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Nova scientia

versión On-line ISSN 2007-0705

Nova scientia vol.10 no.21 León nov. 2018 

Ciencias Naturales e Ingenierías

Existence of global solutions in a model of electrical activity of the monodomain type for a ventricle

Existencia de solución en un modelo de actividad eléctrica de tipo monodominio para un ventrículo

Ozkar Hernández Montero1  * 

Andrés Fraguela Collar1 

Raúl Felipe Sosa1 

1FCMF-BUAP, Puebla, México



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 Rogers-Mculloch model is used to represent the electrical activity through the cell membrane.


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 Faedo-Galerkin type is proposed.


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.


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.

Keywords: monodomain; bidomain; reaction-diffusion; Faedo-Galerkin



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 Rogers-Mculloch para representar la actividad eléctrica a través de la membrana celular.


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 Faedo-Galerkin.


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.


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.

Palabras clave: monodominio; bidominio; reacción-difusión; Faedo-Galerkin


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 ∈ C1(ℝ) satisfies inf 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 ∈ ℝ3 simulating an isolated ventricle surrounded by an insulating medium. The boundary 𝜕Ω of the spatial region is formed by two disjoint components; the component Γ0 imulating the epicardium and the component Γ1 simulating the endocardium. The way Ω is electrically stimulated is by means Purkinje fibers, which directly stimulate only the inner wall Γ1 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 Γ0 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:

ut+fu,w-·σu=0,           t,x0,×Ω, (1)

wt+gu,w=0,                                 t,x 0,×Ω, (2)

σu·η=0,                                          t,x 0,×Γ0, (3)

σu·η=stφx,                            t,x 0,×Γ1, (4)

u0,x=u0x,  w0,x=w0x,      x Ω. (5)

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

0<mξ2ξtσxξMξ2<,  ξR3, (6)

Is met for almost all x ∈ Ω.

(h3) sL(0,+∞).

(h4) φL21).

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

fu,w=a1u-urestu-uthu-upeak+a2u-urestw, (7)

gu,w=b-u+urest+c3w. (8)

(h6) u0, w0 ∈ L2(Ω).

It is convenient to establish some notations that we will follow throughout this work. For convenience, we will denote V = H1(Ω) and H = L2(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

VLpΩHH'Lp'ΩV' (9)

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 1p+1p'=1.

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 uX, we denote


Thus [u] ∈ X/ℝ.

This paper is organized as follows. The spaces Lq(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.


Lq(0,T;X) spaces

Let X be a Banach space, we denote by Lq(0,T;X) the space of the functions t → f(t) of [0,T] → X that are measurable with values in X such that

fLq0,T;X=0Tf(t)Xqdt1/q<, (10)

with this norm Lq(0,T;X) is complete. Observe that


where QT[0,T] × Ω.

It is necessary to give a definition of the derivative of an element of Lq(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

D' 0,T;X=LD0,T;X, (11)

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 ftD'(0,T;X) given by

ftϕ=-fdϕdt,     ϕD0,T. (12)

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

f^ϕ=0TftϕtdtX,     ϕD0,T.  (13)

In this way, we can define the derivative in the sense of distributions of a function fLq(0,T;X) as

f^tϕ=-0Tftϕ'dt,       ϕD0,T.

Theorem 1. Let QT a bounded open in ℝ × ℝN fn and f functions in Lq(QT), 1 < q < ∞, such that

fnLqQTC,     fnf c.p en QT,

for a certain constant C > 0, then,

fnf weakly in LpQT

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 < qi < ∞,i = 0,1,

W1q0,q10,T;V,V'=υ|υLq00,T;V,υ'=dυdtLq10,T;V', (14)

endowed with the norm νLp0(0,T;V)+ν'Lp1(0,T;V'). Then W1,q0,q1 (0,T;V,V') is a Banach space and W1,q0,q1 0,T;V,V' Lq0(0,T;H). The inmersion of W1,q0,q1 0,T;V,V' in Lq0(0,T;H) is compact.

Proposition 1. Let with uLq0(0,T;V) with q0 ≥ 2, then, uW1,q0,q1 0,T;V,V', for some q1 ≥ 2, if and only if there exist afunction u~ Lq1(0,T;V') that satisfies

-0Tu,vϕ'=0Tu~ ,vV'×Vϕ      ϕD0,T,vV,

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 Lq1(0,T;V'), that satisfies

ddtu,v=u~ , vV'×V,   for all vV.

From now on, we write 〈∙,∙〉 instead of 〈∙,∙〉 𝑉′×𝑉 .

Theorem 3. If f ∈ Lq and ∂tf ∈ Lq(0,T;X) (1 ≤ q ≤ ∞),then, f is continuous

almost everywhere from (0,T) to X

Proof. (Lions 1969, lema 1.2, p. 7).


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

au,v=Ωσu·v. (15)

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

au,vMuVvV,         u,vV, (16)

αuV2αu,u+auH2,           uV, (17)

with α,M > 0. There is a growing sequence 0 = λ0 < ⋯ < λi < ⋯ ∈ ℝ and there is an orthonormal basis of H formed by eigenvectors {ψi}i∈ℕ such that, ψi ∈ V y

vV,   aψi,v=λiψi,v. (18)

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 muH2 on both sides of the inequality we get


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 λ0 = 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

au,v=Au,v. (19)

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 ∈ L21) then for v ∈ V the function


defines a linear and continuous functional. This is, we haveφ^V'.

We will denote

fu,w=f1u+f2uw,         gu,w=g1u+g2w, (20)









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

f1uc1+c2up-1,     f2uc3+c4up/2-1,     g1uc5+c6up/2

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

u22u33+13,     uu33+23,     uu22+12







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

λufu,w+wgu,waup-μλu2+w2-c. (21)

Proof. By direct calculation from (20) we have


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

α2u334θu34/3+14α2θ4,     u2w12βu22+12wβ2,

α0uu22+α022,     uwu22+w22,     urestww22+urest22



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 12=ργ, with p<1 y γ=12ρ>2,


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




which is fulfilled for

34θ4/3=a12,     β22=14,

obviously, we can find a ρ small enough to meet such conditions. We have μ = γ, λ > 0 arbitrary, a=λa14 and


Proposition 6. Let u ∈ Lp (Ω) and w ∈ H, Then f(u,w) ∈ Lp′ (Ω) and g(u,w) ∈ H. In addition, the following inequalities are met



where Ai ≥ 0, i = 0,...,3 y Bi ≥ 0, i = 0,...,3 are constants that depend only on ci, i = 1,...,6 and p.

Proof. Let (u,w) ∈ ℝ2, by proposition 4 we have



with B1 =c5,B2 = c6 y B3 = |g2|. On the other hand, by Young’s inequality, with β=2p'>1 and 1β+1β'=1, we have


then, because p2-1β'=p2-12 p-1p-2=p-1 we have


then, once more by Young’s inequality wwββ+1β, therefore we can find constants A1, A2 y A3 such that


If (u,v) ∈ Lp(Ω) × H, by direct calculation and taking into account that (p − 1)p′ = p,βp′ = 2 we have

fu,vLP'ΩA1+A2up-1+A3wβLP' ΩA1Ω1/p'+A2uLpΩp/p'+A3wH2/p'.

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,

Ωtutv+Ωfut,wtv+aut,v=stΓ1φv,        vV, (22)

Ωwtz+Ωgut,wtz=0,          zH, (23)

u0,x=u0x,     w0,x=w0x,           xΩ (24)

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 Vm the linear space generated by {ψ01,…,ψm}, where the functions ψi,i = 0,…,m, are eigenfunction of the bilinear form a(⋅,⋅) as established in the proposition 2. Note that Vm ⊂ V. For each m, we consider the variational problem restricted to the space Vm, that is, instead of v andz we take ψi,i = 0,…,m, and approximate u(t) and w(t) by um(t) and um(t) respectively, with

umt=i=0muimtψi   Vm,       wmt=i=0mwimtψi   Vm. (25)

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

ddtuimt+Ωfumt,wmtψi+λiuimt=stΓ1φψi, (26)

ddtwimt+Ωgumt,wmtψi=0 (27)

um0=um0,    wm0=wm0, (28)

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),

  1. u ∶ [0,T] ↦ and w ∶ [0,T] ↦ H are continuous.

  2. For almost all t ∈ (0,τ), we have u(t) ∈ V, also u ∈ Lp (QT) ∩ L2 (0,T;V) and w ∈ L2 (QT), with QT = (0,T) × Ω.

In addition, the functions u and w satisfy

ddtut,v+aut,v+Ωfut,wtv=stφ^, v,         vV, (29)

ddtwt,z+Ωgut,wtz=0        zH, (30)

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

If, furthermore, given u0 in H, u, w0 in H, u,w in, are weak solutions that satisfy

u0=u0,     w0=w0,     in H,

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 ∈ W1,2,p′(0,T;V′,V) and w ∈ W1,2,2(0,T;H,H), then the equation (29) means that

-0Tu,vϕ'+0Tau,vϕ+0TΩfu,wvϕ=0Tsτφ^,vϕ,   for all ϕD0,T,

thus, by proposition 1, it has


which implies that

0Tdudt+Au+fu,w-sτφ^,vϕdτ=0,     for all ϕD0,T,vV.

From the above it follows that,

dudt+Au+fu,w=stφ^,     for a.a.  t0,T, (31)

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

dwdt=gu,w,     for a.a.  t0,T, (32)

is fulfilled in H.

Definition 5. (Strong Solution). Let be u ∈ W1,2,p′ (0,T;V,V′) and w ∈ W1,2,2(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,

u0=u0,     w0=w0,     in H,

for u0, w0 given, we say that u,w are strong solutions of variational Cauchy problem associated to (1)-(5).


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 um, wm 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 um, wm 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 Ci > 0,i = 1,…,4, such that for all T > 0. The following estimates are met a priori

λumtH2+wmtH2C1,     t0,T, (33)

umLpQTL20,T;VC2, (34)

u'mLp'QT+L20,T;V'C3, (35)

w'mL2QTC4, (36)

where um'=i=0muim'ψi and wm'=i=0mwim'ψi are the derivatives of the functions um ∶ [0,T] ↦ V and wm ∶ [0,T] ↦ H.

Proof. Note that the integrals in (26) and (27) are well defined, in deed, as um (t) ∈ V ⸦ Lp (Ω) and wm(t) ∈ H we have from proposition 6 that f(um(t),wm (t)) ∈ Lp′(Ω) ⊂ V′ and, g(um,wm) ∈ H, then because ψi ∈ V ⊂ Lp(Ω) and ψi ∈ H we have



The terms in (26) and (27) are continuous as functions of uim(t) and wim(t), then the initial value problem formed by (26) - (27) with initial conditions (28) has a unique maximal solution defined for t ∈ [0,tm) with uim and wim in C1, for each initial condition u0m, w0m, (by Cauchy-Peano theorem).

If (um,wm) is not a global solution, this is tm < 1, then it is not bounded in [ 0,tm). Suppose that (um,wm) is a maximal solution of (26)-(28). Multiplying (26) by λuim, (27) by wim and adding on i = 0,…,m we get

λi=0muimtddtuimt+λλiuimtuimt+Ωλfumt,wmtuimtψi=λstφ^,i=0muimtψi (37)

i=0mwimtddtwimt+Ωgumt,wmtwimtψi=0. (38)

Note that for being {ψi} an orthonormal set we have

ddtumH2=2i=0muimtddtujmt,     aum,um=i=0mλiuimtujmt.

Then, by the previous observations, adding (37) and (38) we have for all t ∈ [ 0,tm)

12ddtλumtH2+wmtH2+λaumt,umt+Ωλfumt,wmtum+gumt,wmtwm=λstφ^,umt. (39)

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

λαuv2-uH2λaumt,umt. (40)

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

aΩumtp-μλumtH2+wmtH2-cΩΩλfumt,wmtumt+gumt,wmtwmt. (41)

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


Adding 12ddtλumtH2+umtH2 on both sides of the previous inequality we get from (39) the following


Then, reorganizing terms and adding αwmtH2 to the right side of the previous inequality we get


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.

12ddtλumtH2+wmtH2+λα2umtV2+aΩumtpcΩ+λ2αst2φ^V'2+α+μλumtH2+wmtH2 (42)

From (42) it follows immediately that


Then, integrating with respect to t over the interval [0,tm) on both sides of the previous inequality we get


Recall now that, there exist a constant c > 0, such that ‖um(0)‖H ≤ c y ‖wm(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 C1 that depends only on c, σ, f, g, u0, w0, Ω,s, φ^ and tm, such that

t0,tm,        λumtH2+wmtH2C1.

As a consequence we have that (um,wm) is bounded in any finite interval of time, this is. tm = +∞. For T > 0 fixed we have shown (33).

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


with k1=cΩT+λ2αs2φ^V'2+12(λum0H2+wm0H2). Then, we use (33) on the integral on the right side of the previous inequality,


with k2 = k1 + (α + μ)C1T. Therefore, we have shown inequality (34) with


Integrating (33) on [0,T] we also get a bound for wm in L2(QT).

Now we will obtain the estimates for u’m and w’m. Consider the projection operator Pm ∶ V′ → V′ defined by uV'Pmu=i=0mu,ψiψi. Equivalently, Pmu is defined as the only element in Vm such that 〈𝑢,𝑣〉 = 〈𝑃𝑚𝑢,𝑣〉 for all 𝑣 ∈ 𝑉𝑚. On the other hand, note that for all 𝑣 ∈ 𝑉 and for all 𝑡 > 0 we have

ddtumt,v=u'mt,v,     Ωfumt,wmtv=fumt,wmt,v,

because u’m(t)Vm ⊂ V′, f(um(t),wm(t)) ∈ Lp′(QT) and vV ⊂ Lp(QT). Thus, from (26) it follows that

vVm,t>0,     u'mt,v=-Aumt+fumt,wmt+stφ^,v,

and then

t>0,     u'mt=Pm-Aumt+fumt,wmt+stφ^, (43)

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 Pm. We begin by highlighting that, as Pm(V′) ⊂ VmV, the restriction of Pm to V can be considered as an operator from V on V defined by uVPmu=i=0mu,ψiψi. If u ∈ H, Pmu is the orthogonal projection of u in Vm, and then ‖Pmu‖H ≤ ‖u‖H. The transpose operator PmT of Pm|v identifies with Pm ∶ V′ → V′, and therefore we have ‖Pmu‖ℒ(V′,V′) = ‖Pmu‖ℒ(V,V). If uV we can calculate


Therefore, for all u ϵ V we have


The previous inequality shows that the family of operators Pm 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

vVmH,>0,     w'mt,v=-gumt,wmt,v,

and therefore

t>0,     w'mt=-Pmgumt,wmt,

where we take the operator Pm restricted to the orthogonal projection Pm|H, so ‖Pmℒ(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 (um,wm) 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 um, u'm, wm and w'm such that

umu, weakly in LpQTL20,T;V. (44)

u'mu~, weakly in Lp'QT+L20,T;V'. (45)

umw, weakly in L2QT. (46)

w'mw~, weakly in L2QT. (47)


umu, strongly in L2QT, (48)

wmw, strongly in L2QT. (49)

Proof. Evidently Lp(QT) ∩ L2(0,T;V) is a reflexive space since Lp(QT) is reflexive, see (Brezis 2011, prop. 3.20, p. 60). By inequality (34), um is a bounded sequence in Lp(QT) ∩ L2(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 Lp'(QT) + L2(0,T; V′) ⊂ Lp'(0,T; V′). By lemma 1 we know that u’m is bounded in Lp' (0,T; V′) while um is bounded in L2 (0,T; V) and then um is a bounded sequence in W1,2,p'(0,T; V,V′), see theorem 2. Then, by the compact immersion of W1,2,p'(0,T;V,V′) in L2(QT), there is a subsequence that converge in L2(Qt).

Corollary 1. The subquences um,wm satisfy

ddtum,vddtu,v,   weakly in D'0,T, for all vV, (50)

ddtwm,hddtw,h,   weakly in D'0,T, for all hH, (51)

and also, it has that

ddtu,v=u~,v,   vV,

ddtw,h=w~,h,   hH,

in D′(0,T). That is,u ∈ W1,2,p' ′(0,T; V,V′), and w ∈ W1,2,2 (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 ψi ,i ≥ 0 and the bilinear form a(⋅,⋅) defined in (15) we have

0Taumt,ψiϕ0Taut,ψiϕi     ϕD0,T.

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


is a continuous linear functional on Lp(QT) ∩ L2(0,T;V), and then the result follows immediately from the fact that um converges to u weakly in Lp(QT) ∩ L2(0,T;V).

Corollary 3. For f and g defined in (7)-(8) and for all ψi,i ≥ 0, we have

0TΩfumt,wmtψiϕ0TΩfut,wtϕtψiϕ,      ϕD0,T,

0TΩgumt,wmtψiϕ0TΩgut,wtψiϕ,      ϕD0,T.

Proof. Given that um → u, and wm → w, in L2(QT), it obtains

umu,     a.e.  in  QT,

wmw,     a.e.  in  QT,

and by the continuity of f,

fum,wmfu,w,     a.e.  in  QT,

gum,wmgu,w,     a.e.  in  QT.









Using an argument of dominated convergence type, see (Lions 1969), we can affirm that f(um,wm), converges to f(u,w), and g(um,wm), converges to g(u,w), weakly in Lp'(QT), and L2(QT), respectively, that is, for all ζ ∈ Lp(QT) and η ∈ L2(QT), it has



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


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

ddtut,ψi+aut,ψi+fut,wt,ψi=stφ,ψi (52)

ddtwt,ψi+gut,wt,ψi=0 (53)

where equality is considered in D′(0,T). Then, because functions ψi,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 W1,2,p'(0,T;V,V') and W1,2,2(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, um,wm, 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 u0,w0 be given the systems of Faedo-Galerkin (26)-(27) have initial conditions u0m,w0m which are the projections of u0,w0 in the subspaces, Vm, for each m = 0,1,…, and are uniformly bounded. In fact,

um0Hu0H,     wm0Hw0H,

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


From the previous section we have that u ∈ W1,2,p′(0,T; V,V′) ⊂ W1,2,2(0,T; V′ V′), and w ∈ W1,2,2(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

12ddtutH2=-aut,ut-fut,wt,ut+stφ,ut (54)

so that the function t utH2 is in H1 (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 um0 and wm0 as the orthogonal projections in H of u0 and w0 respectively, we obtain that u(0) = u0 and w(0) = w0.


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 Felipe-Sosa was supported by SEP during the achievement of this work.


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Received: June 01, 2018; Accepted: July 10, 2018

*Autor para correspondencia: Ozkar Hernández Montero, E-mail:

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