Introduction

The bidomain model represents an active myocardium on a macroscopic scale by relating membrane
ionic current, membrane potential, and extracellular potential (^{Henriquez 1993}). Created in 1969 (^{Schmidt 1969}), (^{Clerc 1976}) and first developed formally in 1978 (^{Tung 1978}), (^{Miller 1978,
I}), the bidomain model was initially used to derive forward models, which
compute extracellular and body-surface potentials from given membrane potentials
(^{Miller 1978, I}), (Gulrajani 1983),
(^{Miller 1978, II}) and (^{Gulrajani 1998}). Later, the bidomain model was
used to link multiple membrane models together to form a bidomain reaction-diffusion
(R-D) model (^{Barr 1984}), (^{Roth 1991}), which simulates propagating
activation based on no other premises than those of the membrane model, those of the
bidomain model, and Maxwell’s equations. Other mathematical derivations of the
macroscopic bidomain type models directly from the microscopic properties of tissue
and using asymptotic and homogenization methods along with basic physical principles
are presented in (^{Neu 1993}), (^{Ambrosio 2000}) and (^{Pennacchio 2005}).

Monodomain R-D models, conceived as a simplification of the R-D bidomain models, with
advantages both for mathematical analysis and computation, were actually developed
before the first bidomain R-D models, and few papers have compared monodomain with
bidomain results. Those that did, have shown small differences (^{Vigmond 2002}), and monodomain simulations have
provided realistic results (^{Leon 1991}),
(^{Hren 1997}), (^{Huiskamp 1998}), (^{Bernus
2002}), (^{Trudel 2004}) and (^{Berenfeld 1996}). In (^{Potse 2006}) has been investigated the impact of the monodomain
assumption on simulated propagation in an isolated human heart, by comparing results
with a bidomain model. They have shown that differences between the two models were
extremely small, even if extracellular potentials were influenced considerably by
fluid-filled cavities. All properties of the membrane potentials and extracellular
potentials simulated by the bidomain model have been accurately reproduced by the
monodomain model with a small difference in propagation velocity between both
models, even in abnormal cases with the Na conductivity (^{Bernus 2002}) reduced to 1=10 of its normal value, and have
arrived at the same conclusions. The difference between the results that may be
obtained with one or another model are small enough to be ignored for most
applications, with the exception of simulations involving applied external currents
or in the presence adjacent fluid on within, although these effects seem to be
ignorable on the scale of a human heart. A formal derivation of the monodomain
equation as we present here can be found in (^{Sundnes
2006}). There are few references in the literature dealing with the proof
of the well-posedness of the bidomain model. The most important seem to be ^{Colli-Franzone and Savarés paper (Colli 2002)},
^{Veneroni’s technical report (Veneroni
2009)} and Y. ^{Bourgault, Y. Coudière and C.
Pierre’s paper (Bourgault 2009)}. In (^{Colli 2002}), global existence in time and uniqueness for the solution of
the bidomain model is proven, although their approach applies to particular cases of
ionic models, typically of the form *f*(*u,w*) = *k*(*u*) + *αw* and *g*(*u,w*) = *βu + γw*, where *k* ∈
C^{1}(ℝ) satisfies inf_{ℝ} *k'* > −∞. In practice a common
ionic model reading this form is the cubic-like FitzHugh-Nagumo model (^{Fitzhugh 1961}), which, although it is important
for qualitatively understanding of the action potential propagation, its
applicability to myocardial excitable cells is limited (^{Keener 1998}), (^{Panfilov
1997}).However, from the results of (^{Colli
2002}) is not possible to conclude the existence of solution for other
simple two variable ionic models widely used in the literature for modelling
myocardial cells, such as the ^{Aliev-Panfilov (Aliev
1996)} and MacCulloch (^{Rogers 1994})
models. In (^{Veneroni 2009}), Colli-Franzone
and Savarés results have been extended to more general and more realistic ionic
models, namely those taking the form of the ^{Luo and
Rudy I model (Luo 1991)}, this result still does not include the
Aliev-Panfilov and MacCulloch models. In reference (^{Bourgault 2009}), global in time weak solutions are obtained for ionic
models reading as a single ODE with polynomial nonlinearities. These ionic models
include the FitzHugh-Nagumo model (^{Fitzhugh
1961}) and simple models more adapted to myocardial cells, such as the
Aliev-Panfilov (^{Aliev 1996}) and
Rogers-MacCulloch (^{Rogers 1994}) models.

In this paper, we give a definition of weak solution of the variational Cauchy problem and, from this one, we give a definition of strong solution. We aim to obtain the existence of a global weak solution for a monodomain R-D model when applied to a ventricle isolated from the torso in absence of blood on within, which is activated through the endocardium by a Purkinje current and for simpler ionic models reading as a single ODE with polynomial nonlinearities. Also, it is proved that this weak solution is strong in the sense of the given definition. We will consider a bounded subset Ω ∈ ℝ^{3} simulating an isolated ventricle surrounded by an insulating medium. The boundary 𝜕Ω of the spatial region is formed by two disjoint components; the component Γ_{0} imulating the epicardium and the component Γ_{1} simulating the endocardium. The way Ω is electrically stimulated is by means Purkinje fibers, which directly stimulate only the inner wall Γ_{1} then the excitable nature of the tissue allows this stimulus to propagate by Ω. We will assume that the ventricle is isolated from the heart and torso, that is to say that Γ_{0} is in contact with an electrically insulating medium. We will use the monodomain model and the Rogers-McCulloch model for ion currents through the cell membrane, in this way and for the above considerations this model can be written as one parabolic PDE with boundary conditions, coupled to a ODE, and some initial data:

The unknowns are the scalar functions *u*(*t,x*) and *w*(*t,x*) which are the membrane potential
and an auxiliary variable without physiological interpretation called the recovery
variable, respectvely. We denote by *η* the unit normal to ∂Ω out of Ω. The
anisotropic properties of the tissue are included in the model by the conductivity
tensor σ(*x*). The functions *f*(*u,w*) and *g*(*u,w*) crrespond to the flow of ions
through the cell membrane. The function *s*:(0,+∞) → ℝ represents the electrical
activation of the endocardium by means of Purkinje fibers. The function φ: Ω → ℝ
represents the activation spatial density. Because we consider that Ω is surrounded
by an insulating medium, there is no current flowing out of Ω, this is expressed in
the boundary condition (3).

The specific assumptions we will make about (1) - (5) are as follows:

(h1) Ω has Lipschitz boundary ∂Ω.

(h2) σ(*x*) is a symmetric matrix, function of the spatial variable *x* ∈ Ω, with coefficients in *L ^{∞}*(Ω) and such that there are positive constants

*m*and

*M*such that

Is met for almost all *x* ∈ Ω.

(h3) *s* ∈ *L ^{∞}*(0,+∞).

(h4) *φ* ∈ *L*^{2}(Γ_{1}).

(h5) *f*(*u,w*) y *g*(*u,w*) y stands for Rogers-McCulloch ionic model,

(h6) *u _{0}, w_{0} ∈ L^{2}*(Ω).

It is convenient to establish some notations that we will follow throughout this work. For convenience, we will denote *V = H ^{1}*(Ω) and

*H = L*(Ω) since we will make constant use of these spaces. It is important to note that in the context of this work the following inclusions are fulfilled for 2 ≤

^{2}*p*≤ 6

Note that only *H* is identified with its dual space. In particular, we will consider *p* = 4 from here on. As usual, *p*′ denotes a positive number such that

Let *X* be a Banach space of integrable functions over Ω, we define the subspace

Which is a Banach space with the norm induced by *X*. For any *u* ∈ *X*, we denote

Thus [*u*] ∈ *X*/ℝ.

This paper is organized as follows. The spaces *L ^{q}*(

*0,T;X*) are the functional setting we will work in, so in section 2.1 the definition of this spaces along with some important facts about them are presented. In section 2.2 some preliminary results are established, mainly related to the diffusion term ∇(

*σ∇u*) and with the model for the ionic current

*f*and

*g*. In section 2.3 we state the definition of weak and strong solution, and enunciate some results that allow us to find a relation between them. The existence will be shown in sections 3.1 and 4.1.

Method

*L ^{q}*(

*0,T;X*) spaces

Let *X* be a Banach space, we denote by *L ^{q}*(

*0,T;X*) the space of the functions

*t → f*(

*t*) of [

*0,T*] →

*X*that are measurable with values in

*X*such that

with this norm *L ^{q}*(

*0,T;X*) is complete. Observe that

where *Q _{T}*[

*0,T*] × Ω.

It is necessary to give a definition of the derivative of an element of *L ^{q}*(

*0,T;X*), for this we will consider the space of distributions on [0,

*T*] with values in

*X*, see (

^{Lions 1969, 7}).

**Definition 1.** *We define D'(0,T;X), the space of distributions on [0,T] with values in X, as*

*where D(0,T) is the set of infinitely differentiable functions of compact support in (0,T)*.

If *f ∈ D'*(*0,T;X*) we can define its derivative in the sense of distributions as

If *f ∈ D'*(*0,T;X*) it corresponds a distribution *D’*(*0,T;X*) defined as follows

In this way, we can define the derivative in the sense of distributions of a function *f* ∈ *L ^{q}*(

*0,T;X*) as

**Theorem 1.** *Let Q _{T} a bounded open in ℝ × ℝ^{N} f_{n} and f functions in L^{q}(Q_{T}), 1 < q < ∞, such that*

*for a certain constant C > 0, then*,

*Proof.* (^{Lions 1969, lema 1.3, p. 12}).

For the chain of inclusions (9) and the fact that the immersion *V →H* is compact we can enunciate the following result, which is a particular case of a classic compactness result, see (^{Lions 1969, th. 5.1, p.58}).

**Theorem 2.** *We define for T finite and 0 < q _{i} < ∞,i = 0,1,*

*endowed with the norm *

**Proposition 1.**
*Let with u∈L*

*where (·,·) represents the scalar product in H, and 〈ũ,ν〉 _{
𝑉′×𝑉
} represents the evaluation of functional ũ in u. That is, u is the distributional derivative of u, and is the only function *

*From now on, we write 〈∙,∙〉 instead of 〈∙,∙〉 _{
𝑉′×𝑉
}*.

**Theorem 3.** *If f ∈ L ^{q} and ∂_{t}f ∈ L^{q}(0,T;X) (1 ≤ q ≤ ∞),then, f is continuous*

*almost everywhere from (0,T) to X*

*Proof. ( ^{Lions 1969, lema 1.2, p. 7})*.

Preliminaries

**Definition 2.** *For all u, v ∈ V × V we define the bilinear form*

**Proposition 2.** *The bilinear form 𝑎 (⋅,⋅) is symmetric, continuous and coercitive in V,*

*with α,M > 0. There is a growing sequence 0 = λ _{0} < ⋯ < λ_{i} < ⋯ ∈ ℝ and there is an orthonormal basis of H formed by eigenvectors {ψ_{i}}_{i∈ℕ} such that, ψ_{i} ∈ V y*

*Proof.* The symmetry of *a*(⋅,⋅) is immediate consequence of the symmetry of σ. By (h2). we have that σ is uniformly elliptic and symmetric, then satisfies the following inequality

then, integrating over Ω and adding

which shows (17), the continuity of *a*(⋅,⋅)is also a consequence of (6). The existence of egenvalues and eigenvectors is obtained by a classical result, see (^{Raviart 1992, thm 6.2-1 y rem. 6.2-2, p. 137-138}), taking into account that λ_{0} = 0 because the bilinear form *a*(⋅,⋅) is canceled only for constant functions.

It is important to note that the properties of the bilinear form *a*(⋅,⋅)allow to introduce an operator in a natural way.

**Definition 3.** *By the previous lemma, the hypotheses of the Lax-Milgram theorem for the bilinear form a(⋅,⋅) are fulfilled and therefore there is an operator A:V → V' injective and continuous with continuous inverse such that*

If *v* is a function defined on Ω we denote its trace to the boundary ∂Ω also as *v*, its meaning will always be clear from the context.

**Proposition 3.** *If φ ∈ L ^{2}(Γ_{1}) then for v ∈ V the function*

*defines a linear and continuous functional. This is, we have*

We will denote

with

**Proposition 4.** *For p = 4, there are constants c _{i} ≥ 0, i = 1,…,6, such that for all u ∈ ℝ the following inequalities hold.*

*Proof.* Due to Young’s inequality the following estimates are met

*Then,*

**Proposition 5.** *For =4, there are λ > 0, μ, c ≥ 0 such that for all(u,w) ∈ ℝ we have*

*Proof.* By direct calculation from (20) we have

On the other hand, from Young’s inequality we have

Then,

To continue, it is necessary to extract a common term from the coefficients corresponding to |*u*|^{2} and |*w*|^{2}, for this we can write

To conclude it is necessary to verify that θ, β and ρ can be chosen so that

which is fulfilled for

obviously, we can find a *ρ* small enough to meet such conditions. We have *μ = γ, λ* > 0 arbitrary,

**Proposition 6.** *Let u ∈ L ^{p} (Ω) and w ∈ H, Then f(u,w) ∈ L^{p′} (Ω) and g(u,w) ∈ H. In addition, the following inequalities are met*

*where A _{i} ≥ 0, i = 0,...,3 y B_{i} ≥ 0, i = 0,...,3 are constants that depend only on c_{i}, i = 1,...,6 and p*.

*Proof.* Let (*u,w*) ∈ ℝ^{2}, by proposition 4 we have

with *B _{1} =c_{5},B2 = c_{6} y B_{3}* = |g

_{2}|. On the other hand, by Young’s inequality, with

then, because

then, once more by Young’s inequality _{1}, A_{2} y A_{3} such that

If (*u,v*) ∈ *L ^{p}*(Ω) ×

*H*, by direct calculation and taking into account that (

*p*− 1)

*p′ = p,βp′*= 2 we have

In a similar way

Definition of weak and strong solution

This section establishes the definition of the solution that will be obtained in section 3.1 for the model (1)-(5) of a ventricle. Also, we define strong solution and give a result of selectivity of the weak solution. It will be necessary to consider the weak formulation both in time and space. In order to give a bit of context to this definition we will start by considering the variational formulation in the spatial variable of the original model,

in this way it will be natural to introduce a succession of approximate solutions through a discretization of the space in which we will look for the solution. This procedure is known as the Faedo-Galerkin method.

We will denote as V_{m} the linear space generated by {ψ_{0},ψ_{1},…,ψ_{m}}, where the functions ψ_{i},*i = 0,…,m*, are eigenfunction of the bilinear form a(⋅,⋅) as established in the proposition 2. Note that *V _{m} ⊂ V*. For each

*m*, we consider the variational problem restricted to the space

*V*

_{m}, that is, instead of

*v*and

*z*we take

*ψ*, and approximate

_{i},i = 0,…,m*u*(

*t*) and

*w*(

*t*) by u

_{m}(

*t*) and

*u*(

_{m}*t*) respectively, with

By means of these substitutions we obtain from (22)-(24) the following system

for *i = 0,…,m.*

**Definition 4.** *(Weak Solution). Let τ > 0 and the functions u ∶ t ∈ [0,τ) ↦ u(t) ∈ H, w ∶ t ∈ [0,τ) ↦ w(t) ∈ H. We say that (u,w) is a weak solution of the varitional formulationof the problem (1)-(4) if for any T ∈ (0,τ),*

*u ∶ [0,T] ↦ and w ∶ [0,T] ↦ H are continuous*.*For almost all t ∈ (0,τ), we have u(t) ∈ V, also u ∈ L*.^{p}(Q_{T}) ∩ L^{2}(0,T;V) and w ∈ L^{2}(Q_{T}), with Q_{T}= (0,T) × Ω

*In addition, the functions u and w satisfy*

*where equality is considered in D′(0,T).*

*If, furthermore, given u _{0} in H, u, w_{0} in H, u,w in, are weak solutions that satisfy*

*then we call u,w a weak solution of variational Cauchy problem associated to (1)-(5)*.

**Remark 1.** *The derivatives that appear in the first terms of the equations (29) and (30) refer to derivatives in the sense of distributions, that is, for ϕ ∈ D(0,T) we have*

Now, we can give a definition of strong solution for the variational formulation. Suppose that, u,w are weak solutions, in the sense of definition 4, and furthermore, *u ∈ W ^{1,2,p′}*(

*0,T;V′,V*) and

*w ∈ W*(

^{1,2,2}*0,T;H,H*), then the equation (29) means that

thus, by proposition 1, it has

which implies that

From the above it follows that,

which holds in *V′* In a similar for it is possible to prove that

is fulfilled in *H*.

**Definition 5.** *(Strong Solution). Let be u ∈ W^{1,2,p′} (0,T;V,V′) and w ∈ W^{1,2,2}(0,T,H,H) we call u,w strong solutions of the variational formulation problem (1)-(4), if they satisfy the equation (31)-(32) in V′ and H, respectively.*

*If, besides,*

*for u _{0}, w_{0} given, we say that u,w are strong solutions of variational Cauchy problem associated to (1)-(5).*

Results

Existence of global solution

The main result of this section is the following theorem.

**Theorem 4.** (Existence of weak solution). *Under the hypotheses (h1)-(h5) plus*

*(h6’) the sequences um0, wm0 are bounded in H,*

*the system (1)- (4) has a weak solution (u,w) in the sense of the definition 4 with τ = +∞.*

The demonstration is developed in the following two subsections,

Existence of approximate solutions

The next lemma states that the approximate solutions u_{m}, w_{m} are defined for all *t* > 0, other important estimates are also established to demonstrate later that the succession of approximate solutions converges to a solution. The following norms will be used.

**Lemma 1.** *The Cauchy problem (26) - (28) has solution for all t > 0. In addition, there are*

*constants C _{i} > 0,i = 1,…,4, such that for all T > 0. The following estimates are met a priori*

where * are the derivatives of the functions u _{m} ∶ [0,T] ↦ V and w_{m} ∶ [0,T] ↦ H.*

*Proof.* Note that the integrals in (26) and (27) are well defined, in deed, as *u _{m}* (

*t*) ∈

*V ⸦ L*(Ω) and w

^{p}_{m}(t) ∈ H we have from proposition 6 that

*f(u*and,

_{m}(t),w_{m}(t)) ∈ L^{p′}(Ω) ⊂ V′*g(u*, then because

_{m},w_{m}) ∈ H*ψ*(Ω) and

_{i}∈ V ⊂ L^{p}*ψ*we have

_{i}∈ H

The terms in (26) and (27) are continuous as functions of *u _{im}*(

*t*) and

*w*(

_{im}*t*), then the initial value problem formed by (26) - (27) with initial conditions (28) has a unique maximal solution defined for

*t*∈ [0,

*t*) with

_{m}*u*and

_{im}*w*in

_{im}*C*, for each initial condition

^{1}*u*, (by Cauchy-Peano theorem).

_{0m}, w_{0m}If (*u _{m},w_{m}*) is not a global solution, this is

*t*< 1, then it is not bounded in [ 0,

_{m}*t*). Suppose that (

_{m}*u*) is a maximal solution of (26)-(28). Multiplying (26) by

_{m},w_{m}*λu*, (27) by

_{im}*w*and adding on

_{im}*i = 0,…,m*we get

Note that for being {ψ_{i}} an orthonormal set we have

Then, by the previous observations, adding (37) and (38) we have for all *t* ∈ [ 0,*t _{m}*)

On the other hand, note that for being a(⋅,⋅) coercitive, see (17), we have

Also, from proposition 5, by integrating both sides of (21) on Ω we get

Then, adding (40) and (41) we get

Adding

Then, reorganizing terms and adding

On the other hand, by Young’s inequality we have for all *θ* > 0 the following

then, by taking *θ = λα* we get the following inequality that will be useful a little later.

From (42) it follows immediately that

Then, integrating with respect to *t* over the interval [0,*t _{m}*) on both sides of the previous inequality we get

Recall now that, there exist a constant *c* > 0, such that *‖u _{m}(0)‖_{H} ≤ c* y

*‖w*, y, also we have that Ω is bounded. Then, from the previous inequality and from Gronwall’s inequality it follows that there is a constant C

_{m}(0)‖_{H}≤ c_{1}that depends only on

_{m}, such that

As a consequence we have that (u_{m},w_{m}) is bounded in any finite interval of time, this is. *t _{m} = +∞*. For

*T*> 0 fixed we have shown (33).

In order to get (34) we begin by integrating (42) in the interval [0,*T*]

with

with *k _{2} = k_{1} + (α + μ)C_{1}T*. Therefore, we have shown inequality (34) with

Integrating (33) on [0,*T*] we also get a bound for wm in *L ^{2}*(

*Q*).

_{T}Now we will obtain the estimates for *u’ _{m}* and

*w’*. Consider the projection operator

_{m}*P*defined by

_{m}∶ V′ → V′*P*is defined as the only element in

_{m}u*V*such that 〈𝑢,𝑣〉 = 〈𝑃

_{m}_{𝑚}𝑢,𝑣〉 for all 𝑣 ∈ 𝑉

_{𝑚}. On the other hand, note that for all 𝑣 ∈ 𝑉 and for all 𝑡 > 0 we have

because *u’ _{m}(t)* ∈

*V*and

_{m}⊂ V′, f(u_{m}(t),w_{m}(t)) ∈ L^{p′}(Q_{T})*v*∈

*V ⊂ L*. Thus, from (26) it follows that

^{p}(Q_{T})

and then

where *A* is the weak operator defined in (19). For the continuity of *A* and the estimate (34) we have for all *T* > 0

On the other hand, from the estimates (33), (34) and by lemma 6

The next thing will be to obtain a bound for the projection operator *P _{m}*. We begin by highlighting that, as

*P*(

_{m}*V′*) ⊂

*V*⊂

_{m}*V*, the restriction of

*P*to

_{m}*V*can be considered as an operator from

*V*on

*V*defined by

*u ∈ H, P*is the orthogonal projection of

_{m}u*u*in

*V*, and then ‖P

_{m}_{m}u‖

_{H}≤ ‖u‖

_{H}. The transpose operator

*P*identifies with

_{m}|_{v}*P*, and therefore we have ‖P

_{m}∶ V′ → V′_{m}u‖

_{ℒ(V′,V′)}= ‖P

_{m}u‖

_{ℒ(V,V)}. If

*u*∈

*V*we can calculate

Therefore, for all *u ϵ V* we have

The previous inequality shows that the family of operators *P _{m}* is uniformly bounded in

*V′*,

Then, the following inequalities are met

Inequality (35) is obtained from the previous inequalities and (43). We will proceed similarly to obtain the estimate for *w′ _{m}*. From (27) it follows that

and therefore

where we take the operator *P _{m}* restricted to the orthogonal projection

*P*, so

_{m}|_{H}*‖P*≤ 1. Then, for

_{m}‖_{ℒ(H,H)}*T*> 0 fixed, from (33), (34) and by proposition 6, we have (36)

Convergence of approximate solutions

In the previous section it was shown that the approximate solutions proposed in (25) exist and are defined for all *t* > 0. In this section we will use the a priori estimates (33) - (36) to show that, there exist subsequences of the approximate solutions (*u _{m},w_{m}*) that converge, in a suitable form, to a weak solution according to the definition 4. Furthermore, we prove that this weak solutions is also a strong solution.

**Lemma 2.** *There are subsequences, which for convenience are also denoted as u _{m}, u'_{m}, w_{m} and w'_{m} such that*

*and*

*Proof.* Evidently *L ^{p}*(

*Q*) ∩

_{T}*L*(

^{2}*0,T;V*) is a reflexive space since

*L*(

^{p}*Q*) is reflexive, see (

_{T}^{Brezis 2011, prop. 3.20, p. 60}). By inequality (34),

*u*is a bounded sequence in

_{m}*L*(

^{p}*Q*) ∩

_{T}*L*(

^{2}*0,T;V*), then it has a subsequence that converge weakly, see (

^{Brezis 2011, thm. 3.18, p. 69}). So (44) has been proved. By a similar argument we obtain (45)-(47).

Note that, because 2 ≥ *p′*, we have *L ^{p'}*(

*Q*) +

_{T}*L*(

^{2}*0,T; V′*) ⊂

*L*(

^{p'}*0,T; V′*). By lemma 1 we know that

*u’*is bounded in

_{m}*L*(

^{p'}*0,T; V′*) while

*u*is bounded in

_{m}*L*(

^{2}*0,T; V*) and then

*u*is a bounded sequence in

_{m}*W*(

^{1,2,p'}*0,T; V,V′*), see theorem 2. Then, by the compact immersion of

*W*(

^{1,2,p'}*0,T;V,V′*) in

*L*(

^{2}*Q*), there is a subsequence that converge in

_{T}*L*(

^{2}*Q*).

_{t}**Corollary 1.** *The subquences u _{m},w_{m} satisfy*

*and also, it has that*

*in D′(0,T). That is,u ∈ W ^{1,2,p'} ′(0,T; V,V′), and w ∈ W^{1,2,2} (0,T;H,H).*

*Proof.* Let us take *v ∈ V, ϕ ∈ D*(*0,T*), and note that,

by taking limit in the above equality we obtain

Thus, we have obtained (50). Also, by the weak converge of *u' _{m}*, we get

and, due to the uniqueness the weak limit

that is

In a similar form are proved the affirmations for *w*.

**Corollary 2.** *For ψ _{i} ,i ≥ 0 and the bilinear form a(⋅,⋅) defined in (15) we have*

*Proof.* Because a(⋅,⋅) is a continuous bilinear form, the map

is a continuous linear functional on *L ^{p}*(

*Q*) ∩

_{T}*L*(

^{2}*0,T;V*), and then the result follows immediately from the fact that

*u*converges to

_{m}*u*weakly in

*L*(

^{p}*Q*) ∩

_{T}*L*(

^{2}*0,T;V*).

**Corollary 3.** *For f and g defined in (7)-(8) and for all ψ _{i},i ≥ 0, we have*

*Proof.* Given that *u _{m} → u*, and

*w*, in

_{m}→ w*L*(

^{2}*Q*), it obtains

_{T}

and by the continuity of *f*,

Also,

And

Using an argument of dominated convergence type, see (^{Lions 1969}), we can affirm that *f*(*u _{m},w_{m}*), converges to

*f*(

*u,w*), and

*g*(

*u*), converges to

_{m},w_{m}*g*(

*u,w*), weakly in

*L*(

^{p'}*Q*), and

_{T}*L*(

^{2}*Q*), respectively, that is, for all

_{T}*ζ ∈ L*(

^{p}*Q*) and

_{T}*η ∈ L*(

^{2}*Q*), it has

_{T}

taking *ζ = ϕv, η = ϕℎ* with *ϕ ∈ D'(0,T), v ∈ V* and *ℎ ∈ H*, it has the result.

Conclusion

By the three previous corollaries it is concluded that the functions *u* and *w* satisfy for all *i* ≥ 1 the the following

where equality is considered in *D′*(*0,T*). Then, because functions *ψ _{i},i ≥ 0* are dense in

*V*, it follows that

*u*and

*w*satisfy the equations (29)-(30) in the definition of weak solution 4.

For other hand, by corollary 1, these weak solutions *u,w* belong to *W ^{1,2,p'}*(

*0,T;V,V'*) and

*W*(

^{1,2,2}*0,T;H,H*), thus they are strong solutions , too.

In other words, we have proved that if the systems of Faedo-Galerkin (26)-(27) are considered with uniformly bounded initial conditions the corresponding solutions, *u _{m},w_{m}*, have subsequences that converge, in a suitable form, to a weak solution of the considered problem.

Note that, in the case that the Cauchy problem be considered for the variational formulation, that is, initial conditions *u _{0},w_{0}* be given the systems of Faedo-Galerkin (26)-(27) have initial conditions

*u*which are the projections of

_{0m},w_{0m}*u*in the subspaces,

_{0},w_{0}*V*, for each

_{m}*m*= 0,1,…, and are uniformly bounded. In fact,

thus, by applying the results previously exposed we obtain the existence of weak solution of the variational Cauchy problem.

Continuity

From the previous section we have that *u ∈ W ^{1,2,p′}*(

*0,T; V,V′*) ⊂

*W*(

^{1,2,2}*0,T; V′ V′*), and

*w ∈ W*(

^{1,2,2}*0,T;H,H*). Then, by theorem (3) it follows that the functions

*u: t ∈*[0,

*T*] →

*u*(

*t*)

*∈ V′*and

*w:t ∈*[0,

*T*] →

*w*(

*t*) ∈

*H*are continuous. Regarding

*u*, it only shows that

*u*, it is weakly continuous in

*V*.

By corollary 1 it follows that

where equality is considered in *D′*(0,*T*). Then, from (52), we have

so that the function *H ^{1}* (0, 1), and then is continuous from [0,

*T*] to ℝ. Then, it follows that function

*u: t ∈*[0,

*T*] ↦

*u*(

*t*)

*∈ H*is continuous.

When we consider *u _{m0}* and

*w*as the orthogonal projections in

_{m0}*H*of

*u*and

_{0}*w*respectively, we obtain that

_{0}*u(0) = u*and

_{0}*w(0) = w*

_{0.}