# On the strong convergence of the Faedo-Galerkin approximations to a strong
T-periodic solution of the torso-coupled bi-domain model

Raul Felipe-Sosa<EMAIL_ADDRESS>Andres Fraguela-Collar  Facultad de
Ciencias en Física y Matemáticas-UNACH, Chiapas, México.  Facultad de Ciencias
Físico Matemáticas-BUAP, Puebla, México.

###### Abstract

In this paper, we investigate the convergence of the Faedo-Galerkin
approximations, in a strong sense, to a strong T-periodic solution of the
torso-coupled bidomain model where $T$ is the period of activation of the
inner wall of heart. First, we define the torso-coupled bi-domain operator and
prove some of its more important properties for our work. After, we define the
abstract evolution system of equations associated with torso-coupled bidomain
model and give the definition of strong solution. We prove that the Faedo-
Galerkin’s approximations have the regularity of a strong solution, and we
find that some restrictions can be imposed over the initial conditions, so
that this sequence of Faedo-Galerkin fully converge to a global strong
solution of the Cauchy problem. Finally, this results are used for showing the
existence a strong $T$-periodic solution.

###### keywords:

bidomain model , Faedo-Galerkin scheme , variational formulation , weak and
strong periodic solutions

###### MSC:

[2010] 00-01, 99-00

##  1 Introduction

In this paper, we use the formulation of the bidomain equations given in [1],
where a parabolic PDE coupled to an elliptic one is considered, and both
coupled to a system of ODEs that model the activation variables for the ionic
currents. In this case, the state variables are the membrane potential
$u_{m}=u_{i}-u_{e}$ and the extra and intra cellular potential $u_{i},u_{e}$,
respectively. We consider a region $\Omega$ in $\mathbb{R}^{3}$ to represent
the heart embedded in the torso. In this regard, it is assumed that
$\Omega=\Omega_{H}\cup\Omega_{T}$, where $\Omega_{H}$ is the region
representing the heart and $\Omega_{T}$ represents the torso. The boundary of
$\Omega_{H}$ has two components, that is,
$\partial\Omega_{H}=\Sigma_{endo}\cup\Sigma_{epi}$ whit $\Sigma_{endo}$
representing the cardiac endocardium, where the periodic activation is given,
and $\Sigma_{epi}$ represents the epicardium which is in contact with the
torso. Also, $\Omega_{T}$ has two components,
$\partial\Omega_{T}=\Sigma_{epi}\cup\Sigma_{T}$, where $\Sigma_{T}$ the
exterior boundary of the torso.

The equations of bidomain are performed as

$\displaystyle A_{m}C_{m}\frac{\partial u_{m}}{\partial
t}+A_{m}I_{ion}(u_{m},\mathbf{w})-\nabla\cdot\left(\sigma_{i}\nabla
u_{m}\right)-\nabla\cdot\left(\sigma_{e}\nabla u_{e}\right)=0,\;\;\;\hbox{in
$(0,\infty)\times\Omega_{H}$},$ (1)
$\displaystyle-\nabla\cdot\left(\sigma_{i}\nabla
u_{m}+(\sigma_{i}+\sigma_{e})\nabla u_{e}\right)=0,\;\;\;\hbox{in
$(0,\infty)\times\Omega_{H}$},$
$\displaystyle\frac{\partial\mathbf{w}}{\partial
t}+g(u_{m},\mathbf{w})=0,\;\;\;\hbox{in $(0,\infty)\times\Omega_{H}$},$

where,

  * 1.

$A_{m},C_{m}$ are the rate of cellular membrane area per volume unit and the
cellular membrane capacitance per area unit, respectively,

  * 2.

$I_{ion}$ is the ionic current across the cell membrane per area unit which
depends on the membrane potential and the vector of activation variables
$\mathbf{w}$,

  * 3.

$g$ is a vector field that models the behavior of the ionic channels,

  * 4.

$\sigma_{i}(\mathbf{x}),\sigma_{e}(\mathbf{x})$ are matrices depending on the
space variable$\mathbf{x}\in\Omega_{H}$.

In general, $\mathbf{w}=\left(w_{1},w_{2},\ldots,w_{k}\right)$ is a vector
that have several components, called gate variables. These variables model the
closure or opening of the ionic channels.

These equations are coupled with a model for the torso which are considered as
a passive conductor.

$\displaystyle-\nabla\cdot\left(\sigma_{T}\nabla u_{T}\right)=0,\;\;\;\hbox{in
$(0,\infty)\times\Omega_{T}$},$ $\displaystyle\sigma_{T}\nabla
u_{T}\cdot\mathbf{n}=0,\;\;\;\hbox{in $(0,\infty)\times\Sigma_{T}$}.$

Here, $u_{T}$ is the potential in the torso, $\sigma_{T}$ is its conductivity
which depends on the space variable, and $\mathbf{n}$ is the outward unit
normal to $\Sigma_{T}$. The last condition means that no current can flow
outside the torso across $\Sigma_{T}$.

In addition to the equations of bidomain and the model for the torso, we
should give boundary conditions and coupling conditions with the torso. A
condition widely assumed, see [2] and [3], is that the intracellular current
does not propagate outside the heart:

$\displaystyle\sigma_{i}\nabla u_{i}\cdot\mathbf{n}=0,\;\;\;\hbox{in
$(0,\infty)\times\partial\Omega_{H}$},$

which can be rewritten as

$\displaystyle\sigma_{i}\nabla u_{m}\cdot\mathbf{n}+\sigma_{i}\nabla
u_{e}\cdot\mathbf{n}=0,\;\;\;\hbox{in $(0,\infty)\times\partial\Omega_{H}$}.$

For the extracelullar current, we give the following boundary condition

$\displaystyle\sigma_{e}\nabla u_{e}\cdot\mathbf{n}=s_{e},,\;\;\;\hbox{in
$(0,\infty)\times\Sigma_{endo}$}.$

where $s_{e}$ is a $T$-periodic function in the variable $t$ that represents
the activation of endocardium.

On other hand, perfect electric conditions between the heart and the torso are
considered

$\displaystyle\left\\{\begin{array}[]{c}u_{e}=u_{T},\\\ \sigma_{e}\nabla
u_{e}\cdot\mathbf{n}=\sigma_{T}\nabla
u_{T}\cdot\mathbf{n},\end{array}\right.\hbox{in
$(0,\infty)\times\Sigma_{epi}$}.$

In summary, the Cauchy problem associated with the torso-coupled bidomain
model can be presented as

$\displaystyle\frac{\partial u_{m}}{\partial
t}+f(u_{m},\mathbf{w})-\nabla\cdot\left(\sigma_{i}\nabla
u_{m}\right)-\nabla\cdot\left(\sigma_{e}\nabla u_{e}\right)=0,\;\;\;\hbox{in
$(0,\infty)\times\Omega_{H}$},$ (2)
$\displaystyle-\nabla\cdot\left(\sigma_{i}\nabla
u_{m}+(\sigma_{i}+\sigma_{e})\nabla u_{e}\right)=0,\;\;\;\hbox{in
$(0,\infty)\times\Omega_{H}$},$ (3)
$\displaystyle\frac{\partial\mathbf{w}}{\partial
t}+g(u_{m},\mathbf{w})=0,\;\;\;\hbox{in $(0,\infty)\times\Omega_{H}$},$ (4)
$\displaystyle\sigma_{i}\nabla u_{m}\cdot\mathbf{n}+\sigma_{i}\nabla
u_{e}\cdot\mathbf{n}=0,\;\;\;\hbox{in $(0,\infty)\times\partial\Omega_{H}$},$
(5) $\displaystyle\sigma_{i}\nabla
u_{m}\cdot\mathbf{n}+\left(\sigma_{i}+\sigma_{e}\right)\nabla
u_{e}\cdot\mathbf{n}=s_{e},\;\;\;\hbox{in $(0,\infty)\times\Sigma_{endo}$},$
(6) $\displaystyle\left\\{\begin{array}[]{c}u_{e}=u_{T},\\\ \sigma_{e}\nabla
u_{e}\cdot\mathbf{n}=\sigma_{T}\nabla
u_{T}\cdot\mathbf{n},\end{array}\right.\hbox{in
$(0,\infty)\times\Sigma_{epi}$},$ (9)
$\displaystyle-\nabla\cdot\left(\sigma_{T}\nabla u_{T}\right)=0,\;\;\;\hbox{in
$(0,\infty)\times\Omega_{T}$},$ (10) $\displaystyle\sigma_{T}\nabla
u_{T}\cdot\mathbf{n}=0,\;\;\;\hbox{in $(0,\infty)\times\Sigma_{T}$},$ (11)
$\displaystyle
u_{m}(0,\mathbf{x})=u^{(0)}_{m}(\mathbf{x}),\;\;\mathbf{w}(0,\mathbf{x})=\mathbf{w}^{(0)}(\mathbf{x}),\;\;\;\hbox{en
$\Omega_{H}$}.$ (12)

###### Remark 1

Note that the constants $A_{m},C_{m}$ do not appear in the equation (2). This
is owing to we have divided the equation (1) by the term $A_{m}C_{m}$ and the
terms $I_{ion}/C_{m},\sigma_{i,e}/A_{m}C_{m}$ have been denoted by
$f,\sigma_{i,e}$, respectively.

In this paper, we work with the operational formulation of the bidomain
problem. That is, the boundary problem (2)-(12) is described as an abstract
evolution system, where its linear part is defined from the so-called torso
coupled bidomain operator. Our goal is to obtain strong solutions associated
to this formulation, taking them as limit of a function sequence which have
the regularity of a strong solution. These are solutions are of a non-linear
ODEs systems sequence. The mentioned sequence of functions is called Faedo-
Galerkin approximation.

The results above statement, which are proved in this paper, present certain
advantages with respect to other results found in the literature about the
convergence the Faedo-Galerkin approximations. In general, the Faedo-Galerkin
systems arise in the weak or variational setting. In that case, the principal
result with respect to the convergence of the Faedo-Galerkin approximations
affirms the convergence, in a weak sense, of a subsequence to a weak solution.
However, in this paper, we demonstrate the convergence of the entire sequence
to a strong solution. This result has important implications. For instance,
starting from a result like this one, it is possible to define a method in
order to approximate a strong solution. In this sense, the convergence of the
entire sequence to the sought strong solution allows us to make sure that the
functions of the sequence are every time closer to this solution, when the
subindex towards to infinite. This would not be a fact if we only prove that
one subsequence of the Faedo-Galerkin sequence converges.

##  2 Some general considerations

In the strong context, we assume that boundary of $\Omega$, $\partial\Omega$,
is $C^{2+\nu}$ and the coefficients of matrices $\sigma_{i,e},\sigma_{T}$ are
$C^{1+\nu}(\Omega_{H,T})$, for some $\nu>0$, respectively. Also, the
conductivity matrices are uniformly elliptic, that is, there are positive
constants $M,m$ such that

$\displaystyle m|\xi|\leq\xi^{t}\sigma_{i,e,T}(\mathbf{x})\xi\leq
M|\xi|,\;\;\;\hbox{for all $\xi\in\mathbb{R}^{3}$ and a.e
$\mathbf{x}\in\Omega_{H,T}$}.$

The fiber structure of the heart is another aspect that should be considered.
This structure determines the anisotropic properties of the cardiac muscle. It
is know that velocity of conduction of the depolarization wavefront is faster
in the longitudinal direction to the fiber than in the transverse direction.
This implies the existence of a matrix $M(\mathbf{x})$ whose columns are the
orthogonal vectors
$\left\\{\mathbf{a}_{l}(\mathbf{x}),\mathbf{a}_{t}(\mathbf{x}),\mathbf{a}_{n}(\mathbf{x})\right\\}$
with $\mathbf{a}_{l}$ parallel to the local fiber direction, $\mathbf{a}_{n}$
the vector orthogonal to the fiber lamina and $\mathbf{a}_{t}$ the vector
lying on the lamina and orthogonal to fiber direction, such that

$\displaystyle\sigma_{i,e}(\mathbf{x})=M^{T}(\mathbf{x})M^{*}_{i,e}M(\mathbf{x}).$

Setting

$\displaystyle
M^{*}_{i,e}=\hbox{diag}\left[\sigma^{i,e}_{l},\sigma^{i,e}_{t},\sigma^{i,e}_{n}\right],$

where $\sigma^{i,e}_{l},\sigma^{i,e}_{t},\sigma^{i,e}_{n}$ are local
coefficients of conductivity measured along the corresponding directions, see
[4].

The fibers are tangent to the boundary, thus

$\displaystyle\sigma_{i,e}(\mathbf{x})\mathbf{n}(\mathbf{x})=\sigma^{i,e}_{n}\mathbf{n}(\mathbf{x}),\;\;\;\hbox{for
a.e. $\mathbf{x}\in\partial\Omega_{H}$},$ (13)

with $\sigma^{i,e}_{n}\geq m>0$.

The above has the following consequence

###### Lemma 2

If the boundary of $\Omega_{H}$ is $C^{1}$ and $M,M^{*}_{i,e}$ are
$C^{0}(\overline{\Omega_{H}})$, then the following statements are true

i)

$\nabla u_{i}\cdot\mathbf{n}=0$ and $\nabla u_{e}\cdot\mathbf{n}=0$
$\Leftrightarrow$ $\nabla u_{m}\cdot\mathbf{n}=0.$

ii)

$\nabla u_{i,e}\cdot\mathbf{n}=0$ $\Leftrightarrow$ $\left(\sigma_{i,e}\nabla
u_{i,e}\right)\cdot\mathbf{n}=0$ and $\left(\sigma_{i}\nabla
u_{e}\right)\cdot\mathbf{n}=0$ $\Leftrightarrow$
$\left\\{\begin{array}[]{c}\left(\sigma_{i}\nabla
u_{m}\right)\cdot\mathbf{n}+\left(\sigma_{i}\nabla
u_{e}\right)\cdot\mathbf{n}=0,\\\ \left(\sigma_{i}\nabla
u_{m}\right)\cdot\mathbf{n}+\left((\sigma_{i}+\sigma_{e})\nabla
u_{e}\right)\cdot\mathbf{n}=0.\end{array}\right.$

Proof. The first affirmation i) is immediate. Let us prove ii). Note that, due
to the symmetry of $\sigma_{i,e}$ and (13) we have

$\displaystyle\left(\sigma_{i,e}\nabla
u_{i,e}\right)\cdot\mathbf{n}=\left(\sigma^{T}_{i,e}\mathbf{n}\right)\cdot\nabla
u_{i,e}=\sigma^{i,e}_{n}\nabla u_{i,e}\cdot\mathbf{n}.$

The first equivalence in ii) it is immediately obtained from the above
equalities. The second equivalence is easy obtained from the first
equivalence.

##  3 Operational formulation. Strong periodic solutions

In this section we obtain what we will call torso coupled bidomain operator.
Once it is obtained, the boundary problem associated to the bidomain model is
formulated. The key idea is that we can obtain $u_{e}$ from the membrane
potential $u_{m}$ integrating the equation (3) with suitable boundary
conditions. This allows us to define a parabolic abstract evolution problem
for the pair $(u_{m},\mathbf{w})$ in the form

$\displaystyle\frac{du_{m}}{dt}+f(u_{m},\mathbf{w})+Au_{m}=s(t),$
$\displaystyle\frac{d\mathbf{w}}{dt}+g(u_{m},\mathbf{w})=0$

where $A$ is a integro-differential second order elliptic operator and $s(t)$
is a function defined in a real interval with values in certain functional
space, that depend on the activation function $s_{e}$. From the solution of
this problem we can calculate the total extracellular potential that we denote
by

$\displaystyle u=\left\\{\begin{array}[]{cc}u_{e},&\hbox{in $\Omega_{H}$},\\\
u_{T},&\hbox{in $\Omega_{T}$}.\end{array}\right.$

We need to give the following notation. Given $X$, certain space of integrable
function on a region $\Theta\subset\mathbb{R}^{3}$, we define

$\displaystyle X/\mathbb{R}=\left\\{\phi\in X:\int_{\Theta}\phi=0\right\\}.$

We denote by

$\displaystyle
L_{H}=L_{2}(\Omega_{H}),\;\;V_{H}=H_{1}(\Omega_{H}),\;\;U_{H}:=V_{H}/\mathbb{R},$
$\displaystyle
L_{T}=L_{2}(\Omega_{T}),\;\;V_{T}=H_{1}(\Omega_{T}),\;\;U_{T}:=V_{T}/\mathbb{R},$

and

$\displaystyle H=L_{2}(\Omega),\;\;V=\left\\{\phi\in
H_{1}(\Omega):\int_{\Omega_{H}}\phi=0\right\\},\;\;U:=V/\mathbb{R},$

where $\Omega=\Omega_{H}\cup\Omega_{T}$.

In the spaces $U_{H,T}$ and $U$ we use the equivalent norm to the norm in
$V_{H,T}$ and $V$ given by

$|\phi|_{U_{H,T}}=\left(\int_{\Omega_{H,T}}\left|\nabla\phi\right|^{2}\right)^{1/2},$

and

$|\phi|_{U}=\left(\int_{\Omega}\left|\nabla\phi\right|^{2}\right)^{1/2},$

respectively.

Here, each triple
$\left\\{U_{H,T},L_{H,T}/\mathbb{R},U^{{}^{\prime}}_{H,T}\right\\}$,
$\left\\{V_{H,T},L_{H,T},V^{{}^{\prime}}_{H,T}\right\\}$,
$\left\\{U,H/\mathbb{R},U^{{}^{\prime}}\right\\}$ and
$\left\\{V,H,V^{{}^{\prime}}\right\\}$ is a Gelfand triple, see [5]. This
means that we have

$\displaystyle U_{H,T}\subset L_{H,T}/\mathbb{R}\subset
U^{{}^{\prime}}_{H,T},$ $\displaystyle V_{H,T}\subset L_{H,T}\subset
V^{{}^{\prime}}_{H,T},$ $\displaystyle U\subset H/\mathbb{R}\subset
U^{{}^{\prime}},$ $\displaystyle V\subset H\subset V^{{}^{\prime}},$

where the inclusions are continuous and the first inclusion is compact, in
each case.

For each duality pair $\left\\{X,X^{\prime}\right\\}$, in the above Gelfand
triplets, by $\left\langle u,\phi\right\rangle$, we denote the value of
functional $u\in X^{\prime}$ in the element $\phi\in X$. The spaces $X$ and
$X^{\prime}$ will be recognized from the context. Also, by $(\cdot,\cdot)$, we
denote the inner product in $L_{H}$ or $H$.

Furthermore, in that follows we use two operators: a restriction operator and
other prolongation. By $R_{H}$, we denote the operator defined from $V$ into
$U_{H}$ such that to each $\phi\in V\mapsto\left.\phi\right|_{\Omega_{H}}\in
U_{H}$. $R_{H}$ is bounded, in fact $\left\|R_{H}\right\|\leq 1$ and its
adjoin operator $R^{*}:U^{{}^{\prime}}_{H}\rightarrow V^{{}^{\prime}}$ is
given by the relation

$\displaystyle\left\langle
R_{H}^{*}\phi_{H},\phi\right\rangle=\left\langle\phi_{H},R_{H}\phi\right\rangle,\;\;\;\hbox{for
all $(\phi_{H},\phi)\in U^{\prime}_{H}\times V$}.$

If we assume that $\Omega_{H}$ has boundary $C^{1}$ from Theorem IX.7 of [5],
it is deduced that there is a prolongation operator, which is bounded,
$P_{H}:U_{H}\rightarrow V$, such that, $R_{H}P_{H}=I_{d}:U_{H}\rightarrow
U_{H}$. Its adjoin operator $P_{H}^{*}:V^{\prime}\rightarrow U^{\prime}_{H}$
defined by

$\displaystyle\left\langle
P_{H}^{*}\phi,\phi_{H}\right\rangle_{U^{\prime}_{H}\times
U_{H}}=\left\langle\phi,P_{H}\phi_{H}\right\rangle_{V^{\prime}\times
V},\;\;\;\hbox{for all $(\phi,\phi_{H})\in V\times U_{H}$}.$

In a similar way, we define the operator $R_{T}:U\rightarrow U_{T}$.

In that follows, we obtain the so-called torso-coupled bidomain operator that
is the linear part of the abstract evolution problem that we want to define.
In this sense, we study the following boundary problem

$\displaystyle-\nabla\cdot\left(\sigma_{i}\nabla
u_{m}\right)-\nabla\cdot\left(\sigma_{i}\nabla u_{e}\right)=0,\;\;\;\hbox{in
$\Omega_{H}$},$ (14) $\displaystyle-\nabla\cdot\left(\sigma_{i}\nabla
u_{m}+\left(\sigma_{i}+\sigma_{e}\right)\nabla u_{e}\right)=0,\;\;\;\hbox{in
$\Omega_{H}$},$ (15) $\displaystyle\sigma_{i}\nabla
u_{m}\cdot\mathbf{n}+\sigma_{i}\nabla u_{e}\cdot\mathbf{n}=0,\;\;\;\hbox{in
$\partial\Omega_{H}$},$ (16) $\displaystyle\sigma_{i}\nabla
u_{m}\cdot\mathbf{n}+\left(\sigma_{i}+\sigma_{e}\right)\nabla
u_{e}\cdot\mathbf{n}=s_{e},\;\;\;\hbox{in $\Sigma_{endo}$},$ (17)
$\displaystyle\left\\{\begin{array}[]{c}u_{e}=u_{T},\\\ \sigma_{e}\nabla
u_{e}\cdot\mathbf{n}=\sigma_{T}\nabla
u_{T}\cdot\mathbf{n},\end{array}\right.\;\;\;\hbox{in $\Sigma_{epi}$},$ (20)
$\displaystyle-\nabla\cdot\left(\sigma_{T}\nabla u_{T}\right)=0,\;\;\;\hbox{in
$(0,\infty)\times\Omega_{T}$},$ (21) $\displaystyle\sigma_{T}\nabla
u_{T}\cdot\mathbf{n}=0,\;\;\;\hbox{in $(0,\infty)\times\Sigma_{T}$}.$ (22)

The variational formulation of the problem (14)-(22) has the following form

$\displaystyle a_{i}(u_{m},\phi_{H})+a_{i}(R_{H}u,\phi_{H})=0,\;\;\;\hbox{for
all $\phi_{H}\in U_{H}$},$ (23)
$\displaystyle\left(\tilde{a}_{e}+\tilde{a}_{i}\right)(u,\phi)+a_{i}(u_{m},R_{H}\phi)=\left\langle\bar{s}_{e},R_{H}\phi\right\rangle,\;\;\;\hbox{for
all $\phi\in U$}.$ (24)

where

$\displaystyle u=\left\\{\begin{array}[]{cc}u_{e},&\hbox{in $\Omega_{H}$,}\\\
u_{T},&\hbox{in $\Omega_{T}$,}\\\ \end{array}\right.$
$a_{i,e}(u_{H},\phi_{H})=\int_{\Omega_{H}}\sigma_{i,e}\nabla
u_{H}\nabla\phi_{H},$

are bilinear forms defined in $U_{H}\times U_{H}$. Also,

$\tilde{a}_{i,e}(u,\phi)=\int_{\Omega}\tilde{\sigma}_{i,e}\nabla u\nabla\phi,$

are bilinear forms defined in $U\times U$, where

$\displaystyle\tilde{\sigma}_{e}=\left\\{\begin{array}[]{cc}\sigma_{e}&\hbox{in
$\Omega_{H}$},\\\ \sigma_{T}&\hbox{in $\Omega_{T}$},\end{array}\right.$

and

$\displaystyle\tilde{\sigma}_{i}=\left\\{\begin{array}[]{cc}\sigma_{i}&\hbox{in
$\Omega_{H}$},\\\ 0&\hbox{in $\Omega_{T}$}.\end{array}\right.$

Finally, $\bar{s}_{e}\in U^{\prime}_{H}$ is defined by

$\displaystyle\left\langle\bar{s}_{e},\phi_{H}\right\rangle:=\int_{\Sigma_{endo}}s_{e}\phi_{H},\;\;\;\forall\phi_{H}\in
U_{H}.$

The bilinear forms $\tilde{a}_{i,e}$ and $a_{i,e}$ are symmetric, continuous
and coerciva. Because of this, by the Lax-Milgram theorem, for each $u_{m}\in
U_{H}$ fixed there is an only $\tilde{u}$ that satisfies de variational
equation (24). Besides, we have that there are operators

$\displaystyle\tilde{A}_{i,e}:U\rightarrow U^{\prime},$ $\displaystyle
A_{i,e}:U_{H}\rightarrow U^{\prime}_{H},$

which are one-to-one continuous with continuous inverse, so that

$\displaystyle\left\langle\tilde{A}_{i,e}\phi_{1},\phi_{2}\right\rangle_{U^{\prime}\times
U}=\tilde{a}_{i,e}(\phi_{1},\phi_{2}),\;\;\;\hbox{for all
$\phi_{1},\phi_{2}\in U$},$ $\displaystyle\left\langle
A_{i,e}\phi^{(1)}_{H},\phi^{(2)}_{H}\right\rangle_{U^{\prime}\times
U}=\tilde{a}_{i,e}(\phi^{(1)}_{H},\phi^{(2)}_{H}),\;\;\;\hbox{for all
$\phi^{(1)}_{H},\phi^{(2)}_{H}\in U_{H}$}.$

Equation (24) can be rewritten in as follows,

$\displaystyle\left\langle\left(\tilde{A}_{e}+\tilde{A}_{i}\right)\tilde{u},\phi\right\rangle_{U^{\prime}\times
U}=\left\langle\bar{s}_{e}-A_{i}u_{m},R_{H}\phi\right\rangle_{U^{\prime}_{H}\times
U_{H}}=\left\langle
R_{H}^{*}\bar{s}_{e}-R_{H}^{*}A_{i}u_{m},\phi\right\rangle_{U^{\prime}\times
U},$

thus,

$\displaystyle\tilde{u}=\left(\tilde{A}_{e}+\tilde{A}_{i}\right)^{-1}R_{H}^{*}\bar{s}_{e}-\left(\tilde{A}_{e}+\tilde{A}_{i}\right)^{-1}R_{H}^{*}A_{i}u_{m}=\left(\tilde{A}_{e}+\tilde{A}_{i}\right)^{-1}R^{*}_{H}\left(\bar{s}_{e}-A_{i}u_{m}\right).$

Substituting this expression for $\tilde{u}$ in (23), we obtain:

$\displaystyle\left\langle
A_{i}u_{m},\phi_{H}\right\rangle_{U^{\prime}_{H}\times U_{H}}+\left\langle
A_{i}R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}\bar{s}_{e},\phi_{H}\right\rangle_{U^{\prime}_{H}\times
U_{H}}$ $\displaystyle-\left\langle
A_{i}R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}A_{i}u_{m},\phi_{H}\right\rangle_{U^{\prime}_{H}\times
U_{H}}=0,\;\;\;\hbox{for all $\phi_{H}\in U_{H}$}.$

In terms of the weak operators $\tilde{A}_{i},\tilde{A}_{e},A_{i}$ the above
equation is written as

$\displaystyle\bar{A}u_{m}=\bar{s},$

where $\bar{A}:U_{H}\rightarrow U^{\prime}_{H}$, with

$\displaystyle\bar{A}=A_{i}-A_{i}R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}A_{i}=A_{i}R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}\tilde{A}_{e}P_{H}=A_{i}R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}A_{e},$

and $\bar{s}\in U^{\prime}_{H}$ so that

$\displaystyle\bar{s}=-A_{i}R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}\bar{s}_{e}.$

With the objective of defining and studying the full bidomain evolution
abstract problem, we must to extend the operator $\bar{A}$ and the functional
$\bar{s}$ to $V_{H}$. Then, we define $J:V_{H}\rightarrow U_{H}$ given by
relation
$\phi_{H}\mapsto\phi_{H}-\frac{1}{|\Omega_{H}|}\int_{\Omega_{H}}\phi_{H}$,
where $|\Omega_{H}|$ is the volume of $\Omega_{H}$. Finally, we define

$\displaystyle A:=J^{*}\bar{A}J:V_{H}\rightarrow V^{\prime}_{H},$

and

$\displaystyle s=J^{*}\bar{s}\in V^{\prime}_{H}.$

In order the functions $u\in U$ and $u_{m}\in U_{H}$ to be considered weak
solution of the problem (14)-(22) in $V$ and $V_{H}$ it is necessary to assume
that $\left\langle\bar{s}_{e},1\right\rangle_{U^{\prime}_{H}\times U_{H}}=0$,
that is,

$\displaystyle\int_{\Sigma_{endo}}s_{e}=0.$

Let us give the following proposition.

###### Proposition 3

The bilinear form associated to the operator $\bar{A}$, denoted by us
$\bar{a}$, is defined in $U_{H}\times U_{H}$ and has been given by the
expression

$\displaystyle\bar{a}(u_{H},\phi_{H})=a_{i}\left(u_{H}-R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}A_{i}u_{H},\phi_{H}\right),$

and the bilinear form associated to operator $A$ is
$a(u_{H},\phi_{H})=\bar{a}(Ju_{H},J\phi_{H})$ for all $(u_{H},\phi_{H})\in
V_{H}\times V_{H}$.

Furthermore, $a$ is symmetric, continuous and coercive and there is an
increasing sequence
$0=\lambda_{0}<\lambda_{1}\leq\ldots\leq\lambda_{n}\leq\ldots$ in $\mathbb{R}$
and an orthonormal base $\left(\psi_{i}\right)^{\infty}_{i=0}\subset L_{H}$,
such that $\psi_{i}\in V_{H}$ and

$\displaystyle a(\psi_{i},v)=\lambda_{i}(\psi_{i},v)\;\;\;\hbox{for all $v\in
V_{H},i\in\mathbb{N}$.}$

Proof. Let $u_{H},\phi_{H}\in U_{H}$ be arbitrary, we have:

$\displaystyle\left\langle\bar{A}u_{H},\phi_{H}\right\rangle=\left\langle
A_{i}u_{H},\phi_{H}\right\rangle-\left\langle
A_{i}R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}A_{i}u_{H},\phi_{H}\right\rangle=$
$\displaystyle\left\langle
A_{i}\left(u_{H}-R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}A_{i}u_{H}\right),\phi_{H}\right\rangle=a_{i}\left(u_{H}-R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}A_{i}u_{H},\phi_{H}\right).$

From the above, we prove the first part of Proposition.

Let us prove that $\bar{a}$ is symmetric. We have that

$\displaystyle\bar{a}(u_{H},\phi_{H})=a_{i}\left(R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}A_{i}u_{H},\phi_{H}\right)=a_{i}\left(\phi_{H},R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}A_{i}u_{H}\right)=$
$\displaystyle\left\langle
R^{*}_{H}\phi_{H},\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}A_{i}u_{H}\right\rangle=\left\langle
R^{*}_{H}A_{i}u_{H},\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}\phi_{H}\right\rangle=\left\langle
A_{i}u_{H},R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}\phi_{H}\right\rangle=$
$\displaystyle
a_{i}\left(u_{H},R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}\phi_{H}\right)=a_{i}\left(R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}\phi_{H},u_{H}\right)=\bar{a}(\phi_{H},u_{H}).$

In the above equality chain, we have used the fact, which is easy to prove,
that

$\displaystyle\left\langle\phi,\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}u\right\rangle=\left\langle
u,\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}\phi\right\rangle,\;\;\;\hbox{for
all $u,\phi\in U^{\prime}$}.$

On the other hand, from a simple calculation we get

$\displaystyle\bar{a}(u_{H},u_{H})\geq\frac{m}{3}\left(1+\frac{m}{2M}\right)\left|u_{H}\right|^{2}_{U_{H}},$

and

$\displaystyle\left|\bar{a}(u_{H},\phi_{H})\right|\leq
M\left(1+\frac{M}{2m}\right)\left|u_{H}\right|_{U_{H}}\left|\phi_{H}\right|_{U_{H}}.$

The result about the existence of the sequence
$\left(\lambda_{n}\right)^{\infty}_{n=0}$ and the orthonormal bases
$\left(\psi_{n}\right)^{\infty}_{n=0}\subset V_{H}$ can be found in [6].

Now, we are going to define the strong operators associated to operators
$A_{i,e},\tilde{A}_{i,e}$ and $A$.

We define

$\displaystyle D(A):=\left\\{u_{H}\in H_{2}(\Omega_{H}):\nabla
u_{H}\cdot\mathbf{n}=0\;\;\hbox{in a.e. $\partial\Omega_{H}$}\right\\}\subset
L_{H},$

We have

$\displaystyle D(A_{i})=D(A_{e})=D(A)/\mathbb{R},$

and

$\displaystyle u_{H}\in D(A_{i,e})\mapsto
A_{i,e}u=\nabla\cdot\left(\sigma_{i,e}\nabla u_{H}\right).$

On the other hand, if we denote by

$\displaystyle D(\widetilde{A})=\left\\{u\in
H_{2}(\Omega):\begin{array}[]{c}\nabla R_{H}u\cdot\mathbf{n}=\nabla
R_{T}u\cdot\mathbf{n}=0,\\\ R_{H}u=R_{T}u,\end{array}\hbox{in a.e.
$\partial\Omega_{H}$},\;\;\hbox{and}\;\;\nabla
R_{T}u\cdot\mathbf{n}=0\;\;\hbox{in $\Sigma_{T}$},\right\\}$

we have that

$\displaystyle
D(\tilde{A}_{i,e})=D(\tilde{A}_{i}+\tilde{A}_{e})=D(\widetilde{A})/\mathbb{R}.$

where for all $u\in D(\tilde{A}_{i,e})$ we have

$\displaystyle\tilde{A}_{i,e}u=\nabla\cdot\left(\widetilde{\sigma}_{i,e}\nabla
u\right).$

It is possible to prove that $A_{i}$ and
$\tilde{A}_{i,e},\tilde{A}_{i}+\tilde{A}_{e}$ are maximal monotone, self-
adjoin and have compact inverse in $L_{H}/\mathbb{R}$ and $H/\mathbb{R}$,
respectively.

With respect to the functional $s\in V^{\prime}_{H}$, it is known that there
is an element $\tilde{s}\in L_{H}$ such that
$\left\|\tilde{s}\right\|_{L_{H}}=\left\|s\right\|_{V^{\prime}_{H}}$ and

$\displaystyle\left\langle
s,u_{h}\right\rangle=\left(\tilde{s},u\right),\;\;\;\hbox{for all $u\in
V_{H}.$}$

###### Definition 4

We define the strong bidomain operator $A:D(A)\subset H\rightarrow H$ given by
the relation

$Au_{H}=A_{i}R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}R^{*}_{H}A_{e}Ju_{H},$

for all $u_{H}\in D(A)$.

Furthermore, by $s$ we denote again the element in $L_{H}$ that it is
identified with the functional $s\in V_{H}$ given in the above comment.

Note that, we again have denoted by $A$ this operator, which is well-defined
due to two facts.

First,

$\displaystyle\left.R^{*}_{H}\right|_{L_{H}/\mathbb{R}}:L_{H}/\mathbb{R}\rightarrow
H/\mathbb{R},$

and it is given by

$\displaystyle R^{*}_{H}u=\left\\{\begin{array}[]{cc}u,&\hbox{in
$\Omega_{H}$,}\\\ 0,&\hbox{in $\Omega_{T}$.}\end{array}\right.$

Furthermore,

$\displaystyle
R_{H}\left(\tilde{A}_{i}+\tilde{A}_{e}\right)^{-1}:H/\mathbb{R}\rightarrow
D(A)/\mathbb{R}.$

###### Theorem 5

The strong operator $A$ of Definition 7 is self-adjoin and maximal-monotone,
such that,

$\displaystyle\left(Au_{H},\phi_{H}\right)=a(u_{H},\phi_{H}),\;\;\;\hbox{for
all $u_{H},\phi_{H}\in D(A)$}.$

The sequence $\left(\lambda_{n}\right)^{\infty}_{n=0}$ given in Proposition 3
satisfies that each $\lambda_{i}$ is a eigenvalue of $A$ associated to
eigenvector $\psi_{i}\in D(A)$. Furthermore, one has that

$\displaystyle D(A)=\left\\{u_{H}\in
L_{H}:\sum^{\infty}_{n=0}\lambda^{2}_{n}\left(\psi_{n},u_{H}\right)^{2}<\infty\right\\},\;\;and\;\;Au_{H}=\sum^{\infty}_{n=0}\lambda_{n}\left(\psi_{n},u_{H}\right)\psi_{n}.$

Finally, we have that
$Au_{m}=s,u=\left(\tilde{A}_{e}+\tilde{A}_{i}\right)^{-1}R^{*}_{H}\left(s-A_{i}u_{m}\right)$
and $u_{e}=R_{H}u\in D(A)$, if and only if $(u,u_{m})$ are solutions of
problem (14)-(22) a.e.

On the other hand, for any $a_{1}>0$, $A+a_{1}:D(A)\rightarrow L_{H}$ is also
a sectorial, auto-adjoin, maximal-monotone operator such that its spectrum
consists of the eigenvalues
$\left\\{a_{1}+\lambda_{i}\right\\}^{\infty}_{i=0}$ with eigenvectors
$\left\\{\psi_{i}\right\\}^{\infty}_{i=0}$. The family of bounded operators
$e^{-t(A+a_{1})}$ for all $t\geq 0,$ is well defined and
$e^{-0(A+a_{1})}=I_{d}$ (the identity operator in $L_{H}$).

With respect to this family of operators, it is possible to give several
properties, such as:

  * 1.

for all $t>0,$ the spectrum of operator $e^{-t(A+a_{1})}$ is
$\left\\{e^{-(\lambda_{i}+a_{1})t}\right\\}^{\infty}_{i=0}$ thus,
$1\not\in\sigma\left(e^{-t(A+a_{1})}\right)$. Furthermore,
$\left(\psi_{i}\right)^{\infty}_{i=0}$ are the eigenvectors associated,

  * 2.

$\left(I_{d}-e^{-T(A+a_{1})}\right)^{-1}$ is a linear bounded operator,
densely defined. Its spectrum is
$\left(\left(1-e^{-(\lambda_{i}+a_{1})T}\right)^{-1}\right)^{\infty}_{i=0},$
with eigenvectors $\left(\psi_{i}\right)^{\infty}_{i=0},$

  * 3.

$\left\|e^{-t(A+a_{1})}\right\|\leq 1$, for all $t\geq 0$.

###  3.1 Definition of the strong solution. Uniqueness and local existence

In this section, $f,g$ are locally Lipschitz. That is, for all
$(u_{0},w_{0})\in\mathbb{R}^{2}$, there is an open $U\ni(u_{0},w_{0})$ and a
constant $K$ such that

$\displaystyle\begin{array}[]{c}|f(u_{1},w_{1})-f(u_{2},w_{2})|\leq
K|(u_{1},w_{1})-(u_{2},w_{2})|,\\\ |g(u_{1},w_{1})-g(u_{2},w_{2})|\leq
K|(u_{1},w_{1})-(u_{2},w_{2})|.\end{array}$

Note, the above means that $f+a_{1}u$ is also locally Lipschitz continuous,
for any $a_{1}>0$.

Consider $\mathcal{Z}:=L_{H}\times L_{H}$ with the norm
$\|(u,w)\|_{\mathcal{Z}}=\max\left\\{\|u\|_{L_{H}},\|w\|_{L_{H}}\right\\}$
which is a Banach space. The operator
$\mathcal{A}:D(\mathcal{A})\subset\mathcal{Z}\rightarrow\mathcal{Z}$ is
defined, given by

$\displaystyle\mathcal{A}(u,w)=\left(\begin{array}[]{cc}a_{1}+A&0\\\
0&0\end{array}\right)\left(\begin{array}[]{c}u\\\ w\end{array}\right),$

where $D(\mathcal{A})=D(A)\times L_{H}$.

###### Lemma 6

The operator $\mathcal{A}$ is sectorial, has compact resolvent and
$\mathbf{R}\mathbf{e}\left\\{\sigma(\mathcal{A})\right\\}>0$.

To deal with the non-linearity, we use the fractional power of the operator
$A$, $A^{\alpha}$, and the interpolation spaces $D(A^{\alpha})$ with
$\alpha\geq 0$. In this sense, we have $A^{\alpha}:D(A^{\alpha})\subset
L_{H}\rightarrow L_{H}$ is an unbounded operator, such that

$\displaystyle D(A^{\alpha})=\left\\{u\in
L_{H}:\sum^{\infty}_{i=0}\lambda^{2\alpha}_{i}\left(u,\psi_{i}\right)^{2}<\infty\right\\},\;\;\hbox{and}\;\;A^{\alpha}u=\sum^{\infty}_{i=0}\lambda^{\alpha}_{i}\left(u,\psi_{i}\right)\psi_{i}.$

The space $D(A^{\alpha})$ with the norm

$\displaystyle\left\|u\right\|_{D(A^{\alpha})}=\left\|u+A^{\alpha}u\right\|_{L_{H}}=\sqrt{\sum^{\infty}_{i=0}(1+\lambda^{2\alpha}_{i})(u,\psi_{i})^{2}},$

is a Banach space.

We also can define $\mathcal{A}^{\alpha}$, and the interpolation spaces
$D(\mathcal{A}^{\alpha})$ as

$\displaystyle\mathcal{Z}^{\alpha}:=D(A^{\alpha})\times
D(A^{\alpha}),\;\;\hbox{and}\;\;\mathcal{A}^{\alpha}(u,w)=\left(\sum^{\infty}_{i=0}\lambda^{\alpha}_{i}\left(u,\psi_{i}\right)\psi_{i},0\right),\;\;\;\forall(u,w)\in\mathcal{Z}^{\alpha}.$

The space $\mathcal{Z}^{\alpha}$, equipped with norm
$\|(u,w)\|_{\alpha}=\max\left\\{\left\|u\right\|_{D(A^{\alpha})},\left\|w\right\|_{D(A^{\alpha})}\right\\}$
is also a Banach space. It is possible to prove that for all
$0\leq\alpha<\beta\leq 1,$ $\mathcal{Z}^{\beta}\subset\mathcal{Z}^{\alpha}$
and $\mathcal{Z}^{0}=\mathcal{Z}$, $\mathcal{Z}^{1}=D(\mathcal{A})\subset
H_{2}(\Omega)\times H_{2}(\Omega)$. Since the resolvent of $\mathcal{A}$ is
compact, the inclusion $\mathcal{Z}^{\beta}\subset\mathcal{Z}^{\alpha}$ is
compact, provided that $\alpha<\beta$.

For any $\alpha>0$, the operator $\mathcal{A}^{-\alpha}$ can also defined in
$\mathcal{Z}$. It is linear, bounded, injective, and is the inverse of
operator $\mathcal{A}^{\alpha}$. In this case, $\mathcal{Z}^{\alpha}$
coincides with the range of $\mathcal{A}^{-\alpha}$. We define
$\mathcal{Z}^{-\alpha}$ to the set $\mathcal{Z}$ with norm

$\displaystyle\left\|z\right\|_{-\alpha}=\left\|\mathcal{A}^{-\alpha}z\right\|_{\mathcal{Z}},$

which is a Banach space. It is easy to prove that the following inclusions are
continuous

$\displaystyle\mathcal{Z}^{\alpha}\subset\mathcal{Z}\subset\mathcal{Z}^{-\alpha},\;\;\;\hbox{for
all $\alpha>0$}.$

The proof of these results can be found in [7].

If $3/4<\alpha\leq 1$, then $\mathcal{Z}^{\alpha}\subset
L_{\infty}(\Omega_{H})\times L_{\infty}(\Omega_{H})$ and given that
$f+a_{1},g:\mathbb{R}^{2}\rightarrow\mathbb{R}$ are locally Lipschitz
continuous then, $F=(f+a_{1},g):\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}$ is
locally Lipschitz continuous. That is, given $z_{0}\in\mathcal{Z}^{\alpha}$
there is an open $\mathcal{U}\subset\mathcal{Z}^{\alpha}$ containing to
$z_{0}$ and a positive constants $L$, such that

$\displaystyle\left\|F(z_{1})-F(z_{2})\right\|_{\mathcal{Z}}\leq
L\left\|z_{1}-z_{2}\right\|_{\alpha},\;\;\;\hbox{for all
$z_{1},z_{2}\in\mathcal{U}$.}$

Following [7], it is possible to give a definition of strong solution of the
Cauchy problem associated to the bidomain problem (2)-(12).

######  Definition 7 (Strong solution)

Consider the functions $z(t)=(u(t),w(t)):[0,t_{1})\rightarrow\mathcal{Z}$, and
$u:[0,t_{1})\rightarrow H$ where

$\displaystyle u(t)=\left\\{\begin{array}[]{cc}u_{e}(t),&\hbox{in
$\Omega_{H}$,}\\\ u_{T}(t),&\hbox{in $\Omega_{T}$.}\end{array}\right.$

We say that the pair $(z,u)$ is a local strong solution to (2)-(12) if

  1. 1.

$z:[0,t_{1})\rightarrow\mathcal{Z}$ is continuous such that
$z(0)=z_{0}=(u_{0},w_{0})$, for all $t\in(0,t_{1})$, $z(t)\in\mathcal{U}\cap
D(\mathcal{A})$, and $\dfrac{dz(t)}{dt}$ exists in the Fréchet sense,

  2. 2.

for all $t\in(0,t_{1})$ the following semilinear evolution equation is
satisfied by $z(t)$

$\displaystyle\frac{dz(t)}{dt}=-\mathcal{A}z(t)+F(z(t))+S(t),\;\;\;\hbox{in
$\mathcal{Z},$}$ (25)

and
$u(t)=\left(\tilde{A}_{e}+\tilde{A}_{i}\right)^{-1}R^{*}_{H}\left(s-A_{i}Ju_{m}(t)\right).$

If $t_{1}=\infty$, we say that the solution is global. Furthermore, if the
pair $z,u$ is a global solution such that $z(t)=z(t+T),u(t)=u(t+T)$, we say
that this is a $T$-periodic solution.

From Lemma 3.3.2 of [7], it can be demonstrate that if $z(t)$ is a local
strong solution with initial condition $z_{0}$, then

$\displaystyle
z(t)=e^{-t\mathcal{A}}z_{0}+\int^{t}_{0}e^{-(t-\tau)\mathcal{A}}\left(F(z(\tau))+S(\tau)\right)d\tau,\;\;\;t\in(0,t_{1}).$
(26)

Conversely, if $z:[0,t_{1})\rightarrow\mathcal{Z}^{\alpha}$ is a continuous
function that satisfies the integral equation (26), and

$\displaystyle\int^{\rho}_{0}\left\|F(z(\tau))+S(\tau)\right\|_{\mathcal{Z}}d\tau<\infty,\;\;\;\hbox{for
some $\rho>0$,}$ (27)

then $z(t)$ is a strong solution in $(0,t_{1})$.

The following theorem is a consequence of Theorem 3.2.2 of [7].

###### Theorem 8

If the initial condition $(u_{0},w_{0})\in\mathcal{Z}^{\alpha}$, then the
bidomain model has a only strong solution, $(u,w)$, that satisfies the initial
condition $(u(0),w(0))=(u_{0},w_{0}).$

##  4 Faedo-Galerkin system associated to the operational formulation

In this section, we investigate when the Faedo-Galerkin approximations
converge to a strong solution, in the sense of Definition 7. We obtain the
Faedo-Galerkin systems from the semilinear evolution equation (25). Our idea
is based in the fact that Faedo-Galerkin approximations have the same
regularity of a strong solution. Furthermore, we can achieve that these
converge in a suitable sense such that its limit be a strong solution. We
believe that it is an important result because, until now, only weak solutions
were obtained as the limit of Faedo-Galerkin approximations.

We define $H_{m}=\left\langle\psi_{0},\psi_{1},\ldots,\psi_{m}\right\rangle$
as the subspace generated by the $m+1$-first eigenvectors associated to the
bidomain operator $A$, endowed with the norm of $L_{H}$. We denote by
$\mathcal{Z}_{m}=H_{m}\times H_{m}$.

Note, $\mathcal{Z}_{m}\subset\mathcal{Z}^{\alpha}$, for all $\alpha\geq 0$
which is only considered as a sets inclusion. We can define
$\mathcal{Z}^{\alpha}_{m}=H_{m}\times H_{m}$ endowed with the norm in
$\mathcal{Z}^{\alpha}$, for $\alpha\geq 0$.

We can define
$\mathcal{P}_{m}:\mathcal{Z}\rightarrow\mathcal{Z}_{m}\subset\mathcal{Z}$ by

$\displaystyle\mathcal{P}_{m}z=\left(\sum^{m}_{i=0}\left(u,\psi_{i}\right)\psi_{i},\sum^{m}_{i=0}\left(w,\psi_{i}\right)\psi_{i}\right),\;\;\;\forall
z=(u,w)\in\mathcal{Z},\;\;\;\hbox{for all
$m\in\mathbb{N}\cup\left\\{0\right\\}.$}$

The operator that was defined in the previous equation is called $\mathcal{Z}$
in $\mathcal{Z}_{m}$ projection operator, which satisfies that
$\left\|\mathcal{P}_{m}\right\|_{\mathcal{L}(\mathcal{Z})}\leq 1$. Similarly,
it can be easily proved that

$\left.\mathcal{P}_{m}\right|_{\mathcal{Z}^{\alpha}}:\mathcal{Z}^{\alpha}\rightarrow\mathcal{Z}^{\alpha},$

$\left\|\mathcal{P}_{m}\right\|_{\mathcal{L}(\mathcal{Z}^{\alpha})}\leq 1$,
and

$\displaystyle\left\|\mathcal{P}_{m}z-z\right\|_{\mathcal{Z}^{\alpha}}\rightarrow
0,\;\;\left\|\mathcal{P}_{m}z-z\right\|_{\mathcal{Z}}\rightarrow
0\;\;\;\hbox{when $m\rightarrow\infty$, for all $z\in\mathcal{Z}^{\alpha}.$}$

In other words,
$\left\|\mathcal{P}_{m}-I_{d}\right\|_{\mathcal{L}(\mathcal{Z})}\rightarrow 0$
and
$\left\|\mathcal{P}_{m}-I_{d}\right\|_{\mathcal{L}(\mathcal{Z}^{\alpha})}\rightarrow
0$. Note that, this implies

$\displaystyle\left\|\mathcal{P}_{m}-\mathcal{P}_{n}\right\|_{\mathcal{L}(\mathcal{Z})}\leq\left\|I_{d}-\mathcal{P}_{n}\right\|_{\mathcal{L}(\mathcal{Z})}+\left\|I_{d}-\mathcal{P}_{m}\right\|_{\mathcal{L}(\mathcal{Z})}\rightarrow
0,$

and

$\displaystyle\left\|\mathcal{P}_{m}-\mathcal{P}_{n}\right\|_{\mathcal{L}(\mathcal{Z})^{\alpha}}\rightarrow
0$

when $m,n\rightarrow\infty$.

Define the functions

$\displaystyle
u_{m}(t,x)=\sum^{m}_{i=0}u^{(m)}_{i}(t)\psi_{i}(x),\;\;w_{m}(t,x)=\sum^{m}_{i=0}w^{(m)}_{i}(t)\psi_{i}(x),$

where the vector functions

$\displaystyle
U_{m}(t)=\left(u^{(m)}_{0}(t),u^{(m)}_{1}(t),\ldots,u^{(m)}_{m}(t)\right)\;\;\hbox{and}\;\;W_{m}(t)=\left(w^{(m)}_{0}(t),w^{(m)}_{1}(t),\ldots,w^{(m)}_{m}(t)\right)$
(28)

are in $C^{1}\left(I,\mathbb{R}^{m+1}\right)$ for certain interval
$I\subseteq\mathbb{R}$ containing to $0$. Note,
$z_{m}(t)=(u_{m}(t),w_{m}(t))\in\mathcal{Z}_{m}$, in particular $z_{m}(t)\in
D(\mathcal{A})$ for all $t\in I$. Besides,

$\displaystyle
u^{\prime}_{m}(t,x)=\sum^{m}_{i=0}u^{(m)^{\prime}}_{i}(t)\psi_{i}(x),\;\;w^{\prime}_{m}(t,x)=\sum^{m}_{i=0}w^{(m)^{\prime}}_{i}(t)\psi_{i}(x),$

are the Fréchet derivatives of the function $u_{m},w_{m}$, respectively, thus,
$z^{\prime}_{m}(t)=(u^{\prime}_{m}(t),w^{\prime}_{m}(t))\in\mathcal{Z}_{m}$.

In that follows, we are interested in to study the following problem of
initial conditions: to find a function $z_{m}\in
C^{1}(I;\mathcal{Z}^{\alpha}_{m})$, for certain $0<\alpha<1$, which satisfies

$\displaystyle\left\\{\begin{array}[]{c}\dfrac{dz_{m}(t)}{dt}=-\mathcal{A}z_{m}(t)+\mathcal{P}_{m}F(z_{m}(t))+\mathcal{P}_{m}S(t),\\\
z_{m}(0)=z^{(0)}_{m}\in\mathcal{Z}_{m}^{\alpha}.\end{array}\right.$ (31)

and converges in a suitable form, which will be specified later, to a strong
solution in the sense of Definition 7. Take into account that
$z^{(0)}_{m}=\left(u^{(0)}_{m},w^{(0)}_{m}\right)$, such that,

$\displaystyle
u^{(0)}_{m}=\sum^{m}_{i=0}u^{(m,0)}_{i}\psi_{i},\;\;w^{(0)}_{m}=\sum^{m}_{i=0}w^{(m,0)}_{i}\psi_{i},$

thus, to the initial condition $z^{(0)}_{m}$ we can also associate a pair of
vectors of $\mathbb{R}^{m+1}$

$\displaystyle
U^{(0)}_{m}=\left(u^{(m,0)}_{0},u^{(m,0)}_{1},\ldots,u^{(m,0)}_{m}\right),\;\;W^{(0)}_{m}=\left(w^{(m,0)}_{0},w^{(m,0)}_{1},\ldots,w^{(m,0)}_{m}\right).$
(32)

The sequence $\left\\{z_{m}\right\\}^{\infty}_{m=0}\subset
C^{1}(I;\mathcal{Z}^{\alpha}_{m})$ built from the solution of problem (31) for
each $m\in\mathbb{N}$, will be called by us as strong Faedo-Galerkin
approximations of the strong solution of the torso-coupled bidomain model.
Now, we will show the reason why this terminology is used.

###### Remark 9

In order to achieve this convergence, we suppose that the sequence of initial
conditions $\left\\{z^{(0)}_{m}\right\\}$ converges in $\mathcal{Z}^{\alpha}$
and its limit is $z_{0}\in\mathcal{Z}^{\alpha}$. In this case, there are
constants $m_{0}\in\mathbb{N},\delta_{m_{0}}>0$ and an open
$\mathcal{U}_{m_{0}}\subset\mathcal{Z}^{\alpha}$, such that $F$ is Lipzchitz
continuous in $\mathcal{U}_{m_{0}}$ with Lipschitz constant $L_{m_{0}}$ and

$\displaystyle\left\\{z^{(0)}_{m}\right\\}^{\infty}_{m=m_{0}}\subset\mathcal{B}_{\alpha}\left(z_{0},\frac{\delta_{m_{0}}}{2}\right)\subset\mathcal{U}_{m_{0}}.$

Now, for each of the remaining elements of the sequence,
$\left\\{z^{(0)}_{k}\right\\}^{m_{0}-1}_{k=0}$, there is an open
$\mathcal{U}_{k}\subset\mathcal{Z}^{\alpha}$ and a ball
$\mathcal{B}_{\alpha}\left(z^{(0)}_{k},\frac{\delta_{k}}{2}\right)$, such
that,

$\displaystyle\mathcal{B}_{\alpha}\left(z^{(0)}_{k},\frac{\delta_{k}}{2}\right)\subset\mathcal{U}_{k},$

and $F$ is Lipschitz continuous in $\mathcal{U}_{k}$ with Lipschitz constant
$L_{k}$. Then, we have

$\displaystyle\left\\{z^{(0)}_{m}\right\\}^{\infty}_{m=0}\subset\left(\bigcup^{m_{0}-1}_{m=0}\mathcal{B}_{\alpha}\left(z^{(0)}_{k},\frac{\delta}{2}\right)\right)\bigcup\mathcal{B}_{\alpha}\left(z_{0},\frac{\delta}{2}\right)\subset\mathcal{U}=\bigcup^{m_{0}}_{m=0}\mathcal{U}_{m}$

where
$\delta=\min\left\\{\delta_{0},\delta_{1},\ldots,\delta_{m_{0}}\right\\}$, and
$F$ is Lipschitz continuous, with Lipschitz constant
$L=\max\left\\{L_{0},L_{1},\ldots,L_{m_{0}}\right\\}$.

Summarizing the presented above, we assume that $\left\\{z^{(0)}_{m}\right\\}$
satisfies the following condition

CB

There is a finite family of balls
$\left\\{\mathcal{B}_{\alpha}\left(z_{j},\frac{\delta}{2}\right)\right\\}^{m_{0}}_{j=0}$,
where $z_{j}\in\mathcal{Z}^{\alpha}$ for $j=0,1,\ldots,m_{0}$, such that,

$\left\\{z^{(0)}_{m}\right\\}\subset\mathcal{F}=\bigcup^{m_{0}}_{j=0}\mathcal{B}_{\alpha}\left(z_{j},\frac{\delta}{2}\right),$

and $F$ is Lipschitz continuous in $\mathcal{F}$.

If $z_{m}$ satisfies the problem in (31), then the vectors $U_{m}(t),W_{m}(t)$
associated with $z_{m}(t)=(u_{m}(t),w_{m}(t))$ given in (28), satisfy the
following problem of initial conditions associated to a system of $2m+2$
equations

$\displaystyle\left\\{\begin{array}[]{c}u^{(m)^{\prime}}_{i}(t)=-\left(a_{1}+\lambda_{i}\right)u^{(m)}_{i}(t)+\displaystyle\int_{\Omega_{H}}f(u_{m}(t),w_{m}(t))\psi_{i}+\int_{\Omega_{H}}S(\tau)\psi_{i},\\\
w^{(m)^{\prime}}_{i}(t)=\displaystyle\int_{\Omega_{H}}g(u_{m}(t),w_{m}(t))\psi_{i},\\\
U_{m}(0)=U_{m}^{(0)},\;\;W_{m}(0)=W_{m}^{(0)}.\end{array}\right.i=0,\ldots,m.$
(36)

This means that the Cauchy problem (31) is equivalent to (36), which is
associated to a system of ordinary
differential equation. That is, if (36) has an only solution, then, (36) does,
too. Besides, if the solution of (36) is defined for all $t\geq 0$, then, the
same happens with the solution of (31).

On the other hand, $z_{m}$ is solution of (31) if and only if

$\displaystyle
z_{m}(t)=e^{-t\mathcal{A}}z^{(0)}_{m}+\int^{t}_{0}e^{-(t-\tau)\mathcal{A}}\mathcal{P}_{m}\left[F(z_{m}(\tau))+S(\tau)\right]d\tau.$
(37)

In what follows, we set $3/4<\alpha_{0}\leq 1$, and we are going to prove that
the Faedo-Galerkin problem (31) has a only solution $z_{m}$ in
$L_{2}(0,t_{1};\mathcal{Z}^{\alpha_{0}})$, for certain $t_{1}>0$, when the
sequence of initial conditions satisfies the condition CB.

By $C\left([0,t_{1}];\mathcal{Z}_{m}^{\alpha_{0}}\right)$ we denote the space
of functions $z_{m}(t)$ continuous in $[0,t_{1}]$ with values in
$\mathcal{Z}_{m}^{\alpha_{0}}$, and

$\displaystyle V_{\delta}=\left\\{z_{m}\in
C\left([0,t_{1}];\mathcal{Z}^{\alpha_{0}}\right):z_{m}(t)\in\mathcal{F},\;\;\hbox{for
$t\in[0,t_{1}]$}\right\\}.$

By $\mathcal{K}_{m}:V_{\delta}\subset
C\left([0,t_{1}];\mathcal{Z}^{\alpha_{0}}\right)\rightarrow
C\left([0,t_{1}];\mathcal{Z}^{\alpha_{0}}\right)$, we denote the operator
given by the relation

$\displaystyle
K_{m}(z_{m})(t)=e^{-t\mathcal{A}}z^{(0)}_{m}+\int^{t}_{0}e^{-(t-\tau)\mathcal{A}}\mathcal{P}_{m}\left[F(z_{m}(\tau))+S(\tau)\right]d\tau,\;\;\;z_{m}\in
V_{\delta}.$

Now, $\delta$ and $t_{1}$ are chosen, such that
$K_{m}\left(V_{\delta}\right)\subseteq V_{\delta}$. In fact, if $z_{m}\in
V_{\delta}$, then, for any $t\in[0,t_{1}]$,
$\left\|z_{m}(t)-\widetilde{z}_{t}\right\|_{\alpha_{0}}\leq\frac{\delta}{2}$,
where $\widetilde{z}_{t}\in\left\\{z_{0},z_{1},\ldots,z_{m_{0}}\right\\}$.

We have

$\displaystyle\left\|\left(K_{m}z_{m}\right)(t)-\widetilde{z}_{t}\right\|_{\alpha_{0}}\leq\left\|\left(K_{m}z_{m}\right)(t)-z^{(0)}_{m}\right\|_{\alpha_{0}}+\left\|z^{(0)}_{m}-\widetilde{z}_{t}\right\|_{\alpha_{0}}$
$\displaystyle\leq\left\|\left(e^{-t\mathcal{A}}-I_{d}\right)z^{(0)}_{m}\right\|_{\alpha_{0}}+\left\|\int^{t}_{0}e^{-(t-\tau)\mathcal{A}}\mathcal{P}_{m}\left[F(z_{m}(\tau))+S(\tau)\right]d\tau\right\|_{\alpha_{0}}+\frac{\delta}{2}.$
(38)

Let us estimate the first term of the right-side of (4). In this sense, we
write

$\displaystyle(I_{d}+\mathcal{A}^{\alpha_{0}})\left(e^{-t\mathcal{A}}-I_{d}\right)z^{(0)}_{m}=-\int^{t}_{0}(I_{d}+\mathcal{A}^{\alpha_{0}})\left(e^{-s\mathcal{A}}-I_{d}\right)z^{(0)}_{m}ds=-\int^{t}_{0}\left(e^{-s\mathcal{A}}-I_{d}\right)(I_{d}+\mathcal{A}^{\alpha_{0}})z^{(0)}_{m}ds,$

thus,

$\displaystyle\left\|\left(e^{-t\mathcal{A}}-I_{d}\right)z^{(0)}_{m}\right\|_{\alpha_{0}}\leq\int^{t_{1}}_{0}\left\|\left(e^{-s\mathcal{A}}-I_{d}\right)\right\|_{\mathcal{L}(\mathcal{Z})}\left\|z^{(0)}_{m}\right\|_{\alpha_{0}}ds\leq
2\left\|z^{(0)}_{m}\right\|_{\alpha_{0}}t_{1}.$

We choose $t_{1}>0$, such that

$\displaystyle\sup_{t\in[0,t_{1}]}\left\|\left(e^{-t\mathcal{A}}-I_{d}\right)z^{(0)}_{m}\right\|_{\alpha_{0}}\leq\frac{\delta}{4}.$

To estimate the second term of the right-side of (4), let us see

$\displaystyle\left\|\int^{t}_{0}e^{-(t-\tau)\mathcal{A}}\mathcal{P}_{m}\left[F(z_{m}(\tau))+S(\tau)\right]d\tau\right\|_{\alpha_{0}}\leq\left\|\int^{t_{1}}_{0}\left(I_{d}+A^{\alpha_{0}}\right)e^{-(t-\tau)\mathcal{A}}\mathcal{P}_{m}\left[F(z_{m}(\tau))+S(\tau)\right]d\tau\right\|_{\mathcal{Z}}$
$\displaystyle\leq\int^{t_{1}}_{0}\left\|\left(I_{d}+A^{\alpha_{0}}\right)e^{-(t-\tau)\mathcal{A}}\mathcal{P}_{m}\left[F(z_{0})+S(\tau)\right]\right\|_{\mathcal{Z}}d\tau+\int^{t_{1}}_{0}\left\|\left(I_{d}+A^{\alpha_{0}}\right)e^{-(t-\tau)\mathcal{A}}\mathcal{P}_{m}\left(F(z_{m}(\tau))-F(z_{0})\right)\right\|_{\mathcal{Z}}d\tau$
$\displaystyle\leq
M(B+L\delta)\int^{t_{1}}_{0}\tau^{-\alpha_{0}}e^{a_{1}\tau}d\tau\leq(B+L\delta)\frac{e^{bt_{1}}t^{1-\alpha_{0}}_{1}}{1-\alpha_{0}},$

where we have taken into account that

$\displaystyle\left\|F(z_{0})+S(\tau)\right\|_{\mathcal{Z}}\leq B,$

for certain constant $B>0$. In summary,

$\displaystyle\sup_{t\in[0,t_{1}]}\left\|\int^{t}_{0}e^{-(t-\tau)\mathcal{A}}\mathcal{P}_{m}F(z_{m}(\tau))d\tau\right\|_{\alpha_{0}}\leq
M(B+L\delta)\frac{e^{a_{1}t_{1}}t^{1-\alpha_{0}}_{1}}{1-\alpha_{0}}\leq\frac{\delta}{4},$

for certain $\delta,t_{1}$ appropriately chosen. We have proved that, there
are positive constants $\delta,t_{1}$, such that
$\mathcal{K}_{m}(V_{\delta})\subseteq V_{\delta}$.

In a similar way, it can also be proved that $\mathcal{K}_{m}$ is a
contracting map if we suitably choose $t_{1},\delta$. The previously proved
results can be statemented in the following lemma.

###### Lemma 10

The Cauchy problem (31) has an only solution $z_{m}\in
L_{2}(0,t_{1};\mathcal{Z}^{\alpha_{0}})$ for certain $t_{1}>0$ and
$3/4<\alpha_{0}<1$. Besides, the sequence
$\left\\{z_{m}\right\\}^{\infty}_{m=0}\subset\mathcal{B}_{L_{2}(0,t_{1};\mathcal{Z}^{\alpha_{0}})}(r_{0})$
for certain $r_{0}>0$, where

$\displaystyle\mathcal{B}_{L_{2}(0,t_{1};\mathcal{Z}^{\alpha_{0}})}(r_{0})=\left\\{z\in
L_{2}(0,t_{1};\mathcal{Z}^{\alpha_{0}}):\left\|z\right\|_{L_{2}(0,t_{1};\mathcal{Z}^{\alpha_{0}})}\leq
r_{0}\right\\}.$

Proof. With the previous explanation, the existence of a solution of problem
(31) $z_{m}\in V_{\delta}\subset C([0,t_{1}];\mathcal{Z}_{m}^{\alpha_{0}})$,
has been demonstrated for each $m\in\mathbb{N}\cup\left\\{0\right\\}$. Note
that $C([0,t_{1}];\mathcal{Z}_{m}^{\alpha_{0}})\subset
L_{2}(0,t_{1};\mathcal{Z}^{\alpha_{0}})$, continuously. Moreover, $z_{m}\in
V_{\delta}$, implies that

$\displaystyle\left\|z_{m}(t)\right\|_{\alpha_{0}}\leq\max\\{\left\|z_{0}\right\|_{\alpha_{0}},\left\|z_{1}\right\|_{\alpha_{0}},\ldots,\left\|z_{m_{0}}\right\|_{\alpha_{0}}\\}+\frac{\delta}{2},$

thus

$\displaystyle\left(\int^{t_{1}}_{0}\left\|z_{m}(\tau)\right\|^{2}_{\alpha_{0}}d\tau\right)^{\frac{1}{2}}\leq\sqrt{t_{1}}\left(\max\\{\left\|z_{0}\right\|_{\alpha_{0}},\left\|z_{1}\right\|_{\alpha_{0}},\ldots,\left\|z_{m_{0}}\right\|_{\alpha_{0}}\\}+\frac{\delta}{2}\right).$

That is,
$r_{0}=\sqrt{t_{1}}\left(\max\\{\left\|z_{0}\right\|_{\alpha_{0}},\left\|z_{1}\right\|_{\alpha_{0}},\ldots,\left\|z_{m_{0}}\right\|_{\alpha_{0}}\\}+\frac{\delta}{2}\right)$.

Now, we shall demonstrate the following Proposition.

###### Proposition 11

Let $\left\\{z_{m}\right\\}^{\infty}_{m=0}\subset V_{\delta}$ be the sequence
of solution of the Faedo-Galerkin systems obtained in Lemma 10, is a Cauchy
sequence in $C\left(0,t_{1};\mathcal{Z}^{\alpha_{0}}\right)$.

Proof. We have

$\displaystyle\left\|z_{m}(t)-z_{n}(t)\right\|_{\alpha_{0}}\leq\left\|e^{-t\mathcal{A}}\left(z^{(0)}_{m}-z^{(0)}_{n}\right)\right\|_{\alpha_{0}}+\int^{t}_{0}\left\|e^{-(t-\tau)\mathcal{A}}\left(\mathcal{P}_{m}F(z_{m}(\tau))-\mathcal{P}_{n}F(z_{n}(\tau))\right)\right\|_{\alpha_{0}}d\tau$
$\displaystyle+\int^{t}_{0}\left\|e^{-(t-\tau)\mathcal{A}}\left(\mathcal{P}_{m}-\mathcal{P}_{n}\right)S(\tau)\right\|_{\alpha_{0}}d\tau$
$\displaystyle\leq\left\|z^{(0)}_{m}-z^{(0)}_{n}\right\|_{\alpha_{0}}+\left\|\mathcal{P}_{m}-\mathcal{P}_{n}\right\|_{\mathcal{L}(\mathcal{Z}^{\alpha_{0}})}\widehat{S}\frac{t^{1-\alpha_{0}}_{1}}{1-\alpha_{0}}+\int^{t}_{0}\left\|e^{-(t-\tau)\mathcal{A}}\left(\mathcal{P}_{m}F(z_{m}(\tau))-\mathcal{P}_{n}F(z_{n}(\tau))\right)\right\|_{\alpha_{0}}d\tau.$

Now,

$\displaystyle\int^{t}_{0}\left\|e^{-(t-\tau)\mathcal{A}}\left(\mathcal{P}_{m}F(z_{m}(\tau))-\mathcal{P}_{n}F(z_{n}(\tau))\right)\right\|_{\alpha_{0}}d\tau\leq\int^{t}_{0}e^{-(t-\tau)a_{1}}(t-\tau)^{-\alpha_{0}}\left\|\mathcal{P}_{m}F(z_{m}(\tau))-\mathcal{P}_{n}F(z_{n}(\tau))\right\|_{\mathcal{Z}}d\tau$
$\displaystyle\leq\int^{t}_{0}(t-\tau)^{-\alpha_{0}}\left\|\mathcal{P}_{m}F(z_{m}(\tau))-\mathcal{P}_{n}F(z_{n}(\tau))\right\|_{\mathcal{Z}}d\tau\leq\int^{t}_{0}(t-\tau)^{-\alpha_{0}}\left\|\mathcal{P}_{m}\left(F(z_{m}(\tau))-F(z_{n}(\tau))\right)\right\|_{\mathcal{Z}}d\tau$
$\displaystyle+\int^{t}_{0}(t-\tau)^{-\alpha_{0}}\left\|\left(\mathcal{P}_{m}-\mathcal{P}_{n}\right)F(z_{n}(\tau))\right\|_{\mathcal{Z}}d\tau\leq\left(L\max_{t\in[0,t_{1}]}\left\|z_{m}(t)-z_{n}(t)\right\|_{\alpha_{0}}+\frac{\delta}{2}\left\|\mathcal{P}_{m}-\mathcal{P}_{n}\right\|_{\mathcal{L}(\mathcal{Z})}+\cdots\right.$
$\displaystyle\left.\left\|\mathcal{P}_{m}-\mathcal{P}_{n}\right\|_{\mathcal{L}(\mathcal{Z})}\max\left\\{\left\|F(z_{0})\right\|_{\mathcal{Z}},\ldots,\left\|F(z_{m_{0}})\right\|_{\mathcal{Z}}\right\\}\right)\frac{t^{1-\alpha_{0}}_{1}}{1-\alpha_{0}}.$

Taking into account the previous estimations, we obtain the following

$\displaystyle\left\|z_{m}-z_{n}\right\|_{C([0,t_{1};\mathcal{Z}^{\alpha_{0}})}\leq\frac{1}{1-L\frac{t^{1-\alpha_{0}}_{1}}{1-\alpha_{0}}}\left(\left\|z^{(0)}_{m}-z^{(0)}_{n}\right\|_{\alpha_{0}}+\left\|\mathcal{P}_{m}-\mathcal{P}_{n}\right\|_{\mathcal{L}(\mathcal{Z})}\widehat{S}\frac{t^{1-\alpha_{0}}_{1}}{1-\alpha_{0}}\right.$
$\displaystyle\left.+\left\|\mathcal{P}_{m}-\mathcal{P}_{n}\right\|_{\mathcal{L}(\mathcal{Z})}\max\left\\{\left\|F(z_{0})\right\|_{\mathcal{Z}},\ldots,\left\|F(z_{m_{0}})\right\|_{\mathcal{Z}}\right\\}\frac{t^{1-\alpha_{0}}_{1}}{1-\alpha_{0}}\right),$
(39)

where $t_{1}$ is taken such that
$\frac{t^{1-\alpha_{0}}_{1}}{1-\alpha_{0}}<L$.

Now, we can give the following Theorem.

###### Theorem 12

If the sequence of the initial conditions
$\left\\{z^{(0)}_{m}\right\\}^{\infty}_{m=0}$ converges to $z_{0}$ in
$\mathcal{Z}^{\alpha_{0}}$, with $3/4<\alpha_{0}<1$, then the sequence of
solutions of the problem (31), $z_{m}\in V_{\delta}$ converges in
$C(0,t_{1};\mathcal{Z}^{\alpha_{0}})$ to function $z\in
C(0,t_{1};\mathcal{Z}^{\alpha_{0}})$, which is the only strong solution of the
coupled-torso bidomain model, such that $z(0)=z_{0}$.

Proof. By Proposition 11, we can affirm that sequence of strong Faedo-Galerkin
approximations is a sequence of Cauchy in
$C(0,t_{1},\mathcal{Z}^{\alpha_{0}})$. Furthermore, we can choose $t_{1}$ and
$\delta>0$, such that $\left\\{z_{m}\right\\}^{\infty}_{m=0}\subset
V_{\delta}$, which is a closed subspace of
$C(0,t_{1},\mathcal{Z}^{\alpha_{0}})$, thus there is $z\in V_{\delta}$, such
that

$\displaystyle z_{m}\rightarrow z,\;\;\;\hbox{strongly in
$C(0,t_{1},\mathcal{Z}^{\alpha_{0}})$.}$

To demonstrate that $z$ is a strong solution, we must only prove that it
satisfies (26) and (27). In first place, we have

$\displaystyle
z_{m}(t)=e^{-t\mathcal{A}}z^{(0)}_{m}+\int^{t}_{0}e^{-(t-\tau)\mathcal{A}}\mathcal{P}_{m}\left[F(z_{m}(\tau))+S(\tau)\right]d\tau,\;\;\;\hbox{for
each $m=0,1,\cdots,$ and $t\in[0,t_{1}].$}$

It is very easy to prove that by making the limit on each sum of the equality
it is obtained

$\displaystyle
z(t)=e^{-t\mathcal{A}}z_{0}+\int^{t}_{0}e^{-(t-\tau)\mathcal{A}}\mathcal{P}_{m}\left[F(z(\tau))+S(\tau)\right]d\tau,\;\;\;t\in[0,t_{1}].$

On the other hand, since $z\in V_{\delta}$, it is possible to prove that

$\displaystyle\int^{t_{1}}_{0}\left\|F(z(\tau))+S(\tau)\right\|_{\mathcal{Z}}d\tau<\infty.$

Theorem 12 tells us that the only strong solution of a Cauchy problem
associated to the torso-coupled bidomain model with initial condition
$z_{0}\in\mathcal{Z}^{\alpha_{0}}$, it is obtained as the limit of a sequence
of strong Faedo-Galerkin approximations; which are obtained as solutions of
the Cauchy problem (31), where the initial condition assumed is
$z^{(0)}_{m}=\mathcal{P}_{m}z_{0}$. Note that (31) is essentially a system of
ordinary differential equations, which has important implications from both a
theoretical and computational point of view. In the first place, if each
$z_{m}$ is defined for all $t\geq 0$ so is the strong solution and if each
$z_{m}$ is a $T$-periodic function, then its limit also is $T$-periodic.
Computationally, this result provides a method of numerical resolution of the
torso-coupled bidomain model which is theoretically supporting. Upon achieving
convergence, we can take $z_{m}$ as an approximation of the solution sought by
making $m$ large enough. The inequality (4) provides an estimate of the
velocity of convergence of the sequence of strong Faedo-Galerkin
approximations.

##  5 Existence of strong periodic solution

In this section, we demonstrate the existence a strong $T$-periodic solution,
in the sense of Definition 7, assuming that $S$ is $T$-periodic, which is
continuous from $[0,T]$ in $\mathcal{Z}$.

From now on, we will use the following notation

$\displaystyle C_{T}=\left\\{z\in C([0,T];\mathcal{Z}):z(0)=z(T)\right\\},$
$\displaystyle C^{\alpha_{0}}_{T}=\left\\{z\in
C([0,T];\mathcal{Z}^{\alpha_{0}}):z(0)=z(T)\right\\},$

where, in each case, we consider the following norms

$\displaystyle\left\|f\right\|_{C_{T}}=\sup_{t\in[0,T]}\left\|f(t)\right\|_{\mathcal{Z}},$
$\displaystyle\left\|f\right\|_{C^{\alpha_{0}}_{T}}=\sup_{t\in[0,T]}\left\|f(t)\right\|_{\mathcal{Z}^{\alpha_{0}}},$

for $f\in C_{T}$ or $f\in C^{\alpha_{0}}_{T}$, respectively.

As the previous section, this $T$-periodic solution is obtained as a limit of
a sequence of solutions of system (31) which are $T$-periodic.

Let

$\displaystyle\frac{dz_{m}}{dt}=-\mathcal{A}z_{m}+\mathcal{P}_{m}S$ (40)

be the linear equation obtained from (31), and let $z_{m}\in
C([0,\infty);\mathcal{Z}^{\alpha_{0}}_{m})$ be a solution (40). Then,

$\displaystyle
z_{m}(t)=e^{-t\mathcal{A}}z^{(0)}_{m}+\int^{t}_{0}e^{-(t-\tau)\mathcal{A}}\mathcal{P}_{m}S(\tau)d\tau,$

where $z_{m}(0)=z^{(0)}_{m}$. Due to the uniqueness of the solution of the
Cauchy problem associated with (40) and the periodicity of the function $S$,
it is possible to prove that $z_{m}$ is $T$-periodic if and only if
$z_{m}(0)=z_{m}(T)$. Thus

$\displaystyle
z_{m}(T)=e^{-T\mathcal{A}}z^{(0)}_{m}+\int^{T}_{0}e^{-(T-\tau)\mathcal{A}}\mathcal{P}_{m}S(\tau)d\tau=z^{(0)}_{m},$

that is,

$\displaystyle\left(I_{d}-e^{-T\mathcal{A}}\right)z_{m}^{(0)}=\int^{T}_{0}e^{-(T-\tau)\mathcal{A}}\mathcal{P}_{m}S(\tau)d\tau.$

Let us remember that $0\notin\sigma(\mathcal{A})\Rightarrow
1\notin\sigma\left(e^{-T\mathcal{A}}\right)$, thus the operator
$I_{d}-e^{-T\mathcal{A}}$ is invertible and

$\displaystyle\left\|\left(I_{d}-e^{-T\mathcal{A}}\right)^{-1}\right\|_{\mathcal{L}(\mathcal{Z})}\leq\frac{1}{1-e^{-Ta_{1}}}.$

Thus, we obtain

$\displaystyle
z_{m}^{(0)}=\left(I_{d}-e^{-T\mathcal{A}}\right)^{-1}\int^{T}_{0}e^{-(T-\tau)\mathcal{A}}\mathcal{P}_{m}S(\tau)d\tau.$

Finally, we can affirm that if $z_{m}$ is $T$-periodic, then

$\displaystyle
z_{m}(t)=\int^{T}_{0}\mathcal{R}(t,\tau)\mathcal{P}_{m}S(\tau)d\tau,$

where

$\displaystyle\mathcal{R}(t,\tau)=\left\\{\begin{array}[]{cc}\left(I_{d}-e^{-T\mathcal{A}}\right)^{-1}e^{-(t-\tau)\mathcal{A}},&\hbox{if
$0\leq\tau\leq t\leq T,$}\\\
\left(I_{d}-e^{-T\mathcal{A}}\right)^{-1}e^{-(t+T-\tau)\mathcal{A}},&\hbox{if
$0\leq t\leq\tau\leq T.$}\end{array}\right.$

For convenience, for $f\in C_{T}$ we denote by

$\displaystyle\mathcal{L}^{(p)}(f)(t)=\int^{T}_{0}\mathcal{R}(t,\tau)f(\tau)d\tau.$

Note that, in this case, $z_{m}$ is a $T$-periodic solution of (40) if and
only if

$\displaystyle
z_{m}(t)=\mathcal{L}^{(p)}(\mathcal{P}_{m}S)(t),\;\;\;t\in[0,T].$

Before continuing with the results that interest us, let us give the following
Lemma.

###### Lemma 13

Let $f$ be a function in $C_{T}$, then $\mathcal{L}^{(p)}_{m}(f)\in
C^{\alpha_{0}}_{T}$, and

$\displaystyle\left\|\mathcal{L}^{(p)}(f)\right\|_{C^{\alpha_{0}}_{T}}\leq
2\left(\frac{1}{a_{1}}+\frac{T^{1-\alpha_{0}}}{(1-\alpha_{0})\left(1-e^{-a_{1}T}\right)}\right)\left\|f\right\|_{C_{T}}.$

Proof. If $f\in C^{\alpha_{0}}_{T}$

$\displaystyle\left\|\mathcal{L}^{(p)}(f)(t)\right\|_{\alpha_{0}}\leq\int^{t}_{0}\left\|\left(I_{d}-e^{-T\mathcal{A}}\right)^{-1}e^{-(t-\tau)\mathcal{A}}f(\tau)\right\|_{\alpha_{0}}d\tau+\int^{T}_{t}\left\|\left(I_{d}-e^{-T\mathcal{A}}\right)^{-1}e^{-(t+T-\tau)\mathcal{A}}f(\tau)\right\|_{\alpha_{0}}d\tau.$

Now,

$\displaystyle\int^{t}_{0}\left\|\left(I_{d}-e^{-T\mathcal{A}}\right)^{-1}e^{-(t-\tau)\mathcal{A}}f(\tau)\right\|_{\alpha_{0}}d\tau=\int^{t}_{0}\left\|\left(I_{d}-e^{-T\mathcal{A}}\right)^{-1}\left(I_{d}+A^{\alpha_{0}}\right)e^{-(t-\tau)\mathcal{A}}f(\tau)\right\|_{\mathcal{Z}}d\tau$
$\displaystyle\leq\int^{t}_{0}\left\|\left(I_{d}-e^{-T\mathcal{A}}\right)^{-1}e^{-(t-\tau)\mathcal{A}}f(\tau)\right\|_{\mathcal{Z}}d\tau+\int^{t}_{0}\left\|\left(I_{d}-e^{-T\mathcal{A}}\right)^{-1}A^{\alpha_{0}}e^{-(t-\tau)\mathcal{A}}f(\tau)\right\|_{\mathcal{Z}}d\tau$
$\displaystyle\leq\frac{\left\|f\right\|_{C^{\alpha_{0}}_{T}}}{1-e^{-a_{1}T}}\int^{t}_{0}e^{-a_{1}(t-\tau)}d\tau+\frac{\left\|f\right\|_{C^{\alpha_{0}}_{T}}}{1-e^{-a_{1}T}}\int^{t}_{0}(t-\tau)^{-\alpha_{0}}d\tau$
$\displaystyle\leq\frac{\left\|f\right\|_{C^{\alpha_{0}}_{T}}}{1-e^{-a_{1}T}}\int^{T}_{0}e^{-a_{1}(t-\tau)}d\tau+\frac{\left\|f\right\|_{C^{\alpha_{0}}_{T}}}{1-e^{-a_{1}T}}\int^{T}_{0}(t-\tau)^{-\alpha_{0}}d\tau$
$\displaystyle\leq\left(\frac{1}{a_{1}}+\frac{T^{1-\alpha_{0}}}{(1-\alpha_{0})\left(1-e^{-a_{1}T}\right)}\right)\left\|f\right\|_{C^{\alpha_{0}}_{T}}.$

Similarly, it is possible to demonstrate

$\displaystyle\int^{T}_{t}\left\|\left(I_{d}-e^{-T\mathcal{A}}\right)^{-1}e^{-(t+T-\tau)\mathcal{A}}f(\tau)\right\|_{\alpha_{0}}d\tau\leq\left(\frac{1}{a_{1}}+\frac{T^{1-\alpha_{0}}}{(1-\alpha_{0})\left(1-e^{-a_{1}T}\right)}\right)\left\|f\right\|_{C^{\alpha_{0}}_{T}}.$

Now, consider the equation in (31)

$\displaystyle\frac{dz_{m}}{dt}=-\mathcal{A}z_{m}+\mathcal{P}_{m}F(z_{m})+\mathcal{P}_{m}S.$
(41)

Following the same line of thought, it shows that $z_{m}$ is a $T$-periodic
solution of (41) if it satisfies the integral equation

$\displaystyle
z_{m}(t)=\mathcal{L}^{(p)}\mathcal{P}_{m}\left(F(z_{m})+S\right)(t),\;\;\;t\in[0,T].$

This means that we will search these periodic solutions as the fixed points of
the operator

$\displaystyle\mathcal{K}^{(p)}_{m}(z_{m})(t):=\mathcal{L}^{(p)}\mathcal{P}_{m}\left(F(z_{m})+S\right)(t),$

defined from $C^{\alpha_{0}}_{T}$ in itself.

We consider a ball
$\mathcal{B}_{\alpha_{0}}(r_{0})\subset\mathcal{Z}^{\alpha_{0}}$, such that
$F:\mathcal{B}_{\alpha_{0}}(r_{0})\rightarrow\mathcal{Z}$ be Lipschitz
continuous with Lipschitz constant $L$. Next, we shall prove it is possible
suitably to choose $T,r_{0}$, such that the operator $\mathcal{K}^{(p)}_{m}$
has an only fixed point in

$\displaystyle C^{(\alpha_{0},r_{0})}_{T}=\left\\{z_{m}\in
C([0,T];\mathcal{B}_{\alpha_{0}}(r_{0})):z_{m}(0)=z_{m}(T)\right\\},$

which is endowed with the norm in $C^{\alpha_{0}}_{T}$.

We suppose that $z_{m}\in C^{(\alpha_{0},r_{0})}_{T}$, taking into account
Lemma 13 we have

$\displaystyle\left\|\mathcal{K}^{(p)}_{m}(z_{m})\right\|_{C^{\alpha_{0}}_{T}}\leq
2\left(\frac{1}{a_{1}}+\frac{T^{1-\alpha_{0}}}{(1-\alpha_{0})\left(1-e^{-a_{1}T}\right)}\right)\left\|F(z_{m})+S\right\|_{C_{T}}.$

On the other hand, we have

$\displaystyle\left\|F(z_{m}(t))+S(t)\right\|_{\mathcal{Z}}\leq\left\|F(z_{m}(t))-F(0)\right\|_{\mathcal{Z}}+\left\|F(0)\right\|_{\mathcal{Z}}+\widehat{S}\leq
L\left\|z_{m}(t)\right\|_{\alpha_{0}}+\left\|F(0)\right\|_{\mathcal{Z}}+\widehat{S},$

where $\widehat{S}=\|S\|_{C_{T}}$, from which it is deduced that

$\displaystyle\left\|F(z_{m})+S\right\|_{C_{T}}\leq
Lr_{0}+\left\|F(0)\right\|_{\mathcal{Z}}+\widehat{S}$

From the above, we obtain

$\displaystyle\left\|\mathcal{K}^{(p)}_{m}(z_{m})\right\|_{C^{\alpha_{0}}_{T}}\leq
2\left(Lr_{0}+\|F(0)\|_{\mathcal{Z}}+\widehat{S}\right)\left(\frac{1}{a_{1}}+\frac{T^{1-\alpha_{0}}}{(1-\alpha_{0})(1-e^{-a_{1}T})}\right).$
(42)

From a similar analysis, we have

$\displaystyle\left\|\mathcal{K}^{(p)}_{m}\left(z^{(1)}_{m}\right)-\mathcal{K}^{(p)}_{m}\left(z^{(2)}_{m}\right)\right\|_{C^{\alpha_{0}}_{T}}\leq
2\left(\frac{1}{a_{1}}+\frac{T^{1-\alpha_{0}}}{(1-\alpha_{0})\left(1-e^{-a_{1}T}\right)}\right)Lr_{0}\left\|z^{(1)}_{m}-z^{(2)}_{m}\right\|_{C^{\alpha_{0}}_{T}}.$
(43)

By the inequalities (42) and (43), we can affirm that
$\mathcal{K}^{(p)}_{m}(C^{(\alpha_{0},r_{0})}_{T})\subseteq
C^{(\alpha_{0},r_{0})}_{T}$ if

$\displaystyle\left(Lr_{0}+\|F(0)\|_{\mathcal{Z}}+\widehat{S}\right)\left(\frac{1}{a_{1}}+\frac{T^{1-\alpha_{0}}}{(1-\alpha_{0})(1-e^{-a_{1}T})}\right)\leq\frac{r_{0}}{2},$
(44)

and the map $\mathcal{K}^{(p)}_{m}$ is a contraction when

$\displaystyle\left(\frac{1}{a_{1}}+\frac{T^{1-\alpha_{0}}}{(1-\alpha_{0})\left(1-e^{-a_{1}T}\right)}\right)Lr_{0}<\frac{1}{2}.$
(45)

###### Theorem 14

If conditions (44), (45) are assumed, and besides

$\displaystyle\frac{1}{a_{1}}+\frac{T^{1-\alpha_{0}}}{(1-\alpha_{0})\left(1-e^{-a_{1}T}\right)}<\frac{1}{2},$

then, there is a sequence of strong Faedo-Galerkin approximations
$\left\\{z^{(p)}_{m}\right\\}^{\infty}_{m=0}\subset
C^{(\alpha_{0},r_{0})}_{T}$, whose elements are $T$-periodic functions.
Furthermore, the sequence converge to a $T$-periodic function $z^{(p)}\in
C^{(\alpha_{0},r_{0})}_{T}$, which is a strong solution of the coupled-torso
bidomian model in the sense of Definition 7.

Proof. The inequalities (44) and (45), imply the existence of only fixed point
of $\mathcal{K}^{(p)}_{m}$ in $C^{(\alpha_{0},r_{0})}_{T}$, denoted by
$z^{(p)}_{m}$, which is a $T$-periodic solution of (41).

Now, we prove that $\left\\{z^{(p)}_{m}\right\\}^{\infty}_{m=0}$ is a Cauchy
sequence in $C^{\alpha_{0}}_{T}$. In fact, we have

$\displaystyle\left\|z_{m}-z_{n}\right\|_{C^{\alpha_{0}}_{T}}=\left\|\mathcal{K}^{(p)}_{m}(z_{m})-\mathcal{K}^{(p)}_{n}(z_{n})\right\|_{C^{\alpha_{0}}_{T}}=\left\|\mathcal{L}^{(p)}\left(\mathcal{P}_{m}F(z_{m})-\mathcal{P}_{n}F(z_{n})\right)\right\|_{C^{\alpha_{0}}_{T}}$
$\displaystyle\leq\left\|\mathcal{L}^{(p)}\mathcal{P}_{m}\left(F(z_{m})-F(z_{n})\right)\right\|_{C^{\alpha_{0}}_{T}}+\left\|\mathcal{L}^{(p)}\left(\mathcal{P}_{m}-\mathcal{P}_{n}\right)(F(z_{n})-F(0))\right\|_{C^{\alpha_{0}}_{T}}+\left\|\mathcal{L}^{(p)}\left(\mathcal{P}_{m}-\mathcal{P}_{n}\right)F(0)\right\|_{C^{\alpha_{0}}_{T}}$
$\displaystyle\leq
2\left(\frac{1}{a_{1}}+\frac{T^{1-\alpha_{0}}}{(1-\alpha_{0})\left(1-e^{-a_{1}T}\right)}\right)\left(L\left\|z_{m}-z_{n}\right\|_{C^{\alpha_{0}}_{T}}+\left\|\mathcal{P}_{m}-\mathcal{P}_{n}\right\|_{\mathcal{L}(\mathcal{Z})}(Lr_{0}+\|F(0)\|_{\mathcal{Z}})\right),$

thus

$\displaystyle\left\|z_{m}-z_{n}\right\|_{C^{\alpha_{0}}_{T}}\leq\frac{Lr_{0}+\|F(0)\|_{\mathcal{Z}}}{1-2\left(\frac{1}{a_{1}}+\frac{T^{1-\alpha_{0}}}{(1-\alpha_{0})\left(1-e^{-a_{1}T}\right)}\right)}\left\|\mathcal{P}_{m}-\mathcal{P}_{n}\right\|_{\mathcal{L}(\mathcal{Z})}.$

The right-side of the above inequality converges towards to zero, thus, it is
demonstrated that the sequence $\left\\{z^{(p)}_{m}\right\\}^{\infty}_{m=0}$
is of Cauchy in $C^{\alpha_{0}}_{T}$. Furthermore, since
$C^{(\alpha_{0},r_{0})}_{T}$ is closed in $C^{\alpha_{0}}_{T}$, being this
last a Banach space, we obtain that

$\displaystyle z^{(p)}_{m}\rightarrow z^{(p)}\in C^{(\alpha_{0},r_{0})}_{T}.$

It is easy to see that $z^{(p)}$ is a $T$-periodic strong solution of the
torso-coupled bidomain model.

## Acknowledgments

## References

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