Networks and Heterogeneous Media, Volume 1, Number 1, March 2006, pp

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1 Manuscript submitted to AIMS Journals Volume X, Number X, XX 2X Website: pp. X XX Networks and Heterogeneous Media, Volume 1, Number 1, March 26, pp ANALYSIS OF A CLASS OF DEGENERAE REACION-DIFFUSION SYSEMS AND HE BIDOMAIN MODEL OF CARDIAC ISSUE Mostafa Bendahmane and Kenneth H. Karlsen Centre of Mathematics for Applications, University of Oslo P.O. Box 153, Blindern, N 316 Oslo, Norway Abstract. We prove well-posedness (existence and uniqueness) results for a class of degenerate reaction-diffusion systems. A prototype system belonging to this class is provided by the bidomain model, which is frequently used to study and simulate electrophysiological waves in cardiac tissue. he existence result, which constitutes the main thrust of this paper, is proved by means of a nondegenerate approximation system, the Faedo-Galerkin method, and the compactness method. 1. Introduction. Our point of departure is a widely accepted model, the so-called bidomain model, for describing the cardiac electric activity in a physical domain R 3 (the cardiac muscle) over a time span (, ), >. In this model the cardiac muscle is viewed as two superimposed (anisotropic) continuous media, referred to as the intracellular (i) and extracellular (e), which occupy the same volume and are seperated from each other by the cell membrane. o state the model, we let u i = u i (t, x) and u e = u e (t, x) represent the spatial cellular at time t (, ) and location x of the intracellular and extracellular electric potentials, respectively. he difference v = v(t, x) = u i u e is known as the transmembrane potential. he anisotropic properties of the two media are modeled by conductivity tensors M i (t, x) and M e (t, x). he surface capacitance of the membrane is represented by a constant c m >. he transmembrane ionic current is represented by a nonlinear (cubic polynomial) function h(t, x, v) depending on time t, location x, and the value of the potential v. he stimulation currents applied to the intra- and extracellular space are represented by a function I app = I app (t, x). A prototype system that governs the cardiac electric activity is the following degenerate reaction-diffusion system (known as the bidomain equations) c m t v div (M i (t, x) u i ) + h(t, x, v) = I app, (t, x), c m t v + div (M e (t, x) u e ) + h(t, x, v) = I app, (t, x), where denotes the time-space cylinder (, ). We complete the bidomain system (1) with Dirichlet boundary conditions for both the intra- and extracellular (1) 2 Mathematics Subject Classification. Primary: 35K57, 35M1; Secondary: 35A5. Key words and phrases. Reaction-diffusion system, degenerate, weak solution, existence, uniqueness, bidomain model, cardiac electric field. his research is supported by an Outstanding Young Investigators Award from the Research Council of Norway. Kenneth H. Karlsen is grateful to Aslak veito for having introduced him to the bidomain model and for various discussions about it. 1

2 2 MOSAFA BENDAHMANE, KENNEH H. KARLSEN electric potentials: u j = on (, ), j = i, e, (2) and with initial data for the transmembrane potential: v(, x) = v (x), x. (3) For the boundary we could have dealt with Neumann type conditions as well, which seem to be used frequently in the applicative literature, i.e., (M j (t, x) u j ) η = on (, ), j = i, e, where η denotes the outer unit normal to the boundary of For the sake of completeness we have included a brief derivation of the bidomain model in Section 2, but we refer to the papers [7, 8, 9, 1, 14, 18, 3] and the books [16, 25, 29] for detailed accounts on the bidomain model. If M i λm e for some constant λ R, then the system (1) is equivalent to a scalar parabolic equation for the transmembrane potential v. his nondegenerate case, which assumes an equal anisotropic ratio for the intra- and extracelluar media, is known as the monodomain model. Being a scalar equation, the monodomain model is well understood from a mathematical point of view, see for example [26]. On the other hand, the bidomain system (1) was studied only recently from a well-posedness (existence and uniqueness of solutions) point view [1]. Indeed, standard elliptic/parabolic theory does not apply directly to the bidomain equations due to their degenerate structure, which is a consequence of the unequal anisotropic ratio of the intra- and extracellular media. In fact, a distinguishing feature of the bidomaim model lies in the structure of the coupling between the intra- and extracellular media, which takes into account the anisotropic conductivity of both media. When the degree of anisotropy is different in the two media, we end up with a system (1) that is of degenerate parabolic type. In this paper we shall not exclusively investigate the bidomain system (1) but also a class of systems that are characterized by a combination of general nonlinear diffusivities and the degenerate structure seen in the bidomain equations. hese reaction-diffusion systems read c m t v div M i (t, x, u i ) + h(t, x, v) = I app, (t, x), c m t v + div M e (t, x, u e ) + h(t, x, v) = I app, (t, x), where the nonlinear vector fields M j (t, x, ξ) : R 3 R 3, j = i, e, are assumed to be Leray-Lions operators, p-coercive, and behave like ξ p 1 for large values of ξ R 3 for some p > 1, see Subsection 3.2 precise conditions. We complete the nonlinear system (4) with Dirichlet boundary conditions (2) for the intra- and extracellular potentials and initial data (3) for the transmembrane potential. Formally, by taking M j (t, x, ξ) = M j ξ, j = i, e, in (4) we obtain the bidomain equations (1). An example of a nonlinear diffusion part in (4) is provided by M j (t, x, ξ) = ξ p 2 M j (t, x)ξ, p > 1, j = i, e. (5) Although (4) can be viewed as a generalization of the bidomain equations in view of its more general diffusion part. he bidomain system contains the term h describing the flow of ions accross the cell membrane. his is the simplest possible model, and in this model it is customary to assume that the current is a cubic polynomial of the transmembrane potential. In a more realistic setup the reactiondiffusion system (1) is coupled with a system of ODEs for the ionic gating variables and for the ions concentration. However, since the main interest in this paper lies (4)

3 DEGENERAE REACION-DIFFUSION SYSEMS 3 with the degenerate structure of the system (1), we neglect the ODE coupling and assume that the relevant effects are taken care of by the nonlinear function h. When it comes to well-posedness analyis for the bidomain model we know of only one paper, namely [1] (it treats both microscopic and macroscopic models). In that paper the authors propose a variational formulation of the model and show after an abstract change of variable that it has a structure that fits into the framework of evolution variational inequalities in Hilbert spaces. his allows them to obtain a series of results about existence, uniqueness, and regularity of solutions. Somewhat related, based on the theory in [1] the author of [27] proves error estimates for a Galerkin method for the bidomain model. Let us also mention the paper [1] in which the authors use tools from Γ-convergence theory to study the asymptotic behaviour of anisotropic energies arising in the bidomain model. Let us now put our own contributions into a perspective. With reference to the bidomain equations (1) and the work [1], we give a different and constructive proof for the existence of weak solutions. Our proof is based on introducing nondegenerate approximation systems to which we can apply the Faedo-Galerkin scheme. o prove convergence to weak solutions of the approximate solutions we utilize monotonicity and compactness methods. Additionally, we analyze for the first time the fully nonlinear and degenerate reaction-diffusion system (4). As already alluded to, we prove existence of weak solutions for the bidomain system (1) and the nonlinear system (4) using specific nondegenerate approximation systems. he approximation systems read c m t v + ε t u i div M i (t, x, u i ) + h(t, x, v) = I app, (t, x), c m t v ε t u e + div M e (t, x, u e ) + h(t, x, v) = I app, (t, x), where ε > is a small number. Notationally, we have let (6) cover both the bidomain case p = 2 and the nonlinear case p > 1 with p 2. We supplement (6) with Dirichlet boundary conditions (2) and initial data u j (, x) = u j, (x), x, j = i, e. (7) Since we use the non-degenerate problem (6) to produce approximate solutions, it becomes necessary to decompose the initial condition v in (3) as v = u i, u e, for some functions u i,, u e,, see Sections 6 and 7 for details. We prove existence of solutions to (6) (for each fixed ε > ) by applying the Faedo-Galerkin method, deriving a priori estimates, and then passing to the limit in the approximate solutions using monotonicity and compactness arguments. Having proved existence for the nondegenerate systems, the goal is to send the regularization parameter ε to zero in sequences of such solutions to fabricate weak solutions of the original systems (1), (4). Again convergence is achieved by priori estimates and compactness arguments. On the technical side, we point out that in the nonlinear case (p > 1, p 2) we must prove strong convergence of the gradients of the approximate solutions to ensure that the limit functions in fact solve the orginal system (4), whereas in the linear bidomain model (1) we can achieve this with just weakly converging gradients. Finally, let us mention that it is possible to analyze systems like the bidomain model by means of different methods than the ones utillized in [1] or in this paper, see for example [6, 12] and also the discussion in [1]. he plan of the paper is as follows: In Section 2 we recall briefly the derivation of the bidomain model. In Section 3 we introduce some notations/functional spaces and recall a few basic mathematical facts needed later on for the analysis. Section (6)

4 4 MOSAFA BENDAHMANE, KENNEH H. KARLSEN 4 is devoted to stating the definitions of weak solutions as well as the main results. In Section 5 we prove existence of solutions for the nondegenerate systems. he main results stated in Section 4 are proved in Section 6 for the bidomain system (1) and in Section 7 for the nonlinear system (4). We conclude the paper in Section 8 by proving uniqueness of weak solutions. 2. he bidomain model. We devote this section to a brief derivation of the bidomain model of cardiac tissue. As principal references on this model we use [14, 16, 25, 29]. he cardiac tissue (represented by the domain R 3 ) is conceived as the coupling of two anisotropic continuous superimposed media, one intracellular and the other extracellular, which are separated by the cell membrane. he electrical potentials in these media are denoted by u i, the intracellular potential, and u e, the extracellular potential. Inside each medium the current flows J j are assumed to obey (the local form of) Ohm s law: J j = M j u j, j = i, e, (8) where the matrices M j = M j (x), j = i, e, represent the conductivities in the intraand extracellular media. hese media have preferred directions of conductivity, which is because the cardiac cells are long and thin with a specific direction of alignment. he conductivity matrices are of the form ( ) M j = σ j t I + σ j l σj t a(x)a(x), j = i, e, (9) where I denotes the identity matrix, σ j l and σ j t, j = i, e, are the conductivity coefficients respectively along and across the cardiac fibers for the intracellular (j = i), extracellular (j = e) media, which are assumed to be the positive constants, while a = a(x) is the unit vector tangent to the fibers at a point x. he conductivity is assumed to be greater along than across the fibers, that is, σ j l > σj t, j = i, e. he matrices M j, j = i, e, are symmetric and positive definite, and possess two different positive eigenvalues σ j l,t. he multiplicity of σj l is 1, while it is 2 for σ i,e t. he conductivity of the composite medium is characterized by M := M i + M e. By the law of current conservation we have J i + J e =. (1) he divergence currents in (1) go between the intra- and extracellular media, and are thus crossing the membrane. Hence they must be related to the transmembrane current per unit volume, which we denote by I m, and to the applied stimulation current I app. he transmembrane current I m is most easily expressed in terms of current per unit area of membrane surface. he transmembrane current per unit volume is then obtained by multiplying I m with a scaling factor χ, which is the membrane surface area per unit volume tissue. Since the currents fields can be considered quasi-static, we thus obtain from (1) J i = χi m + I app, J e = χi m I app. (11) As a primary unknown we introduce the transmembrane potential v, which is defined as the difference between the intra- and extracellular potentials: v = u i u e. Now the next step is express the membrane current I m in terms of the unknown v. o this end, we need a model describing the electrical properties of the cell

5 DEGENERAE REACION-DIFFUSION SYSEMS 5 membrane. he model that we adopt here resides in representing the membrane by a capacitor and passive resistor in parallel. We recall that a capacitor is defined by q = c m v, (12) where q and c m denote respectively the amount of charge and the capacitance. he capacitive current, denoted by I c, is the amount of charge that flows per unit time, so by taking derivatives in (12) we bring about I c = t q = c m t v. (13) he transmembrane current I m is the sum of the capacitive current and the transmembrane ionic current, i.e., I m = I c +I ion, where the ionic current I ion is assumed (for simplicity) to depend only the transmembrane potential v. Exploiting (13) we can express the membrane current I m as I m = c m t v + I ion (v). (14) We mention that in [1] (see also [27]) the authors employed the FitzHugh- Nagumo model for the ionic current. he FitzHugh-Nagumo membrane kinetics was introduced first as a simplified version of the membrane model of Hodgkin and Huxley describing the transmission of nervous electric impulses. he ionic current in this model is represented as (see for example [21]) I ion = I ion (v, w) = F (v) + δw, (15) where and F : R R is a cubic polynomial, δ > is a constant, and w is the recovery variable. he recovery variable satisfies a single ODE that depends on v. In this work we assume there is no recovery variable w and the scaling factor χ is set to 1, so that the ionic current can be represented as I ion = I ion (v) = h(v), (16) for some given function h that depends only on the transmembrane potential v. he cell model (I ion ) that we employ herein is simple. Many more advanced models exist, see, e.g., [2, 15, 2, 22, 31]. We refer also to [25] for an overview of many relevant cell models, which consist of systems of ODEs that are coupled to the partial differential equations for the electrical current flow. Finally, combining (16), (14), and (11) we obtain the bidomain system (1). Remark 2.1. here are additional ordinary differential equations governing the evolution of the recovery variable w. In this paper, we focus on the difficulties associated with spatial coupling and assume that the features associated with w are of secondary concern. However, as in [1], we could easily accommodate for the FitzHugh-Nagumo model in our analysis. Remark 2.2. We refer to Subsection 3.2 for precise conditions on the function h in (16). Here it suffices to say that a representative example of h is the cubic polynomial ( h(v) = χ G v 1 v ) ) (1 vvp, v th where we assign the following meanings to the constants χ, G, v th, v p : χ is the ratio of the membrane area per unit tissue, G is the maximum membrane conductance per unit area, and v th, v p are respectively the threshold and plateau values of v.

6 6 MOSAFA BENDAHMANE, KENNEH H. KARLSEN Remark 2.3. he conductivity tensors M j, j = i, e, do not typically depend on time t in the bidomain application, but we have included this dependency in (1) for the sake of generality. he same applies to the (t, x) dependency in h, see (1). Remark 2.4. Although we do not claim any relevance of the nonlinear system (4) when it comes to representing the electrical properties of cardiac tissue, it can be illuminating to observe that (4) can be derived as above by assuming simply that the flows J j are nonlinear functions of the potentials u j (instead of (8)): J j = J j (t, x, u j ), which would correspond to a nonlinear Ohm s law. he bidomain model is based on linear current flows, i.e., the usual Ohm s law J j = M j u j. his law leads to harmonic current flow potentials in which the assumption of linearity simplifies the analysis. Surely, Ohm s law is an approximate empirical law. From the perspective of possible nonlinear models, it is natural to consider power-law currents as the next approximation, i.e., flow vectors of the form J j = u j p 2 M j u j, where p is a constant satisfying p > 1. his means that the magnitudes of the current flows are given by J j = C j u j p 1, for some constants C j. In this case, which yields p-harmonic current flow potentials, the nonlinear function h is a natural generalization of the transmembrane ionic current in the bidomain model. 3. Preliminaries Mathematical preliminaries. he purpose of this subsection is to introduce some notations as well as recalling a few well-known and basic mathematical results. As general books of reference, see [13, 24]. Let be a bounded open subset of R 3 with a smooth (say C 2 ) boundary. For 1 q <, we denote by W 1,q () the Sobolev space of functions u : R for which u L q () and u L q (; R 3 ). We let W 1,q () denote the functions in W 1,q () that vanish on the boundary. For q = 2 we write H 1 () instead of W 1,2 (). If 1 q < and X is a Banach space, then L q (, ; X) denotes the space of measurable function u : (, ) X for which t u(t) X L q (, ). Moreover, C([, ]; X) denotes the space of continuous functions u : [, ] X for which u C([, ];X) := max t (, ) u(t) X is finite. For 1 q <, we denote by q the conjugate exponent of q: q = q q 1. We will use Young s inequality (with ε) frequently: ab εa q + C(ε)b q, C(ε) = 1, a, b, ε >. q (εq) q /q For 1 q < 3, we denote by q the Sobolev conjugate of q, that is q = 3q 3 q. If 3 q <, we take q [q, + ) to be as large as required in the specific context. For u W 1,q () with q [1, ), the Poincaré inequality reads u Lq () { C u Lq (), 1 < q <, C u L3 (), q = 1, (17) for some universal constant C, whereas the Sobolev embeddings read W 1,q () L q () if 1 q < 3, W 1,q () L r () for all r [1, ), if q = 3, W 1,q () L () if 3 < q <. (18)

7 DEGENERAE REACION-DIFFUSION SYSEMS 7 Let H be a Hilbert space equipped with a scalar product (, ) H. Let X be a Banach space such that X H H X and X is dense in H (X denotes the dual of X, etc.). Suppose u L p (, ; X) is such that t u belongs to L p (, ; X ) for some p (1, ). hen u C([, ]; H). Moreover, for every pair (u, v) of such functions we have the integration-by-parts formula (u(t), v(t)) H (u(s), v(s)) H = t s t t u(τ), v(τ) X,X dτ + t v(τ), u(τ) X,X dτ, for all s, t [, ]. Specifically when u = v there holds t u(t) 2 H u(s) 2 H = 2 t u(τ), u(τ) X,X dτ. We will make use of the last two results with X = L p () (p > 1) and H = L 2 (). Next we recall the Aubin-Lions compactness result (see, e.g., [19]). Let X be a Banach space, and let X, X 1 be separable and reflexive Banach spaces. Suppose X X X 1, with a compact embedding of X into X. Let {u n } n 1 be a sequence that is bounded in L α (, ; X ) and for which { t u n } n 1 is bounded in L β (, ; X 1 ), with 1 < α, β <. hen {u n } n 1 is precompact in L α (, ; X). Let us also recall the following well-known compactness result (see, e.g., [28]): Let X Y Z be Banach spaces, with a compact embedding of X into Y. Let {u n } n 1 be a sequence that is bounded in L (, ; X) and equicontinuous as Z-valued distributions. hen the sequence {u n } n 1 is precompact in C([, ]; Y ) Assumptions. In this subsection we intend to provide precise conditions on the data of our problems, which are all posed in a physical domain that is a bounded open subset of R 3 with smooth boundary. Recall that the bidomain system (1) results if specify M j (t, x, ξ) = M j (t, x)ξ in the nonlinear system (4). herefore the conditions stated next for the vector fields M j (t, x, ξ) cover also the bidomain system Conditions on the diffusive vector fields M j (t, x, ξ). Let 1 < p < +. We assume M j = M j (t, x, ξ) : R 3 R 3, j = i, e, are functions that are measurable in (t, x) for each ξ R 3 and continuous in ξ R 3 for a.e. (t, x), i.e., M i, M e are vector-valued Carathéodory functions. For j = i, e our basic requirements are ) M j (t, x, ξ) C M ( ξ p 1 + f 1 (t, x), (19) s s (M j (t, x, ξ) M j (t, x, ξ )) (ξ ξ ) ξ ξ p, if p 2 C M ξ ξ 2 ( ξ + ξ 2 p, if 1 < p < 2, (2) ) M j (t, x, ξ) ξ C M ξ p, (21) for a.e. (t, x), ξ, ξ R 3, and with C M being a positive constant and f 1 belonging to L p ( ). Moreover, we assume there exist Carathéodory functions

8 8 MOSAFA BENDAHMANE, KENNEH H. KARLSEN M j (t, x, ξ) : R 3 R, j = i, e, such that for a.e. (t, x) and ξ R 3 ξ l M j (t, x, ξ) = M j,l (t, x, ξ), l = 1, 2, 3, (22) t M j (t, x, ξ) K 1 M j (t, x, ξ) + f 2, (23) for some constant K 1 and function f 2 L 1 ( ). Remark 3.1. ypical examples of vector fields M j that satisfy conditions (19)-(21) are the p-laplace type operators in (5). Concerning (5), the vector fields M j (t, x, ξ) satisfying (22) are given by 1 p ξ p M j (t, x), and they satisfy (23) trivially if the matrices M j are independent of time t (the representative case). Remark 3.2. Referring to the bidomain model and the above discussion we perceive that conditions (19)-(21) are satisfied with M j = M j (t, x)ξ, p = 2 provided M j L ( ; R N N ), j = i, e, M j (t, x)ξ ξ C M ξ 2, for a.e. (t, x) and ξ R 3, j = i, e Conditions on the ionic current h(t, x, v). We assume h : R R is a Carathéodory function. For 1 < p <, we assume there exist constants C h, K 2 > such that h(t, x, v 1 ) h(t, x, v 2 ) h(t, x, ) =, C h, v 1 v 2, (24) v 1 v 2 t H(t, x, v) K 2 H(t, x, v) + f 3, H(t, x, v) = v h(t, x, ρ) dρ, (25) for a.e. (t, x) and for some function f 3 L 1 ( ). We assume additionally that there is a constant C h > such that (t, x) h(t, x, v) < lim inf lim sup v v 3(p 1) 3 p v < lim inf v h(t, x, v) v q lim sup v h(t, x, ) Lip loc (R), if p > 3. h(t, x, v) C h, if 1 < p < 3, v 3(p 1) 3 p h(t, x, v) v q C h, q 1, if p = 3, Remark 3.3. One should be aware that condition (25) is trivially satisfied when h is independent of time t, which is the representative case for the bidomain model. Remark 3.4. A consequence of (24) and (26) is that for a.e. (t, x) and v R there holds ( ) C v 3(p 1) 3 p h(t, x, v) C v 3(p 1) 3 p + 1, if 1 < p < 3, (27) and C v q h(t, x, v) C ( v q + 1), q 1, if p = 3, (28) for some constants C, C, C >. Remark 3.5. A fact that will be used several times in this paper is (26) (h(t, x, v 1 ) h(t, x, v 2 )) (v 1 v 2 ) + C h (v 1 v 2 ) 2, (29) v 1, v 2 R and for a.e. (t, x). his inequality is an outcome of (24). Remark 3.6. In the fully nonlinear case (p > 1 with p 2), condition (26) is used to prove strong L p convergence of the gradients of the approximate solutions, which is needed in the existence proof, see in particular Section 7.

9 DEGENERAE REACION-DIFFUSION SYSEMS A basis for the Faedo-Galerkin method. Later on we use the Faedo- Galerkin method to prove existence of solutions. For that purpose we need a basis. he material presented in this subsection is standard, and we have included it just for the sake of completeness. Let q > be such that q < p = 3p 3 p and s N satisfy s > 5 2.hen W s,2 () W 1,p () L q () (W s,2 ()), with continuous and dense inclusions. We denote by W s,2 () the higher order Sobolev space { u, D α u L 2 (), α s, u = on }. In particular, the inclusion W 1,p () L q () is compact. he Aubin-Lions compactness criterion says that the inclusion W L p (, ; L q ()) is compact, { ( where W = u L p (, ; W 1,p ()) : t u L p, ; (W s,2 ()) )}. Consider the following spectral problem: Find w W s,2 () and a number λ such that { (w, φ)w s,2 () = λ(w, φ) L 2 (), φ W s,2 (), (3) w =, on, where (, ) W s,2 () and (, ) L 2 () denote the inner products of W s,2 () and L 2 () respectively. By the Riesz representation theorem there is a unique Θe such that s,2 Φ(e) := (e, φ) L 2 () = (Θe, φ) W s,2 (), φ W (). Clearly, the operator L 2 () e Θe L 2 () is linear, symmetric, bounded, and compact. Moreover, Θ is positive since (e, Θe) L 2 () = (Θe, Θe) W s,2 (), Hence, problem (3) posseses a sequence of positive eigenvalues {λ l } and the corresponding eigenfunctions form a sequence {e l } s,2 that is orthogonal in W () and orthonormal in L 2 (), see, e.g., [24, p.267]. 4. Statement of main results. In this section we define what we mean by weak solutions of the bidomain system (1) and the nonlinear system (4), starting with the former model. We also supply our main existence results. Definition 4.1 (Bidomain model). A weak solution of (1), (2), (3) is a triple of functions u i, u e, v L 2 (, ; H 1 ()) with v = u i u e such that t v belongs to L (, 2, ( H 1 () ) ), v() = v a.e. in, and c m t v, ϕ i dt + M i (t, x) u i ϕ i dx dt Q (31) + h(t, x, v)ϕ i dx dt = I app ϕ i dx dt, c m t v, ϕ e dt M e (t, x) u e ϕ e dx dt Q (32) + h(t, x, v)ϕ e dx dt = I app ϕ e dx dt, for all ϕ j L 2 (, ; H 1 ()), j = i, e. Here,, denotes the duality pairing between H 1 () and (H 1 ()).

10 1 MOSAFA BENDAHMANE, KENNEH H. KARLSEN Remark 4.1. In view of (26) with p = 2 and Sobolev s embedding theorem (the latter tells us that H 1 () L 6 ()), we conclude h(t, x, v) L 2 ( ) and thus h(t, x, v)ϕ j dx dt, j = i, e, are well-defined integrals. Moreover, consult Subsection 3.1, it follows from Definition 4.1 that v C([, ]; L 2 ()), and thus the initial condition (3) is valid. heorem 4.1 (Bidomain model, p = 2). Assume conditions (19)-(26) hold with p = 2. If v L 2 () and I app L 2 ( ), then the bidomain problem (1), (2), (3) possesses a unique weak solution. If v = u i, u e, with u i,, u e, H 1 () and I app L 2 ( ), then this weak solution obeys t v L 2 ( ). Definition 4.2 (Nonlinear model, p > 1 with p 2). A weak solution of (4), (2), (3) is a triple of functions u i, u e, v L p (, ; W 1,p ()) with v = u i u e such that t v L p, ; (W 1,p ()) ), v() = v a.e. in, and c m t v, ϕ i dt + M i (t, x, u i ) ϕ i dx dt Q (33) + h(t, x, v)ϕ i dx dt = I app ϕ i dx dt, c m t v, ϕ e dt M e (t, x, u e ) ϕ e dx dt Q (34) + h(t, x, v)ϕ e dx dt = I app ϕ e dx dt, for all ϕ j L p (, ; W 1,p ()), j = i, e. Here,, denotes the duality pairing between W 1,p () and (W 1,p ()). Remark 4.2. Due to (26) with p 2, the equality 3(p 1) 3 p p = p for 1 < p < 3, and (18), it is clear that the function h(t, x, v) belongs to L p ( ), and thus the integrals h(t, x, v)ϕ j dx dt, j = i, e, are well-defined. Moreover, by Definition 4.2, there holds v C([, ]; L 2 ()). Consequently, (3) has a meaning. heorem 4.2 (Nonlinear model, p > 1 with p 2). Assume conditions (19)-(26) hold. If v L 2 () and I app L 2 ( ), then the nonlinear problem (4), (2), (3) possseses a unique weak solution. If v = u i, u e, with u i,, u e, W 1,p () and I app L 2 ( ), then this weak solution obeys t v L 2 ( ). Now we are ready to embark on the proofs of heorem 4.1 and Existence of solutions for the approximate problems. his section is devoted to proving existence of solutions to the approximate problems (6), (2), (7) introduced and discussed in the introduction. he existence proof is based on the Faedo-Galerkin method, a priori estimates, and the compactness method. Definition 5.1 (Approximate problems). A solution of problem (6), (2), (7) is a triple of functions u i, u e, v L p (, ; W 1,p ()) with v = u i u e such that t u j L 2 ( ), u j () = u j, a.e. in, for j = i, e, and c m t vϕ i dx dt + ε t u i ϕ i dx dt + M i (t, x, u i ) ϕ i dx dt Q (35) + h(t, x, v)ϕ i dx dt = I app ϕ i dx dt,

11 DEGENERAE REACION-DIFFUSION SYSEMS 11 c m t vϕ e dx dt ε t u e ϕ e dx dt M e (t, x, u e ) ϕ e dx dt Q + h(t, x, v)ϕ e dx dt = I app ϕ e dx dt, for all ϕ j L p (, ; W 1,p ()), j = i, e. Remark 5.1. Cosmetically speaking, we have chosen to let Definition 5.1 cover both the bidomain case p = 2 and the nonlinear case p > 1 with p 2. Although in this section we keep the same notation for the two cases, we will at various places in the presentation that follows employ different proofs. Supplied with the basis {e l } + introduced in Subsection 3.3, we look for finite dimensional approximate solutions to the regularized problem (6), (2), (7) as sequences {u i,n } n>1, {u e,n } n>1, {v n } n>1 defined for t and x by and u i,n (t, x) = v n (t, x) = n c i,n,l (t)e l (x), u e,n (t, x) = (36) n c e,n,l (t)e l (x), (37) n d n,l (t)e l (x), d n,l (t) := c i,n,l (t) c e,n,l (t). (38) he goal is to determine the coefficients {c i,n,l (t)} n, {c e,n,l(t)} n, {d n,l(t)} n such that for k = 1,..., n (c m t v n, e k ) L2 () + (ε tu i,n, e k ) L 2 () + M i (t, x, u i,n ) e k dx + (c m t v n, e k ) L 2 () (ε tu e,n, e k ) L 2 () M e (t, x, u e,n ) e k dx + and, with reference to the initial conditions (7), u i,n (, x) = u,i,n (x) := u e,n (, x) = u,e,n (x) := v n (, x) = v,n (x) := h(t, x, v)e k dx = h(t, x, v n )e k dx = I app,n e k dx, I app,n e k dx, n c i,n,l ()e l (x), c i,n,l () := (u i,, e l ) L2 (), n c e,n,l ()e l (x), c e,n,l () := (u e,, e l ) L2 (), n d n,l ()e l (x), d n,l () := c i,n,l () c e,n,l (), ln (39), we have used a finite dimensional approximation of I app : I app,n (t, x) = n (I app, e l ) L2 () (t)e l(x). By our choice of basis, u i,n and u e,n satisfy the Dirichlet boundary condition (2). With I app L 2 ( ) and u,j W 1,p (), it is clear that, as n, I app,n I app in L 2 ( ) and u,j,n u,j in W 1,p (), for j = i, e. (39) (4)

12 12 MOSAFA BENDAHMANE, KENNEH H. KARLSEN Using the orthonormality of the basis, we can write (39) more explicitly as a system of ordinary differential equations: c m d n,k(t) + εc i,n,k(t) + M i (t, x, u i,n ) e k dx + h(t, x, v n )e k dx = I app,n e k dx, c m d n,k(t) εc e,n,k(t) M e (t, x, u e,n ) e k dx + h(t, x, v n )e k dx = I app,n e k dx. (41) Adding together the two equations in (41) yields for k = 1,..., n (2c m + ε) d n,k(t) = (M e (t, x, u e,n ) M i (t, x, u i,n )) e k dx 2 h(t, x, v n )e k dx + 2 I app,n e k dx (42) =:F k ( t, {d n,l } n, {c i,n,l} n, {c e,n,l} n ). Plugging the equation (42) for d n,k (t) back into (41), we find for k = 1,..., n εc i,n,k(t) = c m 2c m + ε F k ( t, {d n,l } n, {c i,n,l} n, {c e,n,l} n ) M i (t, x, u i,n ) e k dx h(t, x, v n )e k dx + =: Fi k ( t, {dn,l } n, {c i,n,l} n, {c e,n,l} n ) I app,n e k dx (43) and εc e,n,k(t) = c m 2c m + ε F k ( t, {d n,l } n, {c i,n,l} n, {c e,n,l} n ) M e (t, x, u e,n ) e l dx + h(t, x, v n )e k dx I app,n e k dx =: Fe k ( t, {dn,l } n, {c i,n,l} n, {c e,n,l} n ). (44) he next step is to prove existence of a local solution to the ODE system (42), (43), (44), (4). o this end, let ρ (, ) and set U = [, ρ]. We choose r > so large that the ball B r R 3n contains the three vectors {d n,l ()} n, {c i,n,l()} n, {c e,n,l ()} n, and then we set V := B r. We also set F = { F k} n k=1, F i = { F k i } n k=1, and F e = { F k e } n k=1. hanks to our assumptions (19)-(26) the functions F, F j : U V R n, j = i, e, are Carathéodory functions. Moreover, the components of F

13 DEGENERAE REACION-DIFFUSION SYSEMS 13 and F j can be estimated on U V as follows: F k ( t, {d n,l } n, {c i,n,l} n, {c e,n,l} n ) 2 I app,n L2 () e k L2 () + ( ) n p M j t, x, c j,n,l e l + 2 ( ) n p h t, x, d n,l e l 1/p ( dx 1/p ( ) 1/p e k p ) 1/p e k p (45) and for j = i, e ( F k j t, {dn,l } n, {c i,n,l} n, {c e,n,l} n ) [ c m 2 I app,n 2c m + ε L 2 () e k L2 () + ( n M j t, x, ( n M j t, x, ) p c j,n,l e l ( ) n p h t, x, d n,l e l ) p c j,n,l e l ( ) n p h t, x, d n,l e l dx 1/p ( 1/p ( 1/p ( dx 1/p ( ) ] 1/p e k p ) 1/p e k p ) 1/p e k p ) 1/p e k p + I app,n L2 () e k L 2 (). (46) In view of (19)-(26) and (18), we can uniformly (on U V ) bound (45) and (46): F k ( t, {d n,l } n, {c i,n,l} n, {c e,n,l} n ) C(r, n)m(t), (47) F k j ( t, {dn,l } n, {c i,n,l} n, {c e,n,l} n ) Cj (r, n)m j (t), j = i, e, (48) where C(r, n), C j (r, n) are constants that depend on r, n and M(t), M j (t) are L 1 (U) functions that are independent of k, n, r. Hence, according to standard ODE theory, there exist absolutely continuous functions {d n,l } n, {c i,n,l} n, {c e,n,l} n satisfying (42), (43), (44), (4) for a.e. t [, ρ ) for some ρ >. Moreover, the following equations hold on [, ρ ): d n,l (t) = d n,l () + 1 2c m + ε t F l ( τ, {d n,k (τ)} n k=1, {c i,n,k(τ)} n k=1, {c e,n,k(τ)} n k=1 ) dτ (49)

14 14 MOSAFA BENDAHMANE, KENNEH H. KARLSEN and for j = i, e c j,n,l (t) = c j,n,l () + 1 ε t Fj l ( τ, {dn,k (τ)} n k=1, {c i,n,k(τ)} n k=1, {c e,n,k(τ)} n k=1) dτ. (5) o summarize our findings so far, on [, ρ ) the functions u i,n, u e,n, v n defined by (37) and (38) are well-defined and constitute our approximate solutions to the regularized system (6) with data (2), (7). o prove global existence of the Faedo-Galerkin solutions we derive n-independent a priori estimates bounding v n, u i,n, u e,n in various Banach spaces. Given some (absolutely continuous) coefficients b j,n,l (t), j = i, e, we form the functions ϕ i,n (t, x) := n b i,n,l(t)e l (x) and ϕ e,n (t, x) := n b e,n,l(t)e l (x). From (41) the Faedo-Galerkin solutions satisfy the following weak formulations for each fixed t, which will be the starting point for deriving a series of a priori esitmates: c m t v n ϕ i,n dx + ε t u i,n ϕ i,n dx + M i (t, x, u i,n ) ϕ i,n dx + h(t, x, v n )ϕ i,n dx (51) = I app,n ϕ i,n dx, c m t v n ϕ e,n dx ε t u e,n ϕ e,n dx M e (t, x, u e,n ) ϕ e,n dx + = I app,n ϕ e,n dx. h(t, x, v n )ϕ e,n dx Remark 5.2. From (51) until (69), we will intentionally commit a notational crime by reserving the letter for an arbitrary time in the existence interval [, ρ ) for the Faedo-Galerkin solutions (and not the final time used elsewhere). Lemma 5.1. Assume conditions (19)-(26) hold and p > 1. If u i,, u e, L 2 () and I app L 2 ( ), then there exist constants c 1, c 2, c 3 not depending on n such that v n L (, ;L 2 ()) + εu L j,n (, ;L 2 ()) c 1, (53) (52) u j,n Lp ( ) c 2, (54) u j,n Lp ( ) c 3. (55) If, in addition, u i,, u e, W 1,p (), then there exists a constant c 4 > not depending on n such that t v n L 2 ( ) + ε t u L j,n 2 ( ) c 4. (56)

15 DEGENERAE REACION-DIFFUSION SYSEMS 15 Proof. Substituting ϕ i,n = u i,n and ϕ e,n = u e,n in (51) and (52), respectively, and then summing the resulting equations, we procure the equation c m d v n 2 dx + ε d u j,n 2 dx 2 dt 2 dt + M j (t, x, u j,n ) u j,n dx + h(t, x, v n )v n dx (57) = I app,n v n dx. By Young s inequality, there exist constants C 1, C 2 > independent of n such that I app,n v n dx dt C 1 + C 2 v n 2 dx dt. (58) Integrating (57) over (, ) and then exploiting (58) and also (21), (24), we obtain c m v n (, x) 2 dx + ε u j (, x) 2 dx 2 2 ( + C M u j,n p dx dt + h(t, x, v n )v n + C h v n 2) dx dt C 1 + (C 2 + C h ) v n 2 dx dt (59) + c m v (x) 2 dx + ε u j, (x) 2 dx 2 2 C 1 + (C 2 + C h ) v n 2 dx dt. In view of (29) and Gronwall s inequality, it follows from (59) that v n (, x) 2 dx + ε u j (, x) 2 dx C 3, (6) for some constant C 3 > independent of n, which proves (53). From (59) and (6) we also conclude that C M u j,n p dx dt C 1 + (C h + C 2 ) C 3, ( h(t, x, v n )v n + C h v n 2) dx dt C 1 + (C h + C 2 ) C 3, where the first estimate proves assertion (54). he Poincaré inequality implies the existence of a constant C 4 > independent of n such that for each fixed t u j,n (t, ) Lp () C 4 u j,n (t, ) Lp (), 1 < p <, j = i, e, and therefore, by (61) his concludes the proof of (55). (61) u j,n (t, ) p L p () dt C 5 for 1 < p < and j = i, e. (62)

16 16 MOSAFA BENDAHMANE, KENNEH H. KARLSEN Now we turn to the proof of (56), and start by reminding the reader of the functions M j and H defined respectively in (22) and (25). We substitute ϕ i,n (t, ) = t u i,n (t, ) in (51) and ϕ e,n (t, ) = t u e,n (t, ) in (52), and sum the resulting equations to bring about an equation that is integrated over (, ). he final outcome reads t v n 2 dx dt + ε t u j,n 2 dx dt + M j (t, x, u j,n ) ( t u j,n ) dx dt + h(t, x, v n ) t v n dx dt = t v n 2 dx dt + ε t u j,n 2 dx dt + t M j (t, x, u j,n ) + H(t, x, v n ) dx dt t M j (t, x, u j,n ) + t H(t, x, v n ) dx dt = I app,n t v n dx dt 1 t v n 2 dx dt + C 6, 2 where we have used Young s inequality and the uniform L 2 boundedness of I app,n to derive the last inequality. aking into account (23) and (25) in (63), we conclude that there exist two constants C 7, C 8 > independent of n such that 1 2 t v n 2 dx dt + ε t u j,n 2 dx dt M j (, x, u j,n (, x)) + H(, x, v n (, x)) dx + C 7 + M j (t, x, u j,n ) + H(t, x, v n ) dx dt M j (, x, u j,n (, x)) + H(, x, v n (, x)) dx + C 8. o deal with the H(, x, v n (, x))-term, observe that the following bounds are consequences of (24) and (26): for a.e. (t, x) and v R. ( ) H(t, x, v) C 9 v 2p 3 p + 1, if 1 < p < 3, ) H(t, x, v) C 9 ( v q+1 + 1, q 1, if p = 3, (63) (64) (65)

17 DEGENERAE REACION-DIFFUSION SYSEMS 17 By definitions of M j and H, (19), u i,, u e, W 1,p (), (65) and (26) for p > 3, and (18), we deduce M j (, x, u j,n (, x)) + H(, x, v n (, x)) dx C 1, for some constant C 1 > independent of n. By the monotonicity conditions (21) and (24), M j (, x, u j,n (, x)) dx (66) and H(, x, v n (, x)) dx + C h vn v n (, x) 2 dx ( ) h(, x, ρ) + C h ρ dρ dx. Using (66) and (67) in (64) we obtain M j (, x, u j,n (, x)) + H(, x, v n (, x)) + C h v n (, x) 2 dx C 7 + C h M j (t, x, u j,n ) + H(t, x, v n ) + C h v n 2 dx dt v n (, x) 2 dx + C 1, C 1 = C 8 + C 1. (67) (68) Now (68), (53), and an application of Gronwall s lemma in (68) furnish M j (, x, u j,n (, x)) dx + H(, x, v n (, x)) dx C 11, (69) for some constant C 11 > independent of n. Finally, combining (66), (67), (69) in (64) delivers (56). We want to show that the local solution constructed above can be extended to the whole time interval [, ) (independently of n). o this end, observe that for an arbitrary t in the existence interval [, ρ ) there holds, thanks to (53), {d n,l (t)},...,n 2 R n + {c j,n,l (t)},...,n 2 = v n (t, ) L2 () + (7) u j,n (t, ) L2 () C, where C > is a constant independent of t and n. We continue by introducing S := {t [, ) : there exist a solution of (39), (4) on [, t)}, and observing that S is nonempty due to the above local existence result. We claim that S is an open set. o see this, let t S and < t 1 < t 2 < t. In view of (49), (47) and (5), (48) we then obtain for l = 1,..., n d n,l (t 1 ) d n,l (t 2 ) c(c, n, c m, ε) t2 R n t 1 M(τ) dτ (71)

18 18 MOSAFA BENDAHMANE, KENNEH H. KARLSEN and t2 c j,n,l (t 1 ) c j,n,l (t 2 ) c(c, n, c m, ε) M j (τ) dτ, j = i, e. (72) t 1 Since M, M j L 1, j = i, e, we use (71) and (72) to conclude respectively that t d n,l (t) and t c j,n,l (t), j = i, e, are uniformly continuous. At time t, we solve the ODE system (42), (43), (44) with initial data lim (d n,l(t), c i,n,l (t), c e,n,l (t)), l = 1,..., n, t t which provides us with a solution on [, t + ε) for some ε = ε ( t) >, and thus S is open. It remains to prove that S is closed. We consider a sequence {t l } l>1 S { } n such that t l t as l. Let (d l n,l (t), cl i,n,l (t), cl e,n,l (t)) denote the solution of (42), (43), (44), (4) on [, t l ), and define for l = 1,..., n { d l d l n,l n,l(t) = (t), if t [, t l), d l n,l (t l), if t [t l, t ), and for j = i, e c l j,n,l(t) = { c l j,n,l (t), if t [, t l), c l j,n,l (t l), if t [t l, t ). It follows from what we have said before that the sequences { dl n,l (t)}, l { c j,n,l (t) }, j = i, e, l = 1,..., n, l>1 l>1 are equibounded and equicontinuous on [, t). Hence there exist subsequences that converge uniformly on [, t) to continuous functions d n,k (t) and c j,n,l (t), j = i, e. By (49), (5), and Lebesgue s dominated convergence theorem, it is easy to see that these functions must solve the ODE system (42), (43), (44), (4) on [, t). Hence t S, and we infer that S is closed. Consequently, S = [, ). Having proved that the Faedo-Galerkin solutions (37), (38) are well-defined, we are now ready to prove existence of solutions to our nondegenerate system (6). heorem 5.1 (Regularized system). Assume (19)-(26) hold and p > 1. If u j, W 1,p (), j = i, e, and I app L 2 ( ), then the regularized system (6)-(2)-(7) possesses a solution for each fixed ε >. he remaining part of this section is devoted to proving heorem 5.1. Lemma 5.1 shows that {v n } n>1, {u j,n } n>1, j = i, e, are bounded in L p (, ; W 1,p ()) and { t v n } n>1, { t u j,n } n>1, j = i, e, are bounded in L 2 ( ). herefore, possibly at the cost of extracting subsequences, which we do not bother to relabel, we can assume there exist limit functions u i, u e, v with v = u i u e such that as n u j,n u j a.e. in, strongly in L 2 ( ), and weakly in L p (, ; W 1,p ()), v n v a.e. in, strongly in L 2 ( ), and weakly in L p (, ; W 1,p (73) ()), M j (t, x, u j,n ) Σ j weakly in L p ( ; R 3 ), h(t, x, v n ) h(t, x, v) a.e. in and weakly in L p ( ). Lemma 5.2. As n, h(t, x, v n ) h(t, x, v) strongly in L q ( ) q [1, p ).

19 DEGENERAE REACION-DIFFUSION SYSEMS 19 Proof. Because of (54), (55), (18), and Remarks 4.1 and 4.2, {h(t, x, v n )} n>1 is bounded in L p ( ). he lemma is then a consequence of (73) and Vitali s theorem. Keeping in mind (73) and Lemma 5.2 we infer, by integrating (51) and (52) over (, ) and then letting n, c m t vϕ i dx dt + ε t u i ϕ i dx dt Q + Σ i ϕ i dx dt + h(t, x, v)ϕ i dx dt Q = I app ϕ i dx dt, Q c m t vϕ e dx dt ε t u e ϕ e dx dt Q Σ e ϕ e dx dt + h(t, x, v)ϕ e dx dt Q = I app ϕ e dx dt, (74) (75) for any ϕ j L p (, ; W 1,p ()), j = i, e. o conclude that the limit functions in (73) satisfy the weak form of (6), we need to identify Σ j (t, x) as M j (t, x, u j ), which boils down to proving strong convergence in L p of the gradients u j,n. We remark that in the case p = 2 (i.e., M j (t, x, ξ) = M j (t, x)ξ) we do not need strong convergence of the gradients, so Lemma 5.3 below is needed only in the fully nonlinear case (p > 1 with p 2). Lemma 5.3. For j = i, e, u j,n u j strongly in L p ( ) as n and Σ j (t, x) = M j (t, x, u j ) for a.e. (t, x) and in L p ( ; R 3 ). Proof. Fixing an integer N 1, we consider functions w j = w j (t, x) of the form N w j (t, x) = a j,l (t)e l (x), j = i, e, (76) where {a j,l } N are given C1 ([, ]) functions and {e l } is the basis introduced in Subsection 3.3. We also set w := w i w e. Assuming that n N, we add together (51) with ϕ i (t, ) = (u i,n w i )(t, ) and (52) with ϕ e (t, ) = (u e,n w e )(t, ).

20 2 MOSAFA BENDAHMANE, KENNEH H. KARLSEN Integrating the resulting equation over (, ) and then adding it to (24) we get (M j (t, x, u j,n ) M j (t, x, w j )) ( u j,n w j ) dx dt = c m t v n (v n w) dx dt ε t u j,n (u j,n w j ) dx dt M j (t, x, w j ) ( u j,n w j ) dx dt [ (h(v n ) h(w))(v n w) + C h v n w 2] dx dt Q h(w)(v n w) dx dt + C h v n w 2 dx dt Q (77) + I app,n (v n w) dx dt Q c m t v n (v n w) dx dt ε t u j,n (u j,n w j ) dx dt M j (t, x, w j ) ( u j,n w j ) dx dt h(w)(v n w) dx dt + C h v n w 2 dx dt Q + I app,n (v n w) dx dt =: E 1 + E 2 + E 3 + E 4 + E 5 + E 6. By Lemma 5.1 and (73), we draw the conclusions that lim E 1 = c m t v(v w) dx dt, n lim E 2 = ε t u j (u j w j ) dx dt. n From (19), (26), (18), and (73), it follows that M j (t, x, w j ) L p ( ; R 3 ), j = i, e, h(w) L p ( ), and thus lim E 3 = M j (t, x, w j ) ( u j w j ) dx dt, n lim E 4 = h(w)(v w) dx dt. n he term E 5 is sorted out using the convergence v n v in L 2 ( ), cf. (73): lim E 5 = C h v w 2 dx dt. n Bringing to mind that {I app,n } n>1 is bounded in L 2 ( ) and exploiting again the convergence v n v in L 2 ( ), we deduce lim E 6 = I app,n (v w) dx dt. n

21 DEGENERAE REACION-DIFFUSION SYSEMS 21 Now we can pass to the limit in (77) to obtain, keeping in mind (2), (M j (t, x, u j,n ) M j (t, x, u j )) ( u j,n w j ) dx dt lim n c m t v(v w) dx dt ε t u j,n (u j w j ) dx dt M j (t, x, w j ) ( u j w j ) dx dt h(w)(v w) dx dt + C h v w 2 dx dt Q + I app,n (v w) dx dt. Since functions of the form (76) are dense in L p (, ; W 1,p ()), inequality (78) holds in fact for all functions w j L p (, ; W 1,p ()). Hence, choosing w j = u j in (78) gives us lim E j (n), where n E j (n) := (M j (t, x, u j,n ) M j (t, x, u j )) ( u j,n u j ) dx dt. When p 2, by (2) we have C M u j,n u j p dx dt E j (n). (8) When 1 < p < 2, we employ (2) as follows: C M u j,n u j p dx dt C M C M u j,n u j 2 2 p dx dt ( u j,n + u j ) p 2 E j (n) C M p 2 ( u j,n + u j ) p dx dt 2 p 2 ( u j,n + u j ) p dx dt Since u j,n is bounded in L p (, ; W 1,p ()) for j = i, e and using that E j(n) as n. Hence, sending n in (8) and (81) yields lim n which proves the first part of the lemma. 2 p 2. (78) (79) (81) u j,n u j p dx dt =, 1 < p <, (82)

22 22 MOSAFA BENDAHMANE, KENNEH H. KARLSEN In view of (82), along subsequences the following convergences hold: u j,n u j a.e. in, j = i, e. Hence, Σ j (t, x) = M j (t, x, u j ) a.e. in and also in L p ( ). his concludes the proof of the lemma. Finally, we prove that the limits u i, u e in (73) obey the initial data (7). Lemma 5.4. For j = i, e, there holds u j (, x) = u j, (x) for a.e. x. Proof. he proof adapts a standard argument given in [13]. Pick a test function ϕ e of the form (76) with ϕ e (, ) =. We use ϕ e (t, ) in (52) and then integrate with respect to t (, ). In the resulting equation we send n, followed by an integration by parts in the obtained limit equation, thereby obtaining c m v t ϕ e dx dt + εu e t ϕ e dx dt Q M e (t, x, u e ) ϕ e dx dt + h(t, x, v)ϕ e dx dt (83) Q = I app ϕ e dx dt + c m v(, x)ϕ e (, x) dx εu e (, x)ϕ e (, x) dx. On the other hand, integration by parts in (52) yields c m v n t ϕ e dx dt + εu e,n t ϕ e dx dt Q M e (t, x, u e,n ) ϕ e dx dt + h(t, x, v n )ϕ e dx dt Q = I app,n ϕ e dx dt + c m v n (, x)ϕ e (, x) dx εu e,n (, x)ϕ e (, x) dx, (84) for all ϕ e of the form (76) with ϕ e (, ) =. Since by construction u j,n (, ) u j, ( ) in W 1,p () for j = i, e and in view of the convergences established for the approximate solutions, sending n in (84) delivers c m v n t ϕ e dx dt + εu e,n t ϕ e dx dt Q M e (t, x, u e,n ) ϕ e dx dt + h(t, x, v n )ϕ e dx dt (85) Q = I app,n ϕ e dx dt + c m v (x)ϕ e (, x) dx εu e, (x)ϕ e (, x) dx, for all ϕ e of the form (76) with ϕ e (, ) =. Comparing (83) and (85), using also that functions of the form (76) are dense in L p (, ; W 1,p ()), yields u e (, x) = u e, (x) for a.e. x. Reasoning along the same lines for u i yields u i (, x) = u i, (x) for a.e. x. 6. Existence of solutions for the bidomain model Proof of heorem 4.1.

23 DEGENERAE REACION-DIFFUSION SYSEMS he case v = u i, u e, with u i,, u e, H 1 (). From the previous section we know there exist sequences {u i,ε } ε>, {u e,ε } ε>, and {v ε = u i,ε u e,ε } ε> of solutions to (6), (2), (7), cf. Definition 5.1 (with p = 2). Furthermore, we have immediately at our disposal a series of a priori estimates, which we collect in a lemma. Lemma 6.1. Assume conditions (19)-(26) hold with p = 2. If u i,, u e, L 2 () and I app L 2 ( ), then there exist constants c 1, c 2, c 3 not depending on ε such that v ε L (, ;L 2 ()) + εu L j,ε (, ;L 2 ()) c 1, u j,ε L2 ( ) c 2, u j,ε L2 ( ) c 3, j = i, e. If, in addition, u i,, u e, H 1 (), then there exists a constant c 4 > independent of ε such that t v ε L2 ( ) + ε t u L j,ε 2 ( ) c 4. (86) Proof. By the (weak) lower semicontinuity properties of norms, the estimates in Lemma 5.1 hold with v n, u i,n, u e,n replaced by v ε, u i,ε, u e,ε, respectively. Moreover, the constants c 1, c 2, c 3, c 4 are independent of ε (consult the proof of Lemma 5.1). In view of Lemma 6.1, we can assume there exist limit functions u i, u e, v with v = u i u e such that as ε the following convergences hold (modulo extraction of subsequences, which we do not bother to relabel): v ε v a.e. in, strongly in L 2 ( ), and weakly in L 2 (, ; H 1 ()), u i,ε u i weakly in L 2 (, ; H 1 ()), u e,ε u e weakly in L 2 (, ; H 1 ()), h(t, x, v ε ) h(t, x, v) a.e. in and weakly in L 2 ( ), and, according to (86), v C 1/2 ([, ]; L 2 ()). Additionally, t v ε t v and ε t u j,ε, j = i, e, weakly in L 2 ( ). Arguing as in the proof of Lemma 5.2, we conclude also that h(t, x, v ε ) h(t, x, v) strongly in L q ( ) q [1, 2). hanks to all these convergences and repeating the argument from the previous section to prove that the initial condition (3) is satisfied, it is easy to see that the limit triple (u i, u e, v = u i u e ) is a weak solution of the bidomain model (1), (2), (3), cf. Definition 4.1, thereby proving heorem 4.1 in the case v = u i, u e, with u i,, u e, H 1 () he case v L 2 (). o deal with this case, we approximate the initial data v by a sequence {v,ρ } ρ> of functions satisfying v,ρ C (), v,ρ L2 () v L 2 (), v,ρ v in L 2 () as ρ. For ρ >, we then introduce an artificial decomposition v,ρ = u i,,ρ u e,,ρ with u i,,ρ, u e,,ρ C (). From the previous subsection, there exist sequences {u i,ρ } ρ>, {u e,ρ } ρ>, {v ρ = u i,ρ u e,ρ } ρ> for which u i,ρ, u e,ρ L 2 (, ; H 1 ()),

24 24 MOSAFA BENDAHMANE, KENNEH H. KARLSEN t v ρ L 2 ( ), and c m t v ρ ϕ i dx dt + M i (t, x) u i,ρ ϕ i dx dt Q + h(t, x, v ρ )ϕ i dx dt = I app ϕ i dx dt (87) and c m t v ρ ϕ e dx dt M e (t, x) u e,ρ ϕ e dx dt Q + h(t, x, v ρ )ϕ e dx dt = I app ϕ e dx dt, (88) for any ϕ j L 2 (, ; H 1 ()). o pass to the limit ρ in (87) and (88) we need a priori estimates. he ones from Lemma 5.1 that survive the test of being ρ-independent are v ρ L (, ;L 2 ()) c, u j,ρ L2 ( ) c, u j,ρ L2 ( ) c, j = i, e. (89) We conclude from (89) that the sequences {u i,ρ } ρ>, {u e,ρ } ρ>, {v ρ } ρ> are bounded in L 2 (, ; H 1 ()). In view of the equations satisfied by v ρ this implies that { t v ρ } ρ> is bounded in L 2 (, ; (H 1 ()) ), but there are no bounds on { t u i,ρ } ρ>, { t u e,ρ } ρ>! herefore, possibly at the cost of extracting subsequences (which are not relabeled), we can assume that there exist limits u i, u e, v L 2 (, ; H 1 ()) with v = u i u e and t v L 2 (, ; (H 1 ()) ) such that as ρ v ρ v a.e. in, strongly in L 2 ( ), and weakly in L 2 (, ; H 1 ()), u i,ρ u i weakly in L 2 (, ; H 1 ()), u e,ρ u e weakly in L 2 (, ; H 1 ()), h(t, x, v ρ ) h(t, x, v) a.e. in and weakly in L 2 ( ), and v C([, ]; L 2 ()). In addition, t v ρ t v weakly in L 2 (, ; (H 1 ()) ). Arguing as in the proof of Lemma 5.2, we obtain h(t, x, v ρ ) h(t, x, v) strongly in L q ( ) q [1, 2). Equipped with these convergences it is not difficult to pass to the limit as ρ in (87), (88) to conclude that the limit triple (u i, u e, v = u i u e ) is a weak solution to the bidomain model (1), (2), (3). his proves heorem 4.1 in the case v L 2 (). 7. Existence of solutions for the nonlinear model Proof of heorem he case v = u i, u e, with u i,, u e, W 1,p (). In view of the results in Section 5, there exist sequences {u i,ε } ε>, {u e,ε } ε>, and {v ε = u i,ε u e,ε } ε> of solutions to (6), (2), (7), cf. Definition 5.1, and the following weak formulations