Introduction
A monodomain model of electrical activity for an isolated ventricle is formulated. This model is written as a reaction diffusion PDE coupled to an ODE, The RogersMculloch model is used to represent the electrical activity through the cell membrane.
Method
We give a definition of weak and strong solution of the variational Cauchy problem associated to the monodomain model. A sequence of approximate solutions of FaedoGalerkin type is proposed.
Results
It is shown that the sequence of approximate solutions converge to a weak solution according to the proposed definition. Finally, we have that this weak solution is also a strong solution.
Conclusion
The monodomain model of electrical activity in an isolated ventricle that is proposed has a weak solution in an appropriate sense. In addition, this weak solution is also a strong solution.
Introducción
Se formula un modelo de monodominio de actividad eléctrica en un ventrículo aislado. Este modelo se escribe como una EDP de tipo reacción difusión acoplada a una EDO, se utiliza el modelo de RogersMculloch para representar la actividad eléctrica a través de la membrana celular.
Método
Se proponen definiciones de solución débil y fuerte respectivamente para el problema de Cauchy variacional asociado al modelo de monodominio. Se propone una sucesión de soluciones aproximadas de tipo FaedoGalerkin.
Resultados
Se demuestra que la sucesión de soluciones aproximadas converge a una solución débil según la definición que se propone. Finalmente, se obtiene que la solución débil es también una solución fuerte.
Conclusión
El modelo de monodominio de actividad eléctrica en un ventrículo aislado que se propone tiene solución débil en un sentido apropiado. Además, esta solución débil también es una solución fuerte.
 monodominio;
 bidominio;
 reaccióndifusión;
 FaedoGalerkin.
 monodomain;
 bidomain;
 reactiondiffusion;
 FaedoGalerkin.
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 bodysurface 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 reactiondiffusion (RD) 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 RD models, conceived as a simplification of the RD bidomain models, with advantages both for mathematical analysis and computation, were actually developed before the first bidomain RD 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 fluidfilled 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 wellposedness of the bidomain model. The most important seem to be ColliFranzone 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 cubiclike FitzHughNagumo 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 AlievPanfilov (Aliev 1996) and MacCulloch (Rogers 1994) models. In (Veneroni 2009), ColliFranzone 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 AlievPanfilov 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 FitzHughNagumo model (Fitzhugh 1961) and simple models more adapted to myocardial cells, such as the AlievPanfilov (Aliev 1996) and RogersMacCulloch (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 RD 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 RogersMcCulloch 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 RogersMcCulloch 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^{2}(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 ≤ 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 ∈ 𝐿^{𝑞}(0,𝑇;𝑋) it corresponds a distribution
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 ‖𝑣‖𝐿^{𝑝0} (0,𝑇;𝑉) + ‖𝑣′‖𝐿^{𝑝1} (0,𝑇;𝑉′) . Then, 𝑊^{1,𝑞0,𝑞1} (0,𝑇; 𝑉,𝑉′) is a Banach space and 𝑊^{1,𝑞0,𝑞1} (0,𝑇; 𝑉,𝑉′) ⊂ 𝐿^{𝑞0} (0,𝑇;𝐻). The immersion of 𝑊^{1,𝑞0,𝑞1} (0,𝑇; 𝑉,𝑉′) in 𝐿^{𝑞0} (0,𝑇;𝐻) is compact.
Proposition 1. Let 𝑢 ∈ 𝐿^{𝑞0} (0,𝑇; 𝑉 ) with 𝑞_{0} ≥ 2, then, 𝑢 ∈ 𝑊^{1,𝑞0,𝑞1} (0,𝑇; 𝑉,𝑉′), for some 𝑞_{1} ≥ 2, if and only if there exist a function ũ ∈ 𝐿^{𝑞1} (0,𝑇; 𝑉′) that satisfies
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 in 𝐿^{𝑞1} (0,𝑇; 𝑉′), that satisfies
From now on, we write 〈∙,∙〉 instead of 〈∙,∙〉_{𝑉′×𝑉}.
Theorem 3. If 𝑓 ∈ 𝐿^{𝑞} (0,𝑇;𝑋) and 𝜕_{𝑡} 𝑓 ∈ 𝐿^{𝑞} (0,𝑇;𝑋) (1 ≤ 𝑞 ≤ ,then, 𝑓 is continuous
almost everywhere from (0,𝑇) to 𝑋. 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 {𝜓𝑖}𝑖∈ℕ such that, 𝜓𝑖 ∈ 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 6.21 y rem. 6.22, p. 137138), 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 LaxMilgram 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 𝑎,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 𝐵𝑖 ≥ 0, 𝑖 = 0,…,3, are constants that depend only on 𝑐_{ 𝑖 } ,𝑖 = 1,…,6 and 𝑝..
Proof. Let (𝑢,𝑤) ∈ ℝ2, 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
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 FaedoGalerkin 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_{𝑖},𝑖 = 0,…,𝑚, and approximate 𝑢(𝑡) and 𝑤(𝑡) by 𝑢_{𝑚}(𝑡) and 𝑤_{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 formulation
of the problem (1)(4) if for any 𝑇 ∈ (0,𝜏),
In addition, the functions 𝑢 and 𝑤 satisfy
where equality is considered in 𝒟′(0,𝑇).
If, furthermore, given 𝑢_{ 0 } ,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,
a sequence of approximate solutions 𝑢_{𝑚},𝑤_{𝑚} is defined,
then, it is verified that the approximate solutions converge to a function that satisfies the definition 4.
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
[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 ⊂ 𝐿𝑝(Ω) and 𝑤_{𝑚}(𝑡) ∈ 𝐻 we have from proposition 6 that 𝑓(𝑢_{𝑚}(𝑡),𝑤_{𝑚} (𝑡)) ∈ 𝐿^{𝑝′(}Ω) ⊂ V′ and, 𝑔(𝑢_{𝑚},𝑤_{𝑚}) ∈ 𝐻, then because 𝜓_{𝑖} ∈ 𝑉 ⊂ 𝐿^{𝑝}(Ω) and 𝜓_{𝑖} ∈ 𝐻 we have
The terms in (26) and (27) are continuous as functions of 𝑢𝑖𝑚(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_{𝑚} ) with 𝑢_{𝑖𝑚} and 𝑤_{𝑖𝑚} in 𝐶^{1}, for each initial condition 𝑢_{0𝑚}, 𝑤_{0𝑚}, , (by CauchyPeano theorem).
If (𝑢_{𝑚},𝑤_{𝑚}) is not a global solution, this is 𝑡_{𝑚} < 1, then it is not bounded in [ 0,𝑡_{𝑚} ). Suppose that (𝑢_{𝑚},𝑤_{𝑚}) is a maximal solution of (26)(28). Multiplying (26) by λ𝑢_{𝑖𝑚}, , (27) by 𝑤_{𝑖𝑚} and adding on 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_{𝑚}(0)‖_{𝐻} ≤ 𝑐, 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_{1} that depends only on
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’_{𝑚}. Consider the projection operator 𝑃_{𝑚} ∶ 𝑉′ → 𝑉′ defined by
because 𝑢’_{𝑚} (𝑡) ∈ 𝑉_{𝑚} ⊂ 𝑉′, 𝑓(𝑢_{𝑚}(𝑡),𝑤_{𝑚}(𝑡)) ∈ 𝐿^{𝑝′(}𝑄𝑇) and 𝑣 ∈ 𝑉 ⊂ 𝐿^{𝑝}(𝑄𝑇). Thus, from (26) it follows that
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_{𝑚} to V can be considered as an operator from 𝑉 on 𝑉 defined by
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_{𝑚}_{𝐻}, so ‖𝑃_{𝑚}‖_{ℒ(𝐻,𝐻}) ≤ 1. Then, for 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 ass 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_{𝑇}) is reflexive, see (Brezis 2011, prop. 3.20, p. 60). By inequality (34), 𝑢_{𝑚} is a bounded sequence in 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}) + ??^{2}(0,𝑇; 𝑉′) ⊂ 𝐿^{𝑝}′(0,𝑇; 𝑉′). By lemma 1 we know that u’_{ m } is bounded in 𝐿^{ 𝑝 } ′(0,𝑇; 𝑉′) while 𝑢_{𝑚} is bounded in 𝐿^{ 2 } (0,𝑇; V) and then 𝑢_{𝑚} is a bounded sequence in 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_{𝑇}), there is a subsequence that converge in 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_{𝑚} converges to 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_{𝑚} → 𝑤, in 𝐿^{2}(𝑄_{𝑇}), it obtains
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_{𝑚},𝑤_{𝑚}), converges to 𝑔(𝑢,𝑤), weakly in 𝐿^{𝑝}′(𝑄_{𝑇}), and 𝐿^{2}(𝑄_{𝑇}), respectively, that is, for all 𝜁 ∈ 𝐿^{𝑝}(𝑄_{𝑇}) and 𝜂 ∈ 𝐿^{2}(𝑄_{𝑇}), it has
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 FaedoGalerkin (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 FaedoGalerkin (26)(27) have initial conditions 𝑢_{0m},𝑤_{0m} which are the projections of 𝑢_{0},𝑤_{0} in the subspaces, 𝑉_{𝑚}, for each 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
When we consider 𝑢_{𝑚0} and 𝑤_{𝑚0} as the orthogonal projections in 𝐻 of 𝑢_{0} and 𝑤_{0} respectively, we obtain that 𝑢(0) = 𝑢_{0} and 𝑤(0) = 𝑤_{0.}
Acknowledgements
We thank Dr. Manlio F. Márquez Murillo for his valuable help to contextualize this work. Ozkar Hernández Montero was supported by CONACYT during the achievement of this work, and Raúl FelipeSosa was supported by SEP during the achievement of this work.

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 » Received: 01/06/2018
 » Accepted: 10/07/2018
 » Digital publication: 04/2019