THERMISTOR SYSTEMS OF p ( x )-LAPLACE-TYPE WITH DISCONTINUOUS EXPONENTS VIA ENTROPY SOLUTIONS

. We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two diﬃculties appear: (i) the elliptic term in the current-ﬂow equation is of p ( x )-Laplacian-type with discontinuous exponent p , which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L 1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder’s ﬁxed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.

1. Introduction. In this paper we study a system of partial differential equations, that describes the electrothermal behavior of organic semiconductor devices. It consists of the current-flow equation for the electrostatic potential ϕ coupled to the heat equation with Joule heat term for the temperature θ, namely −∇ · A(x, θ, ∇ϕ) = 0, −∇ · (λ(x)∇θ) = H(x, θ, ∇ϕ) (1b) in a bounded domain Ω ⊂ R d . The current flux A : Ω × R + × R d → R d is given via describes an Arrhenius-like temperature law. The Joule heat term in the right-hand side of (1b) H : where η(x, θ, z) ∈ [0, 1] is the light-outcoupling factor. In particular, the equation (1a) is of p(x)-Laplace-type, where x → p(x) is a measurable function satisfying 1 < ess inf x∈Ω p(x) ≤ ess sup x∈Ω p(x) < ∞.
Systems of the form (1) model materials conducting both heat and electrical current and for which the electrical conductivity in the definition of A strongly depends on the temperature. Devices of this type are called thermistors, see [3,4]. Recently, systems of the form (1) with the specific constitutive law in (2) were introduced in [12] to describe electrothermal effects, such as self-heating and inhomogeneous current distributions, in organic, i.e. carbon-based, semiconductor devices, see also [6]. For example, Organic Light-Emitting Diodes (OLEDs) are thin-film heterostructures based on organic molecules or polymers, where each functional layer has, in general, its own current-voltage characteristics and material parameters. In particular, the exponent p(x), which describes the non-Ohmic behavior of each layer, changes abruptly from one material to another. In electrodes, the typically used parameter is p(x) = 2, while organic layers feature significantly larger values, e.g. p(x) ≈ 9 (see [6]). The material function β, which is related to the so-called activation energy in an Arrhenius law, is positive in organic layers. This, however, leads to a positive feedback as the electrical conductivity increases with rising temperature and in turn the power dissipation increases with the electrical current. This mechanism was identified in [6] as the cause of the appearance of different operation modes and accompanying unpleasant brightness inhomogeneities in large-area OLEDs.
The analytical difficulties of the problem in (1)-(4) arise from two issues: First, the exponent function x → p(x) is discontinous and in general only measurable. Second, the source term H in the right hand side of (1b) is only in the space L 1 (Ω) for functions ϕ in the energy space associated with the differential operator in the left hand side of (1a).
In [9] these issues were overcome in the two-dimensional case and for a piecewise constant x → p(x) satisfying p(x) ≥ 2 by showing improved integrability of the gradient of the electrostatic potential, i.e. ∇ϕ ∈ L sp(·) (Ω) d for some s > 1. The latter is proved using Caccioppoli estimates and a Gehring-type lemma. This significantly helps to deal with the right hand side H in the heat equation in the existence proof, since then we have a priori control of H(·, θ, ∇ϕ) in L s (Ω). In particular, one does not need to face the problem of concentration effects and correspondingly the presence of a singular measure. However, this approach heavily relies on the use of the Poincaré inequality, which does in general not hold for discontinuous exponents p, see [5,Sec. 8.2]. Moreover, the extension to higher dimension and ranges of p that are realistic for organic devices is unclear.
To tackle higher spatial dimensions an approach based on regularization and Galerkin approximation was discussed in [2]. Therein, a regularized version of the system in (1) was introduced, where the crucial term H is approximated so that it remains bounded. The existence of solutions to the regularized problem is proven by Galerkin approximations. By using suitable test functions in the weak formulation of the regularized version of (1) uniform estimates for ϕ and θ independent of the regularization parameter ε > 0 were derived, which allows to pass to the limit ε → 0 and to obtain weak solutions of (1).
In this paper, we present a different existence proof using the concept of entropy solutions with Robin boundary conditions (4b) and Schauder's fixed-point theorem. More precisely, for a given temperature distribution θ we obtain a unique solution ϕ = ϕ( θ) of the current flow equation (1a). Using ϕ and θ in H, we then solve the heat equation to obtain an entropy solution θ. The map θ → θ is continuous and hence has a fixed point in a suitable compact set M ⊂ L 1 (Ω). Here, the crucial point is the continuous dependence of the entropy solutions on given right hand sides, see Lemma 3.4.
Outline of the paper. We start in Section 2 with fixing the notation, introducing the underlying assumptions and function spaces and formulating our main result concerning the existence of weak solutions to the coupled p(x)-thermistor model (1). Section 3 covers the solvability of the two subproblems, i.e. the current-flow and heat equation. Finally, in Section 4 we verify the solvability of the coupled system via Schauder's fixed-point theorem. In the Appendix we collect and proof needed results for entropy solutions to linear elliptic problems with boundary conditions of the form (4b).
2. Preliminaries and main result.

2.1.
Assumptions on the data. Here, we collect the essential assumptions for the analytical investigations: on Ω. The Arrhenius factor is of the form F (x, θ) = exp −β(x) 1 θ − 1 θa with β ∈ L ∞ + (Ω) and the constant θ a > 0 is the ambient temperature. (A5) The heat conductivity λ satisfies λ ∈ L ∞ (Ω) and λ ≥ λ 0 > 0 a.e. on Ω. The heat transfer coefficient κ is such that κ ∈ L ∞ + (Γ) and κ L 1 (Γ) > 0. (A6) The light-outcoupling factor η = η(x, θ, z) is such that η : Ω × R + × R d → R is a Carathéodory function and η(x, θ, z) ∈ [0, 1] holds f.a.a. x ∈ Ω and ∀(θ, z) ∈ R + × R d . The assumptions (A4) ensure important structural properties of the function A. They imply that A is a Carathéodory function with the growth Due to the inequality (see [13,Chapter 10]) for z 1 , z 2 ∈ R d , the function A(x, θ, ·) ist strictly monotone in the third argument, meaning that A(x, θ, z 1 )−A(x, θ, z 2 ) ·(z 1 −z 2 ) > 0 ∀θ ∈ R + , ∀z 1 = z 2 ∈ R d , f.a.a. x ∈ Ω. (6) Additionally, there exists a constant c a > 0 such that is fulfilled. (The lower estimate θ ≥ θ a for solutions θ of the heat equation can be obtained in the setting of entropy solutions for the heat flow equation by Lemma 3.5.) 2.2. Function spaces. For constant p ∈ [1, ∞], we use the classical Lebesgue spaces L p (Ω) and Sobolev spaces W 1,p (Ω). Following [5] we work with the generalized function spaces L p(·) (Ω), where we assume that the bounded variable exponents p ∈ P(Ω) satisfy The generalized Lebesgue space L p(·) (Ω) consists of all measurable functions u for which the modular (Ω) is a Banach space and [5, Lemma 3.2.5] ensures for all u ∈ L p(·) (Ω) the estimates We work with a definition of generalized Sobolev spaces that will be appropriate for our problem. We emphasize that the spaces introduced here are not necessarily equivalent to the standard Sobolev spaces with the variable exponent in [5]. This is necessary since in our case the exponent p is discontinuous and varies over a large range, see also [2]. For a given p ∈ P(Ω) we define the generalized Sobolev space W 1,p(·) (Ω) and equip it with the following norm In the case 1 < p − ≤ p + < ∞ the space W 1,p(·) (Ω) is a separable and reflexive Banach space, since L p(·) has the same properties, see Appendix B. We introduce the subspace W 1,p(·) D (Ω) := {u ∈ W 1,p(·) (Ω) : u = 0 on Γ D }. Since we assume that Γ D is of positive (d−1)-dimensional measure, we have the equivalent norms Indeed, we can use the facts that the classical Sobolev space W

1,p− D
(Ω) satisfies the Poincaré inequality and that the variable exponent Lebesgue space L p(x) (Ω) is continuously embedded into L p− (Ω) to obtain for arbitrary u ∈ W Furthermore, we denote by H 1 (Ω) the usual Hilbert space. By means of the assumption (A5) the estimates with constants α, α > 0 are satisfied. Moreover, the dual space of a Banach space X is denoted by X * . In our estimates, positive constants, which may depend at most on the data of our problem, are denoted by c. In particular, we allow them to change from line to line.
We prove Theorem 2.1 in Section 4 by constructing a suitable fixed-point map for the temperature θ and applying Schauder's fixed-point theorem. Note that Schauder's theorem does not give uniqueness of fixed points and hence of solutions. However, uniqueness of solutions for the system in (1) cannot be expected due to the hysteretic behavior induced by the positive feedback with respect to temperature described in the introduction. Even in the spatially homogeneous setting of selfheating in organic devices (see [7] and [12, Sect. 2.1]) S-shaped current-voltage characteristics occur. Here, three different currents are possible for the same applied voltage.

3.1.
Existence of solutions to the current flow equation. In the first step, we turn our attention to the current-flow equation (1a) for the potential ϕ. In particular, we consider an arbitrary but fixed θ, which is assumed to lie in the set of relevant temperature distributions given by Θ := {θ ∈ L 1 (Ω) : θ ≥ θ a a.e. on Ω}.
For fixed θ ∈ Θ, we introduce the operator A θ : (Ω) * and consider the following problem: which corresponds to finding a weak solution ϕ ∈ ϕ D + W
In summary, the main theorem of monotone operators (see [8]) ensures the existence of a solution to (13). Since A θ is strictly monotone, this solution is unique.
3.2. Entropy solutions of the heat equation. For the solvability of the second equation with right hand side in L 1 (Ω) we use the concept of entropy solutions. In the case of Dirichlet boundary conditions this theory is well presented in the survey [14], for nonlinear problems see [1,11].
We consider the stationary heat flow equation with Robin boundary conditions and right hand side f ∈ L 1 (Ω) as well as boundary data g ∈ L 1 (Γ), namely in Ω, For k > 0, we define the truncation C k : R → [−k, k] by C k (s) := max{−k, min{s, k}} for all k > 0 and all ω ∈ H 1 (Ω) ∩ L ∞ (Ω).
Note that for ω ∈ H 1 (Ω) with ω L ∞ (Ω) ≤ c we have ω L ∞ (Γ) ≤ c with the same constant. Theorem 3.3. We assume (A1) and (A5). Let f ∈ L 1 (Ω), g ∈ L 1 (Γ). Then there exists a unique entropy solution θ to (18). This entropy solution belongs to W 1,q (Ω), for all 1 ≤ q < d d−1 . Especially, there are constants c Eq > 0 not depending on f and g such that We give the proof of Theorem 3.3 in Appendix A and finalize this subsection by two lemmas with special properties of entropy solutions to (18).
Then the corresponding entropy solutions θ l to (18) converge weakly in W 1,q (Ω) to the entropy solution θ for data f and g, 1 ≤ q < d d−1 .
Proof. 1. Since the θ l are entropy solutions to (18) for f l and g l by (19) with ω = 0 we find where the unified constant c > 0 results from the fact that f l → f in L 1 (Ω), g l → g in L 1 (Γ). As in Step 2 of the proof of Theorem 3.3 (see Appendix A) we then verify for all exponents q < d d−1 that In summary we find a θ ∈ W 1,q (Ω) such that for a (non-relabeled) subsequence we have in Ω, (Ω) and a.e. in Ω, and a.e. on Γ for all k > 0.
By the weak lower semicontinuity of the norm and C k (θ l ) To verify that θ is the entropy solution to (18) with data f and g we prove now as in [11] the strong convergence C k (θ l ) → C k (θ) in H 1 (Ω). Let 2h > k and take ω l := C h (θ l ) − C k/2 (θ l ) + C k/2 (θ) as test function for the entropy solution θ l in (19) with data f l and g l . Then for m := h + 2k we have ∇C k (θ l − ω l ) = 0 on the set {x ∈ Ω : |θ l (x)| > m}. Therefore it follows from the entropy formulation Splitting the volume integral on the left hand side in integrals on Ω l k := {x ∈ Ω : Using the entropy formulation we obtain where, due to (21), the last term obviously converges to zero for l → ∞. Additionally, for every fixed h, the convergences (21) ensure the weak convergences for as well as corresponding convergences a.e. in Ω and Γ for a further non-relabeled subsequence. Because of the estimates we therefore conclude with f l → f in L 1 (Ω), g l → g, in L 1 (Γ), (21), and Lebesgue's dominated convergence theorem, in the limit l → ∞ . According to Lebesque's theorem, we have that the terms on the right hand sides in (23) converge to zero for h → ∞. Hence, we can fix a sufficiently large h ε > 0 such that With this h ε and m = m ε = 2k+h ε we estimate the remaining term in the right hand side of (22). Since |∇C m (θ l )| is bounded in L 2 (Ω) and χ {|θ l |>k/2} |∇C k/2 (θ)| → 0 in L 2 (Ω) we verify Therefore, passing to the limit l → ∞ in (22) and using (24) it results where ε > 0 can be made arbitrarily small by increasing h ε . This yields lim l→∞ Ω λ|∇ C k/2 (θ l ) − C k/2 (θ) | 2 dx = 0, and together with λ ≥ λ 0 > 0 and C k/2 (θ l ) → C k/2 (θ) in L 2 (Ω) we obtain C k/2 (θ l ) → C k/2 (θ) in H 1 (Ω) which then implies again for a non-relabeled subsequence ∇C k/2 (θ l ) → ∇C k/2 (θ) a.e. in Ω. These arguments hold true for all k > 0 and enable us to verify similarly to Step 5 in the proof of Theorem 3.3 that θ is an entropy solution for the data f and g.
We fix k > θ a . Since C k (θ n ) → C k (θ) in L 1 (Ω) as n → ∞ (note that θ n → θ in L 1 (Ω) due to Lemma 3.4) we find a subsequence {n l } such that C k (θ n l ) → C k (θ) a.e. in Ω which implies together with θ n l ≥ θ a a.e. on Ω that C k (θ) ≥ θ a a.e. in Ω. Especially this ensures θ ≥ 0 a.e. in Ω because of θ a > 0. Additionally, by our choice k > θ a we have θ ≥ C k (θ) ≥ θ a a.e. in Ω. 4. Solution of the coupled system via Schauder's fixed-point theorem. In this section we proof our main result, Theorem 2.1.
1. Definition of the fixed-point map. Let where c Q > 0 will be fixed in (27). We consider the fixed-point map Q : M → M with θ = Q( θ) defined by θ being the unique entropy solution of where H is defined in (3) and the function ϕ = ϕ( θ) is the unique weak solution to −∇ · A(x, θ, ∇ϕ) = 0 in Ω, Since θ ∈ M we have θ ∈ Θ and the existence and uniqueness of a weak solution ϕ ∈ ϕ D + W
2. Existence of a solution. The continuity of the mapping Q : M → M will be proven in Lemma 4.1. Since for all θ ∈ M the norm θ W 1,q 0 is uniformly bounded, the compact embedding of W 1,q0 (Ω) in L 1 (Ω) gives the desired compactness of the set Q(M) ⊂ L 1 (Ω). Therefore Schauder's fixed-point theorem ensures the solvability of the coupled problem (1)-(4). This finishes the proof of Theorem 2.1. Proof. Let θ, θ n ∈ M with θ n → θ in L 1 (Ω). We denote by ϕ n the unique solution to (26) with θ n instead of θ as fixed argument in A. We have to show that θ n = Q( θ n ) → θ = Q( θ) in L 1 (Ω).
By Theorem 3.1, the weak solution to (26) with θ as second argument in A is unique such that we find ϕ = ϕ = ϕ( θ).
2. Weak convergence of the whole sequence ϕ n ϕ in W 1,p(·) (Ω). To verify the weak convergence ϕ n ϕ in the reflexive Banach space W 1,p(·) (Ω) of the whole sequence and not only of the subsequence given in (29) we apply [8, Lemma 5.4, Chap. 1]. We have to show that for every weakly convergent subsequence ϕ n k ϕ it holds true that ϕ = ϕ: If there is a subsequence ϕ n k ϕ in W 1,p(·) (Ω) then the arguments of Step 1 ensure again non-relabeled subsequences such that ∇ϕ n k → ∇ ϕ a.e. on Ω and A(·, θ n k , ∇ϕ n k ) A(·, θ, ∇ ϕ) in L p (·) (Ω) d . And ϕ would solve (26), with θ as second argument in A. Since the weak solution to (26) is unique -we have ϕ = ϕ. Thus, the convergence ϕ n ϕ in W 1,p(·) (Ω) is valid for the whole sequence and not only a subsequence.
Since a sequence of functions {h n } ⊂ L 1 (Ω), which converges weakly to h ∈ L 1 (Ω) is uniformly equi integrable, the sequence {A(·, θ n , ∇ϕ n ) · ∇ϕ n } is uniformly equi integrable. Due to the pointwise a.e. convergence of θ n (x) → θ(x), ∇ϕ n (x) → ∇ϕ(x) a.e. in Ω and the Caratheodory property of A we find h n → h a.e. in Ω for the subsequence. Next, we apply Vitali's theorem which tells us that for a sequence of functions {h n } ⊂ L 1 (Ω), which converges pointwise a.e. in Ω to h the following two assertions are equivalent: (i) The sequence {h n } is uniformly equi integrable and (ii) h n → h in L 1 (Ω).
According to Lemma 3.4 we find for the entropy solutions θ n and θ of (25) with right hand sides f n and f , respectively, that θ n θ in W 1,q0 (Ω). By Theorem 3.3 the solution to (25) with right hand side f is unique, thus, with (29) it follows that θ n = Q( θ n ) θ = θ = Q( θ) in W 1,q0 (Ω), for the subsequence in (29).
4. Convergences of the whole sequence θ n θ in W 1,q0 (Ω) and θ n → θ in L 1 (Ω). Similar to Step 2, we have to show that for each weakly convergent subsequence θ n k θ in the reflexive Banach space W 1,q0 (Ω) the identity θ = θ is fulfilled. We can argue as in Step 3 to find for not-relabeled subsequences f n k = H(·, θ n k , ∇ϕ n k ) → H(·, θ, ∇ϕ) = f in L 1 (Ω) such that the uniqueness result in Theorem 3.3 and Lemma 3.4 ensure θ n k θ = θ in W 1,q0 (Ω). This leads to the convergence of the whole sequence θ n θ in W 1,q0 (Ω) and by the compact embedding of W 1,q0 (Ω) into L 1 (Ω) we obtain the strong convergence of the whole sequence in L 1 (Ω) which proves the continuity of the operator Q.
Appendix A. Proof of Theorem 3.3. In the case of Dirichlet boundary conditions the theory of entropy solutions is presented in the survey [14], for nonlinear problems see [1,11]. To provide a complete theory for our type of boundary conditions in (18) we give a proof by adapting the techniques in [11,14].
For the uniqueness proof of Theorem 3.3 we need that weak solutions to (18) for more regular data f and g are admissible test functions ω in the sense of (19), which we state first.
Proof. We test (18) by rθ r−1 , r = 2 k , k ∈ N, and use the notation v := θ r 2 , By the Gagliardo-Nirenberg inequality, the trace inequality, and Young's inequality we find for all ε > 0 a c ε > 0 such that (33) it results with the same constant c 0 (f, g) ≥ 2 for r = 2 k , k ∈ N, and β = 2 d+2 the estimate Setting a k := 1 + θ 2 k L 2 k the previous estimates ensure the recursion where we have used that The starting estimate for a 1 is obtained by testing (18) by θ. Applying embedding and trace inequality as well as Young's inequality gives This ensures that a 1 = 1 + θ 2 L 2 ≤ c 1 (f, g) which finishes the proof.
For k > 0 we use for (34) the test function C k (θ n ) ∈ H 1 (Ω) and find Therefore, for all fixed k > 0 the sequence {C k (θ n )} is bounded in H 1 (Ω). 2. According to Sobolev's embedding result, for d > 2 we have additionally Since |C k (θ n )| = k on Ω k,n := {x ∈ Ω : |θ n (x)| ≥ k} it results where c > 0 does not depend on n and k. [10, Chap. 2, p. 105] ensures that For d = 2 this estimate is valid for all q < ∞. Similar to [14, p. 34], but now exploiting (35) in the case of Robin boundary conditions we verify that Ω |∇θ n | q dx ≤ c q ( f L 1 + g L 1 (Γ) ) q for q < d d−1 .
Thus, θ n W 1,q is uniformly bounded and we find a subsequence again denoted by {θ n } and θ ∈ W 1,q (Ω) with θ n θ in W 1,q (Ω), θ n → θ in L q (Ω) and a.e. in Ω. 3. Due to our construction, f n → f in L 1 (Ω) and f n is a Cauchy sequence in L 1 (Ω) and g n → g in L 1 (Γ) and g n is a Cauchy sequence in L 1 (Γ). Repeating Step 2 of the proof for θ n − θ m with f n − f m , g n − g m (weak solutions of a linear problem) we find Ω |∇(θ n − θ m )| q dx ≤ c q ( f n − f m L 1 + g n − g m L 1 (Γ) ) q and similar estimates of the form (36) for the corresponding differences. Therefore {θ n } is a Cauchy sequence in W 1,q (Ω) for all q < d d−1 and {θ n } converges strongly to θ in W 1,q (Ω). For a subsequence, ∇θ n → ∇θ a.e. in Ω and θ n → θ a.e. on Γ.
Next, the weak convergence of C k (θ n ) to C k (θ) in H 1 (Ω) ensures Collecting and balancing all the limit terms we end up with meaning that θ is an entropy solution of (18). 6. Finally, if there would be another entropy solution θ besides θ we argue with a proof by approximation. Let for the subsequence {n} from Step 3 of the above proof f n := C n (f ) ∈ L ∞ (Ω), g n := C n (g) ∈ L ∞ (Γ) and let θ n be the corresponding unique weak solution to (18). According to Lemma A.1, θ n belongs to H 1 (Ω) ∩ L ∞ (Ω) and