}\right)$
such that
$\left\|\left(U_{0},\Phi^{0}\right)\right\|_{\mathcal{H}_{\Omega,\Gamma}^{2,2}}\leq
R,$ for some positive monotone nondecreasing function $K$. Indeed, according
to assumptions (4.1)-(4.3), we can infer that

$0\leq\int_{0}^{\infty}\mu_{S}(s)ds=\mu_{S}^{0}<\infty,\text{ for each
}S\in\left\\{\Omega,\Gamma\right\\},$ (4.111)

such that repeated application of Jensen’s inequality yields

$\displaystyle\left\|\int_{0}^{\infty}\mu_{\Omega}(s)\mathrm{A_{W}^{\alpha,0,0,\omega}}\Phi_{0}(s)ds\right\|_{\mathbb{X}^{2}}^{2}$
$\displaystyle\leq\mu_{\Omega}^{0}\int_{0}^{\infty}\mu_{\Omega}(s)\left\|\mathrm{A_{W}^{\alpha,0,0,\omega}}\Phi_{0}(s)\right\|_{\mathbb{X}^{2}}^{2}ds$
$\displaystyle\leq
C\mu_{\Omega}^{0}\int_{0}^{\infty}\mu_{\Omega}(s)\left\|\Phi_{0}(s)\right\|_{H^{2}}^{2}ds$

and

$\displaystyle\left\|\int_{0}^{\infty}\mu_{\Gamma}(s)\mathrm{C}\xi_{0}(s){\mathrm{d}}s\right\|_{L^{2}\left(\Gamma\right)}^{2}$
$\displaystyle\leq\mu_{\Gamma}^{0}\int_{0}^{\infty}\mu_{\Gamma}(s)\left\|\mathrm{C}\xi_{0}(s)\right\|_{L^{2}\left(\Gamma\right)}^{2}ds$
$\displaystyle\leq
C\mu_{\Gamma}^{0}\int_{0}^{\infty}\mu_{\Gamma}(s)\left\|\Phi_{0}(s)\right\|_{H^{2}\left(\Gamma\right)}^{2}ds.$

Our starting point is the validity of the energy estimate (4.74) which holds
on account of the first assumption of (4.105). Next we proceed to take
$\overline{U}=\partial_{t}U_{n}(t)$ in (4.107) and
$\overline{\Phi}=\partial_{t}\Phi_{n}^{t}\left(s\right)$ in (4.108),
respectively, by noting that this choice
$\left(\overline{U},\overline{\Phi}\right)$ is an admissible test function.
Summing the resulting identities and using (4.4), we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\left\\{\left\|\partial_{t}U_{n}\right\|_{\mathbb{X}^{2}}^{2}+\left\|\partial_{t}\Phi_{n}^{t}\right\|_{\mathcal{M}_{\Omega,\Gamma}^{1}}^{2}\right\\}-\left\langle\mathrm{T}\partial_{t}\Phi_{n}^{t},\partial_{t}\Phi_{n}^{t}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}$
(4.112)
$\displaystyle+\left(\omega\left\|\nabla\partial_{t}u_{n}\right\|_{L^{2}(\Omega)}^{2}+\nu\left\|\nabla_{\Gamma}\partial_{t}u_{n}\right\|_{L^{2}(\Gamma)}^{2}+\beta\left\|\partial_{t}u_{n}\right\|_{L^{2}(\Gamma)}^{2}\right)$
$\displaystyle=-\left\langle
f^{{}^{\prime}}\left(u_{n}\right)\partial_{t}u_{n},\partial_{t}u_{n}\right\rangle_{L^{2}\left(\Omega\right)}-\left\langle\widetilde{g}^{{}^{\prime}}\left(u_{n}\right)\partial_{t}u_{n},\partial_{t}u_{n}\right\rangle_{L^{2}\left(\Gamma\right)}$
$\displaystyle\leq\max\left(M_{f},M_{g}\right)\left\|\partial_{t}U_{n}\right\|_{\mathbb{X}^{2}}^{2}.$

Thus, integrating (4.112) with respect to $\tau\in(0,t),$ by application of
Growall’s inequality, we have the estimate

$\left\|\left(\partial_{t}U_{n}\left(t\right),\partial_{t}\Phi_{n}^{t}\right)\right\|_{\mathcal{H}_{\Omega,\Gamma}^{0,1}}^{2}+\int_{0}^{t}\left(2\left\|\partial_{t}U_{n}(\tau)\right\|_{\mathbb{V}^{1}}^{2}+\left\|\partial_{t}\Phi_{n}^{\tau}\right\|_{L_{k_{\Omega}\oplus
k_{\Gamma}}^{2}\left(\mathbb{R}_{+};\mathbb{V}^{1}\right)}^{2}\right)d\tau\leq
K_{T}\left(R\right),$ (4.113)

for all $t\geq 0$ and all $R>0$ such that
$\left\|\left(U_{0},\Phi^{0}\right)\right\|_{\mathcal{H}_{\Omega,\Gamma}^{2,2}}\leq
R$. Thanks to (4.113), we deduce the uniform bounds

$\displaystyle\partial_{t}U_{n}$ $\displaystyle\in
L^{\infty}\left(0,T;\mathbb{X}^{2}\right)\cap
L^{2}\left(0,T;\mathbb{V}^{1}\right),$ (4.114)
$\displaystyle\partial_{t}\Phi_{n}$ $\displaystyle\in
L^{\infty}\left(0,T;\mathcal{M}_{\Omega,\Gamma}^{1}\right)\cap
L^{2}\left(0,T;L_{k_{\Omega}\oplus
k_{\Gamma}}^{2}\left(\mathbb{R}_{+};\mathbb{V}^{1}\right)\right),$ (4.115)

which establishes (4.27)-(4.28) for the approximate solution
$\left(U_{n},\Phi_{n}\right)$.

We now establish a bound for $U_{n}$ in
$L^{\infty}\left(0,T;\mathbb{V}^{1}\right)$ in a different way from the proof
of Theorem 4.11. For this estimate, the uniform regularity in (4.114)-(4.115)
is crucial. To this end, we proceed to take $\overline{U}=U_{n}(t)$ in (4.107)
in order to derive

$\displaystyle\frac{d}{dt}\left(\left\|U_{n}\right\|_{\mathbb{V}^{1}}^{2}+\left\langle\partial_{t}U_{n},U_{n}\right\rangle_{\mathbb{X}^{2}}+2\int_{\Omega}h_{f}\left(u_{n}\right)dx+2\int_{\Gamma}h_{g}\left(u_{n}\right)d\sigma\right)$
(4.116)
$\displaystyle=2\left\|\partial_{t}{U}_{n}\right\|_{\mathbb{X}^{2}}^{2}-2\left\langle\partial_{t}\Phi_{n}^{t},{U}_{n}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}.$

Moreover, using (4.114) and owing to the Cauchy-Schwarz and Young inequalities
and the second of (4.105), the following basic inequality holds:

$\displaystyle
C_{\ast}\left\|U_{n}\right\|_{\mathbb{V}^{1}}^{2}-K_{T}\left(R\right)$ (4.117)
$\displaystyle\leq\left\|U_{n}\right\|_{\mathbb{V}^{1}}^{2}+\left\langle\partial_{t}U_{n},U_{n}\right\rangle_{\mathbb{X}^{2}}+2\int_{\Omega}h_{f}\left(u_{n}\right)dx+2\int_{\Gamma}h_{g}\left(u_{n}\right)d\sigma$
$\displaystyle\leq
C\left\|U_{n}\right\|_{\mathbb{V}^{1}}^{2}+K_{T}\left(R\right),$

for some constants $C_{\ast},C>0$ and some function $K_{T}>0,$ all independent
of $n$ and $t$. Finally, for any $\eta>0$ we estimate

$\displaystyle-\left\langle\partial_{t}\Phi_{n}^{t},{U}_{n}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}$
$\displaystyle\leq\eta\left\|U_{n}\right\|_{\mathbb{V}^{1}}^{2}+C_{\eta}\int_{0}^{\infty}\mu_{\Omega}(s)\left\|\partial_{t}\eta_{n}(s)\right\|_{H^{1}}^{2}ds+C_{\eta}\int_{0}^{\infty}\mu_{\Gamma}(s)\left\|\partial_{t}\xi_{n}(s)\right\|_{H^{1}\left(\Gamma\right)}^{2}ds$
(4.118)
$\displaystyle\leq\eta\left\|U_{n}\right\|_{\mathbb{V}^{1}}^{2}-C_{\eta}\delta_{\Omega}^{-1}\int_{0}^{\infty}\mu_{\Omega}^{{}^{\prime}}(s)\left\|\partial_{t}\eta_{n}(s)\right\|_{H^{1}}^{2}ds-
C_{\eta}\delta_{\Gamma}^{-1}\int_{0}^{\infty}\mu_{\Gamma}^{{}^{\prime}}(s)\left\|\partial_{t}\xi_{n}(s)\right\|_{H^{1}\left(\Gamma\right)}^{2}ds,$

where in the last line we have employed assumption (4.106). Thus, from (4.116)
we obtain the inequality

$\displaystyle\frac{d}{dt}\left(\left\|U_{n}\right\|_{\mathbb{V}^{1}}^{2}+\left\langle\partial_{t}U_{n},U_{n}\right\rangle_{\mathbb{X}^{2}}+2\int_{\Omega}h_{f}\left(u_{n}\right)dx+2\int_{\Gamma}h_{g}\left(u_{n}\right)d\sigma\right)$
(4.119) $\displaystyle\leq
C_{\eta}\left\|U_{n}\left(t\right)\right\|_{\mathbb{V}^{1}}^{2}+\Lambda_{2}\left(t\right),$

for a.e. $t\in\left(0,T\right),$ where we have set

$\Lambda_{2}\left(t\right):=2\left\|\partial_{t}{U}_{n}\right\|_{\mathbb{X}^{2}}^{2}-2\left\langle\partial_{t}\Phi_{n}^{t},{U}_{n}\right\rangle_{\mathcal{M}_{\Omega,\Gamma}^{1}}.$

We now observe that $\Lambda_{2}\in L^{1}\left(0,T\right)$ on account of
(4.74), (4.114)-(4.115) and (4.117)-(4.118), because
$\partial_{t}U_{n}\left(0\right)\in\mathbb{X}^{2}$ by (4.109). Thus, observing
(4.117), the application of Gronwall’s inequality to (4.119) yields the
desired uniform bound

$U_{n}\in L^{\infty}\left(0,T;\mathbb{V}^{1}\right).$ (4.120)

Finally, by comparison in equation (4.95), by virtue of the uniform bounds
(4.120) and (4.115) we also deduce

$\mathrm{T}\Phi_{n}^{t}\in
L^{\infty}\left(0,T;\mathcal{M}_{\Omega,\Gamma}^{1}\right)$ (4.121)

uniformly with respect to all $n\geq 1$. In particular, it holds
$\Phi_{n}^{t}\in L^{\infty}\left(0,T;D\left(\mathrm{T}\right)\right)$
uniformly. Finally, by (4.120) and assumptions (i)-(ii), we also have

$F\left(U_{n}\right)\in L^{\infty}\left(0,T;\mathbb{X}^{2}\right).$

We can pass to the limit as $n\rightarrow\infty$ in (4.114)-(4.115), (4.120)
and (4.121) to find a limit point $\left(U,\Phi\right)$ with the properties
stated in (4.25)-(4.28). Passage to the limit in equations (4.49)-(4.50) and
in particular, in the nonlinear terms is done in the same fashion as in the
proof of Theorem 4.11. Indeed, exploiting (4.120) and (4.114) we still have
the validity of (4.104) and, hence, the limit solution $\left(U,\Phi\right)$
solves (4.21) in the sense of Definition 4.7.

Uniqueness follows from Proposition 4.10 owing to assumption (4.4). The proof
of theorem is now complete. ∎

Finally, we may conclude with the following.

######  Theorem 4.14.

Let the assumptions of Theorem 4.11 be satisfied. Let $\left(U,\Phi\right)$ be
a unique strong solution corresponding to a given initial datum
$\left(U_{0},\Phi_{0}\right)\in\mathcal{H}_{\Omega,\Gamma}^{2,2}.$ Then, this
solution also satisfies