The thermistor problem with degenerate thermal conductivity and metallic conduction

THE THERMISTOR PROBLEM WITH DEGENERATE THERMAL CONDUCTIVITY AND METALLIC CONDUCTION Mar ´ıa Teresa Gonz´alez Montesinos Departamento de Matem´aticas Universidad de C´adiz Escuela Superior de Ingenier´ıa 11002 C´adiz, SPAIN Francisco Orteg´on Gallego Departamento de Matem´aticas Universidad de C´adiz CASEM, Campus del R´ıo San Pedro 11510 Puerto Real, C´adiz, SPAIN Abstract. The aim of this work is to establish the existence of a capacity solution to the thermistor problem supposing that the thermal and the electrical conductivities are not bounded below by a positive constant value. Furthermore, the thermal conductivity vanishes at points where the temperature is null. These assumptions on data include the case of practical interest of the Wiedemann–Franz law with metallic conduction and lead us to very complex mathematical situations. 1. Introduction. The heat produced by an electrical current passing through a conductor device is governed by the so–called thermistor problem. This problem consists in a coupled system of parabolic–elliptic equations, whose unknowns are the temperature inside the conductor, u, and the electrical potential, ϕ. The conservation laws of both current and energy lead us to the equations that determine the thermistor problem, namely, ∂u ∂t −∇·(a(u)∇u) = σ(u)|∇ϕ|2in ΩT= Ω ×(0, T), ∇ · (σ(u)∇ϕ) = 0 in ΩT, u= 0 on ΓT=∂Ω×(0, T), ϕ=ϕ0on ΓT, u(·,0) = u0in Ω,              (1) whereΩ , the domain occupied by the electrical device, is an open, bounded and smooth enough subset of RN , N ≥ 1, T > 0 and a and σ are the thermal and electrical conductivities. A great deal of attention has been paid by several authors in the study of the thermistor problem during the last two decades [1, 2, 14, 16, 21]. In all these works, and many others, it is assumed that a ≡ a0 ∈ R or a0 ≤ a(s) ≤ a1, for all s ∈ R. Also, σ is taken as a very smooth and bounded function. These hypotheses on the . conductivities together with the regularity supposed on data yield to the existence of weak solutions, and even to the uniqueness of a regular enough weak solution. In [7] the existence of weak solutions is established assuming that the thermal conductivity satisfies the Wiedemann-Franz law, that is, a(u) = Luσ(u), L > 0 being a constant value, and σ∈C(R). This hypothesis on aleads to a very complex mathematical situation and it is precisely the main obstacle in the resolution of the problem, since a(0) = 0 and the parabolic equation becomes degenerate. The Wiedemann–Franz law is also taken into account in [9], where two existence results weak solutions of the steady state of the thermistor problem are achieved. Firstly, aand σare not bounded below far from zero, arising to a doubly non– uniformly elliptic system. Secondly, apart from the hypotheses above, it is assumed that the thermal conductivity blows up for a finite value of the temperature; in this way, the system becomes singular and non–uniformly. The notion of capacity solution was introduced in [18], where the term ∇ · (a(u)∇u) is replaced by ∇ · a(∇u), a:RN→RNbeing a continuous operator such that |a(ξ)| ≤ C|ξ|, for |ξ|>> 1, and [a(ξ)−a(η)] (ξ−η)≥α|ξ−η|2, for all ξ, η ∈RN,α > 0. Also it is assumed that 0 < σ(s)≤¯σfor all s∈R; it is just this hypothesis which hampers the resolution of the problem, because it enables that σ(s)→0 as |s| → ∞. Then, if uis unbounded in ΩT, the elliptic equation becomes degenerate when uis infinite and no a priori estimates of ∇ϕare available and, therefore, ϕmay not belong to a Sobolev space. Then, instead of ϕ, Φ = σ(u)∇ϕ is considered as a single function and it is shown that Φ ∈L2(ΩT)N. This yields a new formulation of the original system and its solution is called capacity solution. Later on, this type of solution was used in [17, 19, 20]. The existence of capacity solutions is proved in [12], where a:Q×R×RN→RN is a Leray–Lions operator which includes the particular case of the p–Laplacian operator, p≥2, and σ∈C(R) is such that 0 < σ(s)≤¯σ, for all s∈R. In most situations of practical interest, σsatisfies this property and also σ(s)→0 as |s| tends to infinity. This paper is organized as follows. In section 2 we show some useful results for the development of the proof of the main result of this paper. In section 3 we introduce the assumptions on data, the notion of capacity solution adapted to our setting and give the existence result. Section 4 develops the proof of the main result, which is divided into several stages. First, we introduce a sequence of approximate problems and derive a priori estimates for the approximate solutions, (un, ϕn). Finally, we pass to the limit and conclude. 2. Notation and some useful results. Throughout this paper, we will denote V=H1 0(Ω), V0=H−1(Ω), H=L2(Ω) and V=L2(0, T;V), V0=L2(0, T;V0) and H=L2(0, T;H). Also, for the sake of simplify the notation, we will write Lp(X) instead of Lp(0, T;X), Xbeing a Banach space. Now we introduce an interpolation result (see Proposition 3.1 in [5] pp. 7 and 8): Lemma 1. Let r, p ≥1, and v∈L∞(Lr(Ω)) ∩Lp(W1,p 0(Ω)). Then v∈Lq(ΩT), with q=pN+r N, and there exists a constant value C=C(N, p, r)>0such that ZΩT |v|q≤Cess sup t∈(0,T )ZΩ |v|r!p/N ZΩT |∇v|p. Secondly, the following compacity result ([15]) will be required: Lemma 2. Let X,Band Ybe three Banach spaces so that X ,→B ,→Y, every embedding being continuous and the inclusion X ,→Bcompact. For 1≤p, q ≤ ∞, let Wbe the Banach space defined as W=v∈Lp(X)/dv dt∈Lq(Y). Then, if 1≤p, q < ∞, the inclusion W,→Lp(B)holds and is compact. On the other hand, if p=∞and q > 1the inclusion W,→C([0, T ]; B)holds and is compact. The next lemma ([12, 18]) is a fundamental tool in order to obtain the strong convergence in L1(ΩT) of a suitable subsequence of certain approximate solutions. Lemma 3. Let (vn)be a bounded sequence in Vand relatively compact in H. Then there exists a subsequence (vn(k))⊂(vn)such that, for every ε > 0, there are a constant value K=K(ε)>0and a function ψ∈L1(W1,1(Ω)) satisfying 0≤ψ≤1,kψ−1kL1(ΩT)+k∇ψkL1(ΩT)N≤ε, (2) |v|,|vn(k)| ≤ Kin {ψ > 0},for all k≥1,(3) where v∈ V is such that vn(k)* v weakly in V. 3. Definition of capacity solution and main result. Our purpose in this paper is to study system (1) under the following hypotheses: (H.1) u0∈L2(Ω) is such that u0≥0 almost everywhere in Ω. (H.2) ϕ0∈L2(H1(Ω)) ∩L∞(ΩT). (H.3) σ∈C(R) and 0 < σ(s)≤σ0, for all s∈R. (H.4) a∈C(R) and 0 < a(s)≤a0, for all s6= 0, a(0) = 0. (H.5) For every δ > 0 there exists a constant value aδ>0 so that ess inf |s|>δ a(s)≥aδ. Remark 1. In view of (H.4) and (H.5) we have that aδ→0 as δ↓0. This situation was already analyzed in [7] under the same assumptions above, except for a small but crucial detail: now the function σis not bounded below by a positive constant. Consequently, the parabolic and the elliptic equations of (1) are degenerate and non–uniformly elliptic, respectively, and the survey of the problem becomes much more complex. In fact, the existence of a weak solution is not assured and we have to deal with the notion of the capacity solution (see [12, 18]). Notice that (H.3)–(H.5) includes the cases physically important of metallic conduction, that is, σ(u) = O(u−1) when u→ ∞, whereas a(u) may be given by the Wiedemann-Franz law. Now let A(s) = Rs 0a(τ) dτ. It is clear that A(0) = 0, A∈C1(R), Ais strictly increasing and Lipschitz–continuous. Furthermore, ∇A(φ) = a(φ)∇φfor all φ∈ L2(H1(Ω)). Then, instead of system (1), we consider the problem ∂u ∂t −∆A(u) = σ(u)|∇ϕ|2in ΩT, ∇ · (σ(u)∇ϕ) = 0 in ΩT, u= 0 on ΓT, ϕ=ϕ0on ΓT, u(·,0) = u0in Ω.              (4) Remark 2. If ϕ∈L2(H1(Ω)) is a solution of the elliptic equation, then it is very easy to check that ∇ · (σ(u)ϕ∇ϕ) = σ(u)|∇ϕ|2in L2((V∩L∞(Ω))0). Definition 1. A triplet (u, ϕ, Φ) is said to be a capacity solution to problem (1) (or (4)) if the following conditions are fulfilled: (C.1) u∈L∞(L1(Ω)), du dt∈ V0,A(u)∈ V∩Lq(ΩT), for all q < 2+2/N,ϕ∈L∞(ΩT) and Φ ∈L∞(HN). (C.2) (u, ϕ, Φ) verifies the system of differential equations du dt−∆A(u) = ∇ · (ϕΦ) and ∇ · Φ = 0 in V0.(5) (C.3) For every S∈W1,∞(R) with supp Scompact, S(A(u))ϕ−S(0)ϕ0∈ V and S(A(u))Φ = σ(u) [∇(S(A(u))ϕ)−ϕ∇S(A(u))] .(6) (C.4) u(·,0) = u0. Remark 3. A capacity solution basically differs from a weak solution in that, in the first case, ∇ϕis considered in the almost everywhere sense whereas in the second one, ∇ϕis regarded in the sense of distributions. However, Definition 1 is somewhat different from the one given in other works (for instance [12, 18]). More precisely, instead of (6), a capacity solution should verify S(u)Φ = σ(u) [∇(S(u)ϕ)−ϕ∇S(u)]. But this condition is pointless because, in general, u /∈ V. That is the reason why we use A(u) in (6) instead of u. We remark that the definition of capacity solution given by Xu in [18] (or by the authors in [12]) and Definition 1 both lead to the following identification: Φ = σ(u)∇ϕalmost everywhere in ΩT, (7) where ∇ϕmust be suitably defined pointwise almost everywhere in Ω (for more details, see [13]). This means that though a capacity solution involves three unknowns, namely, u,ϕand Φ, there are in fact only two, since Φ is actually related to uand ϕaccording to the identity (7). The main result now follows. Theorem 1. Under the hypotheses (H.1)–(H.5), system (4) admits a capacity solution (u, ϕ, Φ) in the sense of Definition 1. Moreover, u≥0almost everywhere in ΩT, the gradient of uis defined almost everywhere in ΩTand ∇uχ{u>δ}∈L2(ΩT)for all δ > 0. Finally, if S∈L1 loc(R)is such that S0∈L∞(R)and supp S⊂R\(−δ0, δ0)for some δ0>0, then S(u)∈ V and ∇S(u) = S0(u)∇uin ΩT. 4. Proof of the main result. The proof of Theorem 1 is divided in four steps: firstly, we consider a sequence of approximate problems. Then a priori estimates for the approximate solutions (un, ϕn) are derived. Finally, we pass to the limit and conclude. One of the major difficulties in this stage lies in showing the strong convergence of (ϕn) for a suitable subsequence. 4.1. Approximate problems. For every n∈Nwe define the regularized functions an(s) = a(s) + 1 nand σn(s) = σ(s) + 1 n; also, set An(s) = Rs 0an(τ)dτ=A(s) + s n. Then the approximate problem of (1) (or (4)) is defined as ∂un ∂t −∇·(an(un)∇un) = σn(un)|∇ϕn|2in ΩT, ∇ · (σn(un)∇ϕn) = 0 in ΩT, un= 0 on ΓT, ϕn=ϕ0on ΓT, un(·,0) = u0in Ω.              (8) By (H.3) and (H.4), respectively, 1 n≤σn(s)≤σ0+ 1 = σ1and 1 n≤an(s)≤ a0+ 1 = a1, for all s∈R. Then, classic results ([1]) lead us to the existence of a weak solution (un, ϕn) to (8) such that un∈ V ∩ C0([0, T]; L2(Ω)),dun dt∈ V0, ϕn−ϕ0∈L∞(V).(9) Furthermore, it is straightforward that kϕnkL∞(ΩT)≤ kϕ0kL∞(ΩT),(10) ZΩ σn(un)|∇ϕn|2≤C, a.e. t∈(0, T ).(11) On the other hand, in view of (H.1), it is easy to show that un≥0 a.e. in ΩT.(12) Moreover, it is shown that hσn(un)|∇ϕn|2, φiV0,V=−ZΩT σn(un)ϕn∇ϕn∇φ, (13) where (∇ · (σn(un)ϕn∇ϕn)) is bounded in V0thanks to (H.3), (10) and (11). 4.2. Estimates of the approximate solutions. Taking An(un) as a test function in the parabolic equation of (8) and bearing in mind (13), we obtain Zt 0ZΩ |∇An(un)|2≤C, for all t∈[0, T ].(14) Since the sequences (∆An(un)) and (∇ · (σn(un)ϕn∇ϕn)) are bounded in V0, dun dtis bounded in V0.(15) Also, choosing as a test function 1 εTε(un)∈ V ∩ L∞(ΩT), with ε > 0, one has that (un) is bounded in L∞(0, T;L1(Ω)).(16) On the other hand, as |∇A(un)| ≤ |∇An(un)|, in view of (14) we have that (A(un)) is bounded in V.(17) Moreover, |A(s)| ≤ a0|s|and |An(s)| ≤ a1|s|, for all s∈R, whereupon (A(un)) and (An(un)) are bounded in L∞(L1(Ω)).(18) Thanks to (17) and (18), the above sequences satisfy Lema 1; then (A(un)) and (An(un)) are bounded in L2+2/N (ΩT).(19) For every δ > 0, we define the function gδ(s) =      s+δif s < −δ, 0 if |s| ≤ δ, s−δif s > δ. Take gδ(un)∈ V as a test function in the parabolic equation of (8); as ∇gδ(un) = g0 δ(un)∇un=χ{|un|>δ}∇un, one has that ∇un∇gδ(un) = |∇gδ(un)|2=|∇un|2in the set {|un|> δ}. Hence, applying Young’s inequality, ZΩT |∇gδ(un)|2≤ku0k2 L2(Ω) 2aδ +1 2ZΩT |∇gδ(un)|2+σ1kϕ0k2 L∞(ΩT) 2a2 δZΩT σn(un)|∇ϕn|2. From (11) we have ZΩT |∇gδ(un)|2≤Cδ=a−1 δku0k2 L2(Ω) +a−2 δσ1kϕ0kL∞(ΩT)CT. (20) We now introduce a regularized function of gδ, namely, γδ∈C∞(R). To this end, let γ∈C∞(R) be a function satisfying the following properties: γ(s) = 0 if s∈[0,1/2], γ(s) = s−1 if s > 3/2, γ(s) is convex in [0,+∞), γ(s) sis increasing in [1/2,3/2] and γ(−s) = −γ(s),for all s∈R. Then for every δ∈(0,1], γδ(s) is defined as γδ(s) = δγ(s/δ). Notice that 0≤γδ(s)≤γδ0(s),for all s≥0 and 0 < δ ≤δ0,(21) |γ0 δ(s)| ≤ 1,|γ00 δ(s)| ≤ K δ,for all s∈R. (22) For δ∈(0,1] and φ∈ D(ΩT) we take γ0 δ(un)φas a test function in the parabolic equation of (8). Bearing in mind that ∇(γ0 δ(un)φ) = γ00 δ(un)∇unφ+γ0 δ(un)∇φand after some calculations it yields dγδ(un) dtis bounded in L1(ΩT) + V0, for every δ∈(0,1].(23) Furthermore, (20) and (22) imply that (γδ(un)) is bounded in V, for every δ∈(0,1].(24) Also, from (16) and the definition of γδwe obtain ZΩ |γδ(un)| ≤ ZΩ |un| ≤ C, a.e. t∈(0, T), for all n∈Nand δ∈(0,1].(25) We can then apply Lemma 1 and deduce (γδ(un)) is bounded in L2+2/N (ΩT), for every δ∈(0,1]. (26) 4.3. Passing to the limit. Applying Lemma 2 with X=H1 0(Ω) ,→B=H, the inclusion being compact, and Y=W−1,r0(Ω), for all r0<N N−1if N≥2, r0<∞ if N= 1, one has that the space V=nv∈ V /dv dt∈L1(W−1,r0(Ω))ois such that the inclusion V,→L2(ΩT) is compact. Thus, thanks to (23) and (24), (γδ(un)) is relatively compact in L2(ΩT), for every δ∈(0,1], wherefrom, for every δ∈(0,1], there is a function Γδ∈L2(ΩT) so that, up to a subsequence, γδ(un)→Γδstrongly in L2(ΩT) and a.e. in ΩT.(27) Moreover, by (25) and (27), it is easy to check that ZΩ |Γδ| ≤ C, a.e. t∈(0, T ) and for all δ∈(0,1].(28) Also, (26) and (27) imply the convergences γδ(un)*Γδweakly in L2+2/N (ΩT), (29) γδ(un)→Γδstrongly in Lq(ΩT), for all q < 2 + 2 N. (30) Since the sequence (γδ(un))δis increasing as δ↓0, also we have that Γδ≤Γδ0 almost everywhere in ΩT, for all δ0≤δ, that is, the sequence (Γδ) is increasing too as δ↓0. Consequently, there exists a measurable function u: ΩT7→ Rsuch that limδ↓0Γδ=ualmost everywhere in ΩT; and by applying the monotone convergence theorem, using (28), we deduce that u∈L1(ΩT) and Γδ→ustrongly in L1(ΩT) and a.e. in ΩT(31) up to a subsequence. Moreover, according to (28) and (31), we may conclude that u∈L∞(L1(Ω)).(32) Also, taking into account the definition of γδ, we obtain un→ustrongly in L1(ΩT) and a.e. in ΩT, (33) which implies that γδ(un)→γδ(u) almost everywhere in ΩTand, then, γδ(u) = Γδ. Notice that (24) yields u≥0. Also (20) and (33) imply in particular that gδ(u)∈ V for all δ∈(0,1]. (34) Accordance with the definition of gδand (34), we may define the gradient of u, at least in the almost everywhere sense. Indeed, ∇gδ(u) = ∇uχ{u>δ}∈L2(ΩT) for all δ∈(0,1]. (35) Expression (35) allows us to define ∇uin {u > 0}; in the level set {u= 0}we just put ∇u= 0. Estimates derived till now lead us to the following convergences for suitable subsequences: dun dt*du dtweakly in V0,(36) A(un), An(un)* A(u) weakly in Vand a.e. in ΩT,(37) An(un), A(un)* A(u) weakly in L2+2/N (ΩT),(38) An(un), A(un)→A(u) strongly in Lq(ΩT), for all q < 2 + 2 N,(39) ϕn* ϕ weakly–∗in L∞(ΩT),(40) σn(un)∇ϕn*Φ weakly–∗in L∞(HN), (41) σn(un)* σ(u) weakly–∗in L∞(ΩT) and a.e. in ΩT, (42) where ϕand Φ are some functions belonging to the spaces L∞(ΩT) and L∞(HN), respectively. All these convergences lead us directly to conditions (C.1) and (C.2) of Definition 1. Moreover; notice that we can not guarantee the convergence ϕn→ϕ almost everywhere in ΩT, up to a subsequence. Nevertheless, the strong convergence in L1(ΩT) holds for a suitable subsequence. 4.3.1. Strong convergence of (ϕn)in L1.Along this section we show the existence of a subsequence (ϕn(k))⊂(ϕn) such that ϕn(k)→ϕstrongly in L1(ΩT). (43) To this end, let S∈W1,∞(R) such that supp Sis compact. Then, for a suitable subsequence, still denoted in the same way, one has S(γδ(un)) ϕn* S (γδ(u)) ϕweakly in L2(H1(Ω)), for every δ∈(0,1].(44) Now consider S∈W1,∞(R) such that supp Sis compact and 0 ≤S≤1. Then, there exists a constant value C0>0, independent of S, such that lim sup n→∞ ZΩT σn(un)|∇ [S(γδ(un)) ϕn−S(γδ(u))ϕ]|2 ≤C0C1/2 δ/2kS0k∞(1 + C1/2 δ/2kS0k∞),for every δ > 0.(45) On the other hand, it is easy to check that, for every δ∈(0,1], the sequence (γδ(un)) verifies the hypotheses of Lemma 3. Thus, taking into account (44) and (45), it can be shown that there exists a subsequence (ϕn(k))⊂(ϕn) such that lim k→∞ ZΩT |ϕn(k)−ϕ|= 0. Remark 4. Since |ϕn(k)| ≤ kϕ0kL∞(ΩT)for all k≥1, the strongly convergence in L1(ΩT), together with the almost everywhere convergence, imply that ϕn(k)→ϕ strongly in Lp(ΩT) for all p < ∞. 4.3.2. For all S∈W1,∞(R)with supp Scompact, S(A(un(k)))ϕn(k)→S(A(u))ϕ weakly in L2(H1(Ω)).We may assume that ϕn(k)→ϕalmost everywhere in ΩT. Let vk=A(un(k))∈ V. Clearly, S(vk)ϕn(k)∈L2(H1(Ω)) and thanks to (33) we have S(vk)ϕn(k)→S(A(u))ϕalmost everywhere in ΩT. Thus it is enough to show that S(vk)ϕn(k)is bounded in L2(H1(Ω)). 4.3.3. Condition (C.3) of Definition 1. Let S∈W1,∞(R), supp Sbeing compact, and consider the identity S(A(un(k)))σn(k)(un(k))∇ϕn(k) =σn(k)(un(k))∇S(A(un(k)))ϕn(k)−ϕn(k)∇S(A(un(k))).(46) Using (41), the strong convergence of (ϕn(k)) in L2(ΩT), the weakly convergence of S(A(un(k))) and S(A(un(k)))ϕn(k)in L2(H1(Ω)), it is straightforward that we can pass to the limit in (46), which yields the desired identity (6). 4.3.4. Condition u(0) = u0.Owing to (C.1), u∈W=v∈L∞(L1(Ω)) /dv dt∈ V0. Also we have W⊂Wr⊂C([0, T ]; W−1,r0(Ω)), every inclusion being continuous, and Wrthe space defined as Wr=nv∈L∞(W−1,r0(Ω)) /dv dt∈L2(W−1,r0(Ω))o. This means that we may expect that the initial condition u(0) = u0makes sense at least in W−1,r0(Ω), for all r0<N N−1if N≥2, r0<∞if N= 1. 4.3.5. More regularity on u.Now, we show the last assertion of Theorem 1. To do so, let S∈L1 loc(R) such that S0∈L∞(R) and supp S⊂R\(−δ0, δ0) for some δ0>0. Consider the functions S(gδ(u)) ∈ V for δ∈(0, δ0). Then, S(gδ(u)) →S(u) when δ↓0 almost everywhere in ΩT. On the other hand, using (35), |∇S(gδ(u))| ≤ kS0k∞|∇gδ(u)|χ{u>δ0}=kS0k∞|∇u|χ{u>δ0}, and thus, S(gδ(u)) is bounded in Vfor δ∈(0, δ0) . This ends the proof of theorem 1. 4.4. Identification of ∇A(u).Since in general u6∈ V, we cannot assure that ∇A(u) = a(u)∇uin ΩT. This is due to the fact that A−1is not globally Lipschitzcontinuous. However, if (u, ϕ, Φ) is a capacity solution obtained in Theorem 1, it can be shown that a(u)∇u∈L2(ΩT)Nand ∇A(u) = a(u)∇uin ΩT. 5. Concluding remarks. The assumptions on the diffusion coefficients include as a particular case the Wiedemann–Franz law and also metallic conduction. In previous works by the authors ([6]-[11]) only one of these two hypotheses was assumed. From the mathematical point of view, these two simultaneous assumptions have led to a very complex situation. The existence of a capacity solution (u, ϕ, Φ) to system (1) has been established in the preceding sections. The functions uand ϕ verify the thermistor problem in divergence form, that is, ∂u ∂t −∇·(a(u)∇u) = ∇ · (σ(u)ϕ∇ϕ) in ΩT, ∇ · (σ(u)∇ϕ) = 0 in ΩT, u= 0 on ΓT, ϕ=ϕ0on ΓT, u(·,0) = u0in Ω,              (47) where u∈L∞(L1(Ω)), ∇ϕmay be defined in an almost everywhere sense (see Remark 3 and [13]), du dt∈ V0,ϕ∈L∞(ΩT), σ(u)∇ϕ∈L∞(0, T;L2(Ω)), ∇uχ{u>δ}∈ L2(ΩT) for all δ > 0, and a(u)∇u∈L2(ΩT). We may consider if the identity ∇ · (σ(u)ϕ∇ϕ) = σ(u)|∇ϕ|2holds true in a certain sense. It is very well known that this is true in N= 1, or when uor ϕare smooth enough, for instance, u∈L∞(ΩT), or ϕ∈L2(0, T;H1(Ω)). In the general case N≥2, this question is far from being trivial. The uniqueness of capacity solutions to system (1) has not been analyzed in this paper. Notice that here the assumptions on data are weaker than the ones considered in other works where uniqueness is established ([1, 2, 3, 14, 16, 21]). In those works there is no need to search for capacity solutions: the regularity of the solutions leads to the usual setting of weak solutions. Acknowledgements. This research was partially supported by Ministerio de Ciencia y Tecnolog´ıa under grant BFM2003-01187 with the participation of FEDER fund and Consejer´ıa de Educaci´on y Ciencia de la Junta de Andaluc´ıa group FQM–315. REFERENCES [1] S. N. Antontsev and M. Chipot, The thermistor problem: existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal., 25 (1994), 1128–1156. [2] M. Chipot and G. Cimatti, A uniqueness result for the thermistor problem, European J. Appl. Math., 2(1991), 97–103. [3] G. Cimatti, Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Quart. Appl. Math., XLVII(1) (1989), 117–121. [4] G. 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