The periodic version of the Da Prato–Grisvard theorem and applications to the bidomain equations with FitzHugh–Nagumo transport

In this article, the periodic version of the classical Da Prato–Grisvard theorem on maximal L p -regularity in real interpolation spaces is developed, as well as its extension to semilinear evolution equations. Applying this technique to the bidomain equations subject to ionic transport described by the models of FitzHugh–Nagumo, Aliev–Panfilov, or Rogers– McCulloch, it is proved that this set of equations admits a unique, strong T -periodic solution in a neighborhood of stable equilibrium points provided it is innervated by T -periodic forces.


Introduction
Given a sectorial operator A on a Banach space X, the problem of maximal regularity for the inhomogenous Cauchy problem received a lot of attention since the pioneering articles by Da Prato-Grisvard [10], Dore-Venni [13] and Lunardi [22]. Nowadays, due to a result due to Weis [26], it is well known that for X being a UMD-space and 1 < p < ∞ , maximal L p -regularity of (ACP) can be characterized in terms of the R-boundedness of the resolvent of A . For further information, see also [11,12] and [23].
The aim of this article is twofold: we first study maximal regularity of the inhomogeneous Cauchy problem within the periodic setting as well as for its semilinear counterpart. The results are then applied to the bidomain equations with FitzHugh-Nagumo transport. This is a well-established system of equations describing the electrical activities of the heart, see, e.g., Colli Franzone et al. [7] and the monograph by Keener and Sneyd [21].
Let us start by describing the problem of maximal periodic regularity for the period T = 2 . We say that A admits maximal periodic L p -regularity, if for all f ∈ L p (0, 2 ;X) Eq. (PACP) with T = 2 admits a unique solution u within the class It was proved by Arendt-Bu [4] that, given 1 < p < ∞ , X a UMD-space, and A ∶ D(A) → X a closed operator, A admits maximal periodic L p -regularity if and only if For definitions of UMD-spaces and R-bounded families of operators as well as their properties, we refer, e.g., to [11] or [20].
In this article, we have the following two extensions of this result in mind: The first is, that some situations make it necessary to handle the case p = 1 . However, it is known that if maximal regularity holds for (ACP) for p = 1 , then X is necessarily a non-reflexive space and that contradicts the UMD-property. The second extension we have in mind is to provide an elegant way to treat polynomial nonlinearities. These can be estimated easily if the ground space X is a Banach algebra. The largest Banach algebra within the scale of Sobolev spaces is the Besov space B n∕p p,1 , which is again not reflexive and thus does not enjoy the UMD-property. In the classical maximal regularity theory for (ACP), a result accounting for these issues is the well-known theorem of Da Prato and Grisvard. In fact, let −A be the generator of a bounded analytic semigroup e −tA on a Banach space X with domain D(A) . For ∈ (0, 1) and 1 ≤ p < ∞ , we denote by D A ( , p) the space defined as u ∈ W 1,p (0, 2 ;X) ∩ L p (0, 2 ;D(A)).
(1.1) As counterpart to the Arendt-Bu theorem characterizing maximal periodic L p -regularity for 1 < p < ∞ and X being a UMD-space, we provide a short derivation of a periodic version of Theorem 1.1. As the classical Da Prato-Grisvard theorem, it holds for 1 ≤ p < ∞ and arbitrary Banach spaces X.
To formulate this result, we define the periodicity of a measurable function as follows. For 0 < T < ∞ , we call a measurable function f ∶ ℝ → X periodic of period T if f (t) = f (t + T) holds true for almost all t ∈ (−∞, ∞) . Then, formally, a candidate for a solution u to (PACP) is given by The periodic version of the Da Prato-Grisvard theorem now states that, under certain assumptions on A and f, u is indeed well-defined, continuous, and periodic and satisfies a maximal regularity estimate in the sense of Da Prato-Grisvard. More precisely, let −A be the generator of a bounded analytic semigroup on X with 0 ∈ (A) , let ∈ (0, 1) , 1 ≤ p < ∞ , and 0 < T < ∞ . We then show that there exists a constant C > 0 such that for

and satisfies
Our second aim is to apply the above technique to the bidomain equations with FitzHugh-Nagumo (or related) transport terms as well as to the Allen-Cahn equations. Note that these equations are, however, semilinear evolution equations. Hence, in order to follow the above strategy, we first need to develop a semilinear version of the periodic Da Prato-Grisvard theorem. This will be done in Sect. 3.
The bidomain system is given by subject to the boundary conditions and the initial data Here ⊂ ℝ n denotes a domain describing the myocardium, the functions u i and u e model the intra-and extracellular electric potentials, u ∶= u i − u e denotes the transmembrane potential, and denotes the outward unit normal vector to . The anisotropic properties of the intra-and extracellular tissue parts will be described by the conductivity matrices i (x) and e (x) . Furthermore, I i and I e stand for the intra-and extracellular stimulation current, respectively. The variable w, the so-called gating variable, corresponds to the ionic transport through the cell membrane.
Mathematical models describing the propagation of impulses in electrophysiology have a long tradition starting with the classical model by Hodgkin and Huxley in the 1950s, see, e.g., [1,3,8,14,24] and the recent survey article of Stevens [25]. In this article, we consider the bidomain model subject to various models for the ionic transport including the models by FitzHugh-Nagumo, Aliev-Panfilov, and Rogers-McCulloch. The FitzHugh-Nagumo model reads as where 0 < a < 1 and b, c > 0 . The Aliev-Panfilov model as well as the Rogers-McCulloch model are described in detail in Sect. 4.
Note that the so-called bidomain operator is a very non-local operator, which makes the analysis of this equation seriously more complicated compared, e.g., to the classical Allen-Cahn equation.
The rigorous mathematical analysis of this system started with the work of Colli Franzone and Savaré [9], who introduced a variational formulation of the problem and showed the global existence and uniqueness of weak and strong solutions for the FitzHugh-Nagumo model. In 2009, a new approach to this system was presented by Bourgault, Coudière, and Pierre in [6]. They introduced for the first time the so-called bidomain operator A within the L 2 -setting and showed that it is a nonnegative and self-adjoint operator. They further showed the existence and uniqueness of a local strong solution and the existence of a global, weak solution to the system above for a large class of ionic models.
New impetus to the field was recently given by Giga and Kajiwara [15], who investigated the bidomain equations within the L p -setting for 1 < p ≤ ∞ . They showed that the negative of bidomain operator A is the generator of an analytic semigroup on L p ( ) for p ∈ (1, ∞] and constructed a local, strong solution to the bidomain system within this setting. Very recently the bidomain operator A was shown to admit a bounded H ∞ -calculus on L p 0 ( ) , see [19]. Moreover, using the theory of critical spaces, it was shown in [18] that the bidomain equations admit global, strong solutions for d = 2, 3 for initial data lying in critical Besov spaces. In this context, it is now a natural question to ask, whether the bidomain equations admit time-periodic solutions. Of course, periodic solutions can be formulated in various ways and various regularity classes, ranging from weak over mild to strong solutions. In this article, we are interested in unique, strong periodic solutions. We show that the innervated system admits a unique, strong time-periodic solution of period T provided the outer forces I i and I e are both time-periodic of period T > 0 . Our approach to the existence of unique, strong T-periodic solutions to the bidomain equations in a neighborhood of stable equilibrium points is the semilinear version of the periodic Da Prato-Grisvard theorem in Sect. 3.
Let us emphasize that we consider here the full bidomain model taking into account the anisotropic phenomena and not only the so-called monodomain approximation.
Recently, a very different approach to periodic solutions to the bidomain equations was developed by Giga, Kajiwara, and Kress [16]. They showed the existence of a strong, periodic solution to the bidomain equations for arbitrary large f ∈ L 2 ( ) based on a weak-strong uniqueness argument. Observe, however, that this approach does, by its nature, not yield uniqueness of the periodic solution. On the other hand, the semilinear version of the periodic Da Prato-Grisvard theorem given in Sect. 3 yields existence and uniqueness of a periodic solution and allows also to consider forcing terms This paper is organized as follows: Sect. 2 presents the periodic version of the Da Prato-Grisvard theorem, which will be extended in Sect. 3 to the semilinear setting. In Sect. 4, we present our second main result, the existence of a unique, strong T-periodic solution to the bidomain equations subject to a large class of models for the ionic transport. The proofs of the main results concerning the bidomain equations will be given in Sect. 5.

A periodic version of the Da Prato-Grisvard theorem
Let X be a Banach space and −A be the generator of a bounded analytic semigroup on X, i.e., A is closed and densely defined and there exists ∈ ( ∕2, ) and C > 0 such that and such that for all ∈ we have In order to formulate the periodic version of the Da Prato-Grisvard theorem let ∈ (0, 1) and 1 ≤ p < ∞ and recall the space We recall that the periodic version of (ACP) reads as Formally, a candidate for a solution u to (PACP) is given by (1.3). The following lemma shows that, under certain assumptions on A and f, u is indeed well-defined, continuous, and periodic.
) and assume that 0 ∈ (A) . Then, the function u defined by (1.3) is well-defined, satisfies u ∈ C(ℝ;D A ( , p)) , and is T-periodic.
Using Hölder's inequality, the periodicity of f, and the exponential decay of e −tA , we obtain for some > 0 . It follows that u is well defined. For the continuity of u, we write for h > 0 By the boundedness of the semigroup it suffices to consider the second integral. This resembles the expression from the first part of the proof but with f being replaced by [e −hA − Id ]f . Thus, and the right-hand side tends to zero as h → 0 by Lebesgue's theorem. The periodicity of u directly follows by using the transformation s � = s + T and the periodicity of f. ◻ We now state the periodic version of the Da Prato-Grisvard theorem.

is periodic of period T, satisfies u(t) ∈ D(A) for almost every t ∈ ℝ , and satisfies
Proof The continuity and periodicity of u are proven in Lemma 2.1. Let t ∈ [0, T) and use the transformation s � = s + (k + 1)T for k ∈ ℕ 0 as well as that f is periodic to write

Remark 2.3
We remark that estimate (2.2) can be obtained not only by interpolation methods as explained in the proof above but also by elementary considerations based on Hölder's inequality and Fubini's theorem. For the convenience of the reader, we present here also this elementary proof. To this end, let 1 , 2 ∈ (0, 1) with 1 + 2 = 1 and 1∕p � < 2 < 1 − + 1∕p � , where p ′ denotes the Hölder conjugate exponent to p. Then, the boundedness and the analyticity of the semigroup and Hölder's inequality imply An application of (2.3) and Fubini's theorem yields as well as Another application of Fubini's theorem yields We finally obtain

Proposition 2.4 Under the hypotheses of Proposition 2.2 the function u defined by (1.3)
is the unique strong solution to (PACP) , i.e., u is the unique periodic function of period T in C(ℝ;X) that is for almost every t ∈ ℝ differentiable in t, satisfies u(t) ∈ D(A) , and Au ∈ L p (0, T;X) , and u solves Proof First of all, u is periodic by Lemma 2.1 and since D A ( , p) continuously embeds into X the very same lemma implies u ∈ C(ℝ;X).
Assume first that f |(0,T) ∈ L p (0, T;D(A)) . Then, by a direct calculation, u defined by (1.3) is differentiable, satisfies u(t) ∈ D(A) , and solves for every t ∈ ℝ . The density of L p (0, T;D(A)) in L p (0, T;D A ( , p)) and the estimate proven in Proposition 2.2 imply that all these properties carry over to all right-hand sides in L p (0, T;D A ( , p)) (but only for almost every t ∈ ℝ ) by an approximation argument.
For the uniqueness, assume that v ∈ C(ℝ;X) with v � , Av ∈ L p (0, T;X) is another periodic function of period T which satisfies the equation for almost every t ∈ ℝ . Let w ∶= u − v . Then w satisfies In this case, for t > 0 , w can be written by means of the semigroup as w(t) = e −tA (u(0) − v(0)) . Now, the exponential decay of the semigroup and the periodicity of w imply that w must be zero for all t ∈ ℝ . ◻ For 0 < T < ∞ , we define the solution space

An existence theorem for general types of nonlinearities
Assumption N There exists R > 0 such that the nonlinear term F is a mapping from R into A and satisfies where DF ∶ R → L( per A , A ) denotes the Fréchet derivative.
The following theorem proves existence and uniqueness of solutions to (NACP) in the class per A for small forcings f. We prove that this solution operator is a contraction on r for some r ≤ R . Let M > 0 denote the infimum of all constants C satisfying (2.6). Choose r > 0 small enough such that which is possible by Assumption N. By virtue of (2.6) as well as the mean value theorem, estimate for any v ∈ r and f satisfying ‖f ‖ A ≤ r∕(2M) =∶ c, Consequently, the solution operator S is a contraction on r and the contraction mapping theorem is applicable. The solution to (NACP) is defined as follows. Let u be the unique fixed point of S. Since Su = u , u satisfies u(0) = u(T) and thus can be extended periodically to the whole real line. This function solves (NACP) . ◻

Periodic solutions to the bidomain equations
Let the space dimension n ≥ 2 be fixed and let ⊂ ℝ n denote a bounded domain with boundary of class C 2 . For the conductivity matrices i and e , we make the following assumptions.
Assumption E The conductivity matrices i , e ∶ → ℝ n×n are symmetric matrices and are functions of class C 1 ( ) . Ellipticity is imposed by means of the following condition: there exist constants , with 0 < < such that for all x ∈ and all ∈ ℝ n . Moreover, it is assumed that It is known that (4.2) is a biological reasonable assumption. Next, we define the bidomain operator in the L q -setting for 1 < q < ∞ . To this end, let Following [15], we define the bidomain operator as follows. Let i and e satisfy Assumption E. Then the bidomain operator A is defined as with domain The following result for A was proven by Giga and Kajiwara in [15] using a contradiction argument. For a direct proof and the H ∞ -calculus property of A, see [19]. Proposition 4.1 [15,Theorem 4.7,Theorem 4.9], [19] Let 1 < q < ∞ , be a bounded Under the assumption of the conservation of currents, i.e., and assuming moreover ∫ u e dx = 0 , the bidomain equations (BDE) may be equivalently rewritten as an evolution equation [6,15] of the form where is the modified source term. The functions u e and u i can be recovered from u by virtue of the following relations For the situation of the bidomain operator A, the solution space for u reads as The solution space for the gating variable w is defined as Then, the solution space for the periodic bidomain system is defined as the product space Finally, for a Banach space X we denote by X (u * , R) the closed ball in X with center u * ∈ X and radius R > 0 , i.e., We now state our main results concerning the existence and uniqueness of strong T-periodic solutions to the bidomain equations subject to various models of the ionic transport. Notice that the respective models treated here are slightly more general as described in the introduction, as an additional parameter > 0 is introduced, that incorporates the phenomenon of fast and slow diffusion.
Additionally to Assumption E on the conductivity matrices of the bidomain operator A, we require the following regularity and periodicity conditions on the forcing term I. Assumption P Let 2 ∈ (0, 1) , 1 ≤ p < ∞ , and 1 < q < ∞ satisfy 2 > n∕q or, if p = 1 let 2 ≥ n∕q . Assume I ∶ ℝ → D A ( , p) is a T-periodic function satisfying I |(0,T) ∈ A for some ∈ (0, 1∕2) and T > 0.

Remark 4.2 If
has a C 4 -boundary and if the conductivity matrices i and e lie in W 3,∞ ( ;ℝ n×n ) , then Assumption P is satisfied by virtue of (4.5) if I i , I e ∶ ℝ → D A ( , p) are T-periodic functions satisfying I i| (0,T) and I e| (0,T) ∈ A . Indeed, this follows by real interpolation since A i (A i + A e ) −1 is bounded on L q av ( ) and from D(A) into W 2,q ( ) ∩ L q av ( ).
We start with the most classical model due to FitzHugh and Nagumo.

The periodic bidomain FitzHugh-Nagumo model
For T > 0 , 0 < a < 1 , and b, c, > 0 , the periodic bidomain FitzHugh-Nagumo equations are given by This system has three equilibrium points, the trivial one (u 1 , w 1 ) = (0, 0) and two others given by (u 2 , w 2 ) and (u 3 , w 3 ) , where . We assume that the following stability condition (S FN ) on the coefficients is satisfied: Our result on strong periodic solutions to the bidomain FitzHugh-Nagumo equations reads then as follows.

The periodic bidomain Aliev-Panfilov model
For T > 0 , 0 < a < 1 , and d, k, > 0 , the periodic bidomain Aliev-Panfilov equations are given by This system has only one stable equilibrium point, namely the trivial solution (u 1 , w 1 ) = (0, 0) . Our theorem on the existence and uniqueness of strong, periodic solutions to the periodic bidomain Aliev-Panfilov equations reads as follows.

The periodic bidomain Allen-Cahn equation
For T > 0 , the periodic bidomain Allen-Cahn equation is given by (4.10) This system has three equilibrium points, u 1 = −1 , u 2 = 0 , and u 3 = 1 and our theorem on the existence and uniqueness of strong, periodic solutions to the periodic bidomain Allen-Cahn equation reads as follows.

Proofs of the theorems related to the bidomain equations
The proofs of the results presented in Sect. 4 will be subdivided in several subsections. We start with a result on the linear part of the bidomain systems. The linear part of the bidomain system will be represented as an operator matrix and it will be eminent that the negative of this operator matrix generates a bounded analytic semigroup.
whenever + bc +d ∈ (−B) . To determine the angle for which + bc +d ∈ (−B) distinguish between the cases | | < M and | | ≥ M for some suitable constant M > 0 . Notice that only the case b, c > 0 is of interest. Let C > 0 be a constant depending solely on such that | + d| ≥ C (| | + d

The bidomain equations as semilinear evolution equations
A short glimpse toward the models presented in Sects. 4.1-4.4 reveals that one of the following situations occurs: • The bidomain operator A appears only in the first but not in the second equation of the bidomain models and the nonlinearity depends linearly on the gating variable w. As a consequence, in the first situation the operator associated with the linearization of the bidomain models can be written as an operator matrix whose first component of the domain embeds into a W 2,q -space. Since the dynamics of the gating variable is described only by an ODE, there appears no smoothing in the spatial variables of w. However, as we aim to employ Theorem 3.1 and as the nonlinearity of the first equation depends linearly on w, at least in the models of Aliev-Panfilov and Rogers-McCulloch, w must be contained in D A ( , p) . Otherwise, one cannot view the nonlinearity as a suitable right-hand side as it is done in Sect. 3.1. Hence, we choose D A ( , p) as the ground space for the gating variable.
To describe this situation in our setup, assume in the following, that −A is the generator of a bounded analytic semigroup on a Banach space , and 0 ∈ (A) . We further set for some 1 < q < ∞ , 1 ≤ p < ∞ , and ∈ (0, 1) Furthermore, define two types of nonlinearities as follows: For a 1 , a 2 , a 3 , a 4 ∈ ℝ let and for b 1 , b 2 ∈ ℝ let Here, F 1 will be a prototype of the nonlinearities considered in Sects. 4.1-4.3 and F 2 for the one considered in Sect. 4.4. For the moment, the condition 0 ∈ (A) seems inappropriate as 0 ∉ (A) . However, we will linearize the bidomain equations around suitable stable stationary solutions and in this situation 0 ∈ (A) will be achieved. In the following, we concentrate only on F 1 , since the results for F 2 may be proved in a similar way. To derive conditions on p, q, and ensuring that F 1 satisfies Assumption N, the following lemma is essential.   this readily proves the proposition. ◻ Finally, by definition of F 1 it is clear that F 1 (0, 0) = 0 . Moreover, due to the polynomial structure of F 1 it is clear that F 1 is Fréchet differentiable with DF 1 (0, 0) = 0 . Hence, we have the following proposition.

Proposition 5.4
With the definitions of this subsection the nonlinearities F 1 and F 2 satisfy Assumption N. Now, we are ready to prove the main results presented in Sect. 4. To do so, the equilibrium points of the nonlinearities are calculated for the respective models. Afterward, the solutions to the bidomain models are written as the sum of the equilibrium solution and a perturbation. This results in an equation for the perturbation which is shown via Theorem 3.1 to have strong periodic solutions for suitable equilibrium points.