2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 127–137. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Existence of weak solutions for the thermistor problem with degeneracy ^{∗}

Abderrahmane El Hachimi & Moulay Rachid Sidi Ammi

Abstract

We prove the existence of weak solutions for the thermistor problem with degeneracy by using a regularization and truncation process. The solution of the regularized-truncated problem is obtained by using Schauder’s fixed point theorem. Then the solutions of the thermistor problem are obtained by applying the monotonicity-compacity method of Lions.

1 Introduction

This paper is devoted to the study of coupled parabolic-elliptic system of partial differential equations related to the often so called thermistor problem. More precisely, we are interested in the existence of solutions of problem

\frac{∂u}{∂t} − △θ(u) = σ(u)|∇φ|^{2} in Ω × (0, T),

div(σ(u)∇φ) = 0 in Ω × (0, T),
u = \overline{u} on Γ_{D}^{u} × (0, T),

\frac{∂θ(u)}{∂n} + β(x, t)(u − \overline{u}) = 0 on Γ_{N}^{u} × (0, T),

φ = \overline{φ} on Γ_{D}^{φ} × (0, T),

\frac{∂φ}{∂n} = 0 on Γ_{N}^{φ} × (0, T),

u(x, 0) = \overline{u}(x, 0) in Ω,

(1.1)

where Ω is a regular open bounded subset of R^{N}, N ≥ 1, with smooth boundary ∂Ω and T a positive real. Here Γ_{D}^{u}, and Γ_{D}^{φ} are two nonempty open subsets of ∂Ω with smooth boundaries, Γ_{N}^{u} = ∂Ω − \overline{Γ_{D}^{u}}, Γ_{N}^{φ} = ∂Ω − Γ_{D}^{φ} and \frac{∂}{∂n} is the outward normal derivative to ∂Ω. While θ, σ, β, \overline{u}, and \overline{φ} are known functions of their arguments.

∗Mathematics Subject Classifications: 35K20, 35K35, 35K45, 35K60.

Key words: Thermistor, degeneracy, existence, regularization, theorem of Leray Schauder,

monotonicity-compacity method.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

These problems arise from many applications in the automotive industry and in the field of physics, especially in the study of electrical heating of a conductor. In this situation, u is the temperature of the conductor, φ the potential and σ denotes the electrical conductivity. Problems of this type, under various assumptions on θ and σ and coupled with different types of boundary conditions, have received a lot of attention in the last decade by numerous authors. We quote in particular [3],[4], [5], [6],[7] ... and the references therein concerning problem (1.1) or it’s corresponding stationary problems.

Our main goal here is to prove existence of weak solutions to (1.1), under the following hypotheses:

(H1) θ is a continuous nondecreasing function from ℝ to ℝ, with θ(0) = 0.

(H2) σ is a real positive continuous function.

(H3) β is a continuous function from Ω × [0, ∞[ to [0, ∞[.

(H4) \overline{u} ∈ W^{1,∞}(Ω × (0, T)) and \overline{φ} ∈ L^{∞}(0, T, W^{1,∞}(Ω)) with 0 ≤ \overline{u}(x, t) ≤ M a.e. in Ω × (0, T), where M is positive constant.

In [6] Xu obtained existence of weak solutions of (1.1) under (H2)–(H4) and the hypothesis

(H1’) θ is an increasing C^{1}-function from ℝ to ℝ, with θ(0) = 0.

The result of Xu states that for each M > ∥\overline{u}∥_{L^{∞}(Ω×(0,T))}, there exists a δ > 0 such that if ∥\overline{φ}∥_{L^{∞}(Ω×(0,T))} < δ one can find a weak solution (u, φ) with ∥u∥_{L^{∞}(Ω×(0,T))} ≤ M. That is to keep temperature from exceeding a certain value, it is enough to make the electrical potential drop applied suitably small.

Here, we obtain existence and boundedness of the temperature regardless to the smallness of the potential, provided that \overline{u} is bounded. Moreover our result generalizes the one of Xu to the case where θ is not differentiable. The solution (u, φ) is obtained as a limit of a sequence of weak solutions (u_{k}, φ_{k}) of some regularized-truncated problem (3.1) associated with (P).

This paper is organized as follows: In section 2, we state our existence result concerning solutions of (1.1). Section 3 is devoted to the existence of weak solutions for problem (3.1). Then section 4 deals with a-priori estimates for solutions of (3.1). Finally section 5 is devoted to the proof of the main result.

2 Statement of the main Result

Let Q_{T} = Ω × (0, T). Define the space

V = \{v ∈ H^{1}(Ω), v = \overline{u} on Γ_{u}^{D}\}

with inner product ((u, v)) = Σ_{i=1}^{N} ∫_{Ω} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{i}} ds = ∫_{Ω} ∇u∇v ds, and ⟨, ⟩ the duality bracket between V′ and V. Also define the space

W (0, T) = \{v ∈ L^{2}(0, T, V) : \frac{∂v}{∂t} ∈ L^{2}(0, T, V′)\}.

Definition 2.1 By a weak solution of (1.1), we mean a pair of function (u, φ) such that u ∈ L^{2}(0, T, V), \frac{∂u}{∂t} ∈ L^{2}(0, T, V′), φ ∈ L^{2}(0, T, H^{1}(Ω)), and for a.e. t

⟨\frac{∂u}{∂t}, v⟩ + ∫_{Ω} ∇θ(u)∇v ds + ∫_{Γ_{N}^{u}} β(u − \overline{u})v ds − ∫_{Γ_{D}^{u}} \frac{∂θ(\overline{u})}{∂n}v ds

= − ∫_{Ω} σ(u)φ∇φ∇v ds + ∫_{Γ_{D}^{φ}} σ(u)\overline{φ}\frac{∂\overline{φ}}{∂n}v ds, for v ∈ V ∩ C^{1}(\overline{Ω}),

∫_{Ω} σ(u)φ∇ψ ds = ∫_{Γ_{D}^{φ}} σ(u)\frac{∂\overline{φ}}{∂n}ψ ds, for ψ ∈ H^{1}(Ω).

The main result of this section is the following.

Theorem 2.2 Under hypotheses (H1)–(H4), Problem (1.1) has a weak solution (u, φ) such that

u ∈ L^{2}(0, T, V) ∩ L^{2}(0, T, H^{1}(Ω)) ∩ L^{2}(0, T, W^{s,2}(Ω)), ∀s : 0 < s < 1,

\frac{∂u}{∂t} ∈ L^{2}(0, T, V′), θ(u) ∈ L^{2}(0, T, H^{1}(Ω)) and φ ∈ L^{2}(0, T, H^{1}(Ω)).

Moreover, 0 ≤ u(x, t) ≤ M a.e. in Q_{T}.

Remark. If (u, φ) is a solution of problem (1.1), then u ∈ W (0, T) which is compactly embedded in L^{2}(0, T, L^{2}(Ω)). Thus from Lion’s lemma of compacity (see [2], p. 58) we deduce that the initial condition of (1.1) makes sense.

3 Existence of solutions for regularized truncated problem

From θ, we construct a sequence θ_{k} ∈ C^{∞} such that \frac{1}{k} ≤ θ′_{k}, θ_{k}(0) = 0 and θ_{k} → θ in C_{loc}(ℝ), and from σ we introduce the truncated function

σ̃(s) = \begin{cases}σ(M) & if s > M, \\ σ(s) & if 0 ≤ s ≤ M, \\ σ(0) & if s < 0. \end{cases}

Now, we define the regularized-truncated problem (3.1) associated with (1.1):

\frac{∂u_{k}}{∂t} − △θ_{k}(u_{k}) = σ̃(u_{k})|∇φ_{k}|^{2} in Q_{T} ,

div(σ̃(u_{k})∇φ_{k}) = 0 in Q_{T} ,
u_{k} = \overline{u} on Γ_{D}^{u} × (0, T),

\frac{∂θ_{k}(u_{k})}{∂n} + β(x, t)(u_{k} − \overline{u}) = 0 on Γ_{N}^{u} × (0, T),

φ_{k} = \overline{φ} on Γ_{D}^{φ} × (0, T),

\frac{∂φ_{k}}{∂n} = 0 on Γ_{N}^{φ} × (0, T),

u_{k}(x, 0) = \overline{u}(x, 0) in Ω.

(3.1)

Lemma 3.1 Let u ∈ L^{2}(0, T, H^{1}(Ω)) and φ ∈ L^{2}(0, T, H^{1}(Ω)). Then

div(σ̃(u)φ∇φ) = σ̃(u)|∇φ|^{2}

in the distributional sense.

For the proof of this lemma see ([6], p. 205). Remarks. 1) By Definition 2.1, (u_{k}, φ_{k}) is a solution of (3.1) if and only

\begin{matrix}⟨\frac{∂u_{k}}{∂t} , v⟩ + ((θ_{k}(u_{k}), v)) + ∫_{Γ_{N}^{u}} β(u_{k} − \overline{u})v ds − ∫_{Γ_{D}^{u}} θ′_{k}(\overline{u})\frac{∂\overline{u}}{∂n}v ds \\ = − ∫_{Ω} σ̃(u_{k})φ_{k}∇φ_{k}∇v ds + ∫_{Γ_{D}^{φ}} σ̃(u_{k})\overline{φ}\frac{∂\overline{φ}}{∂n}v ds, \end{matrix}  (3.2)

for all v ∈ V ∩ C^{1}(\overline{Ω}), and

∫_{Ω} σ̃(u_{k})∇φ_{k}∇ψ ds = ∫_{Γ_{D}^{φ}} σ̃(u_{k})\frac{∂\overline{φ}}{∂n}ψ ds, for all ψ ∈ H^{1}(Ω).  (3.3)

2) The boundary integral in the right term of (3.3) makes sense since the operator trace from H^{1}(Ω) to the boundary space L^{2}(∂Ω) is linear and compact. In fact, one can show that for each φ ∈ L^{∞}(0, T, H^{1}(Ω)), the restriction of φ to ∂Ω × (0, T) belongs to the space L^{2}(0, T, L^{2}(∂Ω)).

For the rest of this paper, we shall denote by c_{i} different constants depending only on Ω and the data but not on k.

Theorem 3.2 Under Hypotheses (H1)–(H4), there exists at least a weak solution (u_{k}, φ_{k}) of (3.1), such that

u_{k} ∈ W (0, T), φ_{k} ∈ L^{∞}(0, T, H^{1}(Ω)),
u_{k}(x, 0) = \overline{u}(x, 0) a.e. in Ω,

and satisfying (3.2)–(3.3). Moreover, 0 ≤ u_{k} ≤ M, a.e. in Q_{T}.

Proof of Theorem 3.2 This is based on Schauder’s fixed point theorem. We shall construct an appropriate mapping whose fixed points are solutions of (3.1). To this end let U_{k}(w) = u_{k,w} where u_{k,w} is the unique solution of

⟨\frac{∂u_{k,w}}{∂t} , v⟩ + ∫_{Ω} θ′_{k}(w)∇u_{k,w}∇v ds + ∫_{Γ_{N}^{u}} β(u_{k,w} − \overline{u})v ds − ∫_{Γ_{D}^{u}} θ′_{k}(\overline{u})\frac{∂\overline{u}}{∂n}v ds

= − ∫_{Ω} σ̃(w)φ_{k,w}∇φ_{k,w}∇v ds + ∫_{Γ_{D}^{φ}} σ̃(w)\overline{φ}\frac{∂\overline{φ}}{∂n}v ds, for allv ∈ V ∩ C^{1}(\overline{Ω}).

(3.4)

(It is easy to show that such a solution exists and is unique.) Let S_{k} : W (0, T) → W (0, T) be the operator defined by S_{k}(w) = φ_{k,w} where φ_{k,w} is the unique solution of

∫_{Ω} σ̃(w)∇φ_{k,w}∇ψ ds = ∫_{Γ_{D}^{φ}} σ̃(w)\frac{∂\overline{φ}}{∂n}ψ ds, for ψ ∈ H^{1}(Ω).  (3.5)

By [3, Lemma 2.2], we have φ_{k,w} ∈ L^{∞}(Ω) and the H^{1}-norm of φ_{k,w} is bounded by a constant which is independent of w. Then, all terms in equations (3.4) and (3.5) are well defined.

To continue the proof of theorem 3.2, we need the following a-priori estimates.

Lemma 3.3 Let (u_{k,w}, φ_{k,w}) be the solution of (3.4)–(3.5). Then, we have the following a-priori estimates

∥φ_{k,w}∥_{L^{2}(0,T,H^{1}(Ω))} ≤ c_{1},  (3.6)
∥u_{k,w}∥_{L^{2}(0,T,V)} ≤ c_{2},  (3.7)
∥u_{k,w}∥_{L^{2}(0,T,L^{2}(Γ_{N}^{u}))} ≤ c_{3},  (3.8)
∥\frac{∂u_{k,w}}{∂t} ∥_{L^{2}(0,T,V)} ≤ c_{4},  (3.9)

where the positive constants c_{i} (i = 1 … 4) are not depending on w, nor on k.

Proof. (i) Taking ψ = φ_{k,w} − \overline{φ} in (2.11) and using the properties of σ̃, the conditions of \overline{φ} and young’s inequality, we deduce that

∫_{Ω} σ̃(w)|∇φ_{k,w}|^{2} ≤ c_{5} ∫_{Ω} |∇\overline{φ}|^{2},

where c_{5} is a positive constant. Which obviously implies (3.6). (ii) Taking v = u_{k,w} in (3.4), it follows that

⟨\frac{∂u_{k,w}}{∂t} , u_{k,w}⟩ + ∫_{Ω} θ′_{k}(w)|∇u_{k,w}|^{2} ds + ∫_{Γ_{N}^{u}} β(u_{k,w} − \overline{u})u_{k,w} ds

= − ∫_{Ω} σ̃(w)φ_{k,w}∇φ_{k,w}∇u_{k,w} ds + ∫_{Γ_{D}^{φ}} σ̃(w)\overline{φ}\frac{∂\overline{φ}}{∂n}u_{k,w} ds

+ ∫_{Γ_{D}^{u}} θ′_{k}(\overline{u})\frac{∂\overline{u}}{∂n}u_{k,w} ds.

So, we get

\frac{1}{2} \frac{∂}{∂t}|u_{k,w}|_{L^{2}(Ω)}^{2} + ∫_{Ω} θ′_{k}(w)|∇u_{k,w}|^{2} ds + ∫_{Γ_{N}^{u}} β|u_{k,w}|^{2} ds

= ∫_{Γ_{N}^{u}} βu_{k,w}\overline{u} ds − ∫_{Ω} σ̃(w)φ_{k,w}∇φ_{k,w}∇u_{k,w} ds

+ ∫_{Γ_{D}^{φ}} σ̃(w)\overline{φ}\frac{∂\overline{φ}}{∂n}u_{k,w} ds + ∫_{Γ_{D}^{u}} θ′_{k}(\overline{u})\frac{∂\overline{u}}{∂n}\overline{u} ds.

Using a lemma3.3 in [3, p. 245] and the hypotheses on θ′_{k} and β, and applying Young’s inequality, there exist positive constants c_{i}, (i = 6 … 12) such that

\frac{1}{2} \frac{∂}{∂t}|u_{k,w}|_{L^{2}(Ω)}^{2} + c_{6} ∫_{Ω} |∇u_{k,w}|^{2} ds + c_{7} ∫_{Γ_{N}^{u}} |u_{k,w}|^{2} ds

≤ ∫_{Γ_{N}^{u}} βu_{k,w}\overline{u} ds − ∫_{Ω} σ̃(w)φ_{k,w}∇φ_{k,w}∇u_{k,w} ds

+ ∫_{Γ_{D}^{φ}} σ̃(w)\overline{φ}\frac{∂\overline{φ}}{∂n}u_{k,w} ds + ∫_{Γ_{D}^{u}} θ′_{k}(\overline{u})\frac{∂\overline{u}}{∂n}\overline{u} ds

≤ c_{8} ∫_{Ω} |∇φ_{k,w}||∇u_{k,w}| ds + ∫_{Γ_{D}^{φ}} σ̃(w)\overline{φ}\frac{∂\overline{φ}}{∂n}u_{k,w} ds

+ ∫_{Γ_{D}^{u}} θ′_{k}(\overline{u})\frac{∂\overline{u}}{∂n}\overline{u} ds + ∫_{Γ_{N}^{u}} βu_{k,w}\overline{u}

≤ \frac{c_{6}}{2} ∫_{Ω} |∇u_{k,w}|^{2} ds + c_{9} ∫_{Ω} |∇φ_{k,w}|^{2} ds + c_{10} ∫_{Γ_{D}^{φ}} |\frac{∂\overline{φ}}{∂n}|^{2} ds

+ \frac{c_{7}}{4} ∫_{Γ_{D}^{φ}} |u_{k,w}|^{2} ds + \frac{c_{7}}{4} ∫_{Γ_{N}^{u}} |u_{k,w}|^{2} ds + c_{11} ∫_{Γ_{N}^{u}} |\overline{u}|^{2} ds + c_{12}.

Then we have

\frac{1}{2} \frac{∂}{∂t}|u_{k,w}|_{L^{2}(Ω)}^{2} + \frac{c_{6}}{2} ∫_{Ω} |∇u_{k,w}|^{2} ds + c_{7} ∫_{Γ_{N}^{u}} |u_{k,w}|^{2} ds

≤ c_{13} + \frac{c_{7}}{4} ∫_{Γ_{D}^{φ}} |u_{k,w}|^{2} ds + \frac{c_{7}}{4} ∫_{Γ_{N}^{u}} |u_{k,w}|^{2} ds

≤ c_{13} + \frac{c_{7}}{4} ∫_{∂Ω} |u_{k,w}|^{2} ds + \frac{c_{7}}{4} ∫_{Γ_{N}^{u}} |u_{k,w}|^{2} ds

≤ c_{13} + \frac{c_{7}}{4} ∫_{Γ_{D}^{u}} |u_{k,w}|^{2} ds + \frac{c_{7}}{2} ∫_{Γ_{N}^{u}} |u_{k,w}|^{2} ds.

Therefore,

\frac{1}{2} \frac{∂}{∂t}|u_{k,w}|_{L^{2}(Ω)}^{2} + \frac{c_{6}}{2} ∫_{Ω} |∇u_{k,w}|^{2} ds + \frac{c_{7}}{2} ∫_{Γ_{N}^{u}} |u_{k,w}|^{2} ds

≤ c_{13} + \frac{c_{7}}{4} ∫_{Γ_{D}^{u}} |\overline{u}|^{2} ds ≤ c_{14}. (3.10)

Integrating (3.10) on (0, T), we conclude to the estimates (3.7) and (3.8) . (iii) According to (3.4), (3.6), (3.7), (3.8) and a lemma3.3 in [3, p. 245], we obtain

∥\frac{∂u_{k,w}}{∂t} ∥_{L^{2}(0,T,V′)} ≤ c_{4}.

Hence Lemma 3.3 is proved. □

Now, we define the space

W_{0} := \{ \begin{matrix}v ∈ W (0, T), ∥v∥_{L^{2}(0,T,V)} ≤ c_{2}, ∥\frac{∂v}{∂t} ∥_{L^{2}(0,T,V′)} ≤ c_{4}, \\ 0 ≤ v(x, t) ≤ M a.e. in Q_{T} , v(0) = \overline{u}(x, 0) in Ω \end{matrix} \} .

Note that W_{0} is a non empty convex set, and by Lemma 3.3, the operator U_{k} : W_{0} → W_{0} is well defined. To use Schauder’s fixed point theorem, it remains to show that U_{k} is continuous with respect to the weak topology of W (0, T). Then, using the weak compactness of W_{0} we shall conclude that U_{k} has a fixed point w in the set W_{0}. To prove the weak continuity, assume that (w_{j})_{j} is a sequence in W_{0} satisfying w_{j} → w weakly in W (0, T) and let (u_{k,w_{j}}, φ_{k,w_{j}}) the corresponding sequence of solutions of (3.4)–(3.5). By estimates (3.6)–(3.9), there exists at least a subsequence denoted again by w_{j} such that as j → ∞, we have

w_{j} → w weak in L^{2}(0, T, V),

\frac{∂w_{j}}{∂t} → \frac{∂w}{∂t} weak in L^{2}(0, T, V′),

u_{k,w_{j}} → u_{k} weak in L^{2}(0, T, V),

u_{k,w_{j}} → u_{k} weak in L^{2}(0, T, L^{2}(Γ_{N}^{u})),

\frac{∂u_{k,w_{j}}}{∂t} → \frac{∂u_{k}}{∂t} weak in L^{2}(0, T, V′),

φ_{k,w_{j}} → φ_{k} weak in L^{2}(0, T, H^{1}(Ω)).

Note that one may assume, without loss of generality, that w_{j} → w strongly in L^{2}(Q_{T}) and a.e. in Q_{T}. Since σ̃ is continuous and bounded, then, thanks to the dominated convergence theorem of Lebesgue, it follows that

σ̃(w_{j}) → σ̃(w) strongly in L^{2}(Q_{T}).

By the trace theorem, we derive that

σ̃(w_{j})\frac{∂φ_{k,w_{j}}}{∂n} → σ̃(w)\frac{∂φ_{k}}{∂n} weak in L^{2}(Γ_{D}^{φ}).

On the other hand, by vertue of the estimates of Lemma 3.3, we deduce that there exist functions α_{1}, α_{2}, α_{3}, such that

σ̃(w_{j})∇φ_{k,w_{j}} → α_{1} weak in L^{2}(Q_{T}),  (3.11)
θ′_{k}(w_{j})∇u_{k,w_{j}} → α_{2} weak in L^{2}(Q_{T}),  (3.12)
σ̃(w_{j})φ_{k,w_{j}}∇φ_{k,w_{j}} → α_{3} weak in L^{2}(Q_{T}).  (3.13)

Now, as ∇φ_{k,w_{j}} → ∇φ_{k} weak in L^{2}(Q_{T}), we have

σ̃(w_{j})∇φ_{k,w_{j}} → σ̃(w)∇φ_{k} in D′(Q_{T}).

Consequently,

α_{1} = σ̃(w)∇φ_{k}.

In a similar way, we obtain

α_{2} = θ′_{k}(w)∇u_{k},
α_{3} = σ̃(w)φ_{k}∇φ_{k}.

Then, passing to the limit as j → ∞ in the relations (3.4) and (3.5) satisfied by (u_{k,w_{j}}, φ_{k,w_{j}}), we deduce immediately that u_{k} = U_{k}(w) and φ_{k} = S_{k}(w). By the unique solvability of (3.4), all the sequence u_{k,w_{j}} converges to u_{k} = U_{k}(w) weakly in W (0, T). This completes the proof. □

Remark By theorem 3.2, we have 0 ≤ u_{k} ≤ M. Hence σ̃(u_{k}) = σ(u_{k}).

4 Estimates on solutions of the regularized truncated problem

In this section, we obtain appropriate estimates on the solutions (u_{k}, φ_{k}) of the regularized-truncated problem (3.1).

Lemma 4.1 Let (u_{k}, φ_{k}) be a solution of (3.1). Under the hypotheses (H1)– (H4), there exist constants c_{i} (i = 15 … 19) such that, for any k ≥ 1, the following estimates hold

|u_{k}(t)|_{L^{2}(Ω)}^{2} ≤ |\overline{u}(x, 0)|_{L^{2}(Ω)}^{2} + c_{15},  (4.1)
∥u_{k}∥_{L^{2}(0,T,V)}^{2} ≤ |\overline{u}(x, 0)|_{L^{2}(Ω)}^{2} + c_{15},  (4.2)
∥\frac{1}{k} u_{k}∥_{L^{2}(0,T,V)}^{2} ≤ \frac{c_{16}}{2k} (|\overline{u}(x, 0)|_{L^{2}(Ω)}^{2} + c_{17}),  (4.3)
∥θ_{k}(u_{k})∥_{L^{2}(0,T,H^{1}(Ω))}^{2} ≤ c_{18},  (4.4)
∥\frac{∂u_{k}}{∂t} ∥_{L^{2}(0,T,V′)} ≤ c_{19}.  (4.5)

The different constants are positive and not depending on k.

Proof. Choosing v = u_{k} as a function test in (3.2), using the hypotheses on θ′_{k} and β and applying a lemma3.3 in [3, p. 245], we obtain

\frac{1}{2} \frac{∂}{∂t}|u_{k}(t)|_{L^{2}(Ω)}^{2} + c_{20} ∫_{Ω} |∇u_{k}|^{2} ds + c_{21} ∫_{Γ_{N}^{u}} |u_{k}|^{2} ds

≤ c_{22} ∫_{Ω} |∇φ_{k}||∇u_{k}| ds + c_{23} ∫_{Γ_{N}^{u}} |u_{k}||\overline{u}| ds

+ c_{24} ∫_{Γ_{D}^{φ}} |u_{k}||\frac{∂\overline{φ}}{∂n}| ds + c_{25} ∫_{Γ_{D}^{u}} |\frac{∂\overline{u}}{∂n}||\overline{u}| ds.

The same arguments as in the proof of Lemma 3.3 lead to

\frac{∂}{∂t}|u_{k}(t)|_{L^{2}(Ω)}^{2} + ∫_{Ω} |∇u_{k}|^{2} ds + ∫_{Γ_{N}^{u}} |u_{k}|^{2} ds ≤ c_{26}.  (4.6)

Integrating (4.6) on (0, t) yiel ds

|u_{k}(t)|_{L^{2}(Ω)}^{2} + ∫_{Q_{t}} |∇u_{k}|^{2} ds + ∫_{0}^{t} ∫_{Γ_{N}^{u}} |u_{k}|^{2} ds ≤ |\overline{u}(x, 0)|_{L^{2}(Ω)}^{2} + c_{15}.

Hence (4.1) and (4.2) are satisfied. On the other hand, by (3.2) we have

⟨\frac{∂u_{k}}{∂t} , u_{k}⟩ + ((θ_{k}(u_{k}), u_{k})) + ∫_{Γ_{N}^{u}} β|u_{k}|^{2}

= − ∫_{Ω} σ̃(u_{k})φ_{k}∇φ_{k}∇u_{k} ds + ∫_{Γ_{N}^{u}} βu_{k}\overline{u} ds

+ ∫_{Γ_{D}^{φ}} σ̃(u_{k})\overline{φ}\frac{∂\overline{φ}}{∂n}u_{k} ds + ∫_{Γ_{D}^{u}} θ′_{k}(\overline{u})\frac{∂\overline{u}}{∂n}\overline{u} ds.

Therefore, arguing exactly as above, we deduce that

|u_{k}(t)|_{L^{2}(Ω)}^{2} +2 ∫_{0}^{T} ((θ_{k}(u_{k}), u_{k}))dt+∫_{0}^{T} ∫_{Γ_{N}^{u}} |u_{k}|^{2} ds ≤ c_{16}(|\overline{u}(x, 0)|_{L^{2}(Ω)}^{2} +c_{17}).

Furthermore,

∫_{0}^{T} ((θ_{k}(u_{k}), u_{k}))dt = ∫_{0}^{T} ( ∫_{Ω} θ′_{k}(u_{k})|∇u_{k}|^{2} ds)dt

and 0 < \frac{1}{k} ≤ θ′_{k}(u_{k}). Then

\frac{1}{k} ∥u_{k}∥_{L^{2}(0,T,H^{1}(Ω))}^{2} ≤ \frac{c_{16}}{2} (|\overline{u}(x, 0)|_{L^{2}(Ω)}^{2} + c_{17}).

Which gives estimate (4.3).

To obtain an estimate on (θ_{k}(u_{k}))_{k≥1}, we take θ_{k}(u_{k}) as a test function in (3.2). We get

⟨\frac{∂u_{k}}{∂t} , θ_{k}(u_{k})⟩ + ∥θ_{k}(u_{k})∥_{H^{1}(Ω)}^{2} + ∫_{Γ_{N}^{u}} β(u_{k} − \overline{u})θ_{k}(u_{k}) ds

= − ∫_{Ω} σ̃(u_{k})φ_{k}∇φ_{k}∇θ_{k}(u_{k}) ds + ∫_{Γ_{D}^{u}} θ′_{k}(\overline{u})\frac{∂\overline{u}}{∂n}θ_{k}(u_{k}) ds

+ ∫_{Γ_{D}^{φ}} σ̃(u_{k})\overline{φ}\frac{∂\overline{φ}}{∂n}θ_{k}(u_{k}) ds.

Straightforward calculations and Bamberger’s lemma [8, p. 8] give

\frac{d}{dt}\{ ∫_{Ω} ( ∫_{0}^{u_{k}(x,.)} θ_{k}(r)dr) ds\} + \frac{1}{2}∥θ_{k}(u_{k})∥_{H^{1}(Ω)}^{2} ≤ c_{27}.  (4.7)

Integrating (4.7) from 0 to T yiel ds

∫_{Ω} ( ∫_{0}^{u_{k}(x,T)} θ_{k}(r)dr) ds + \frac{1}{2}∥θ_{k}(u_{k})∥_{L^{2}(0,T,H^{1}(Ω))}^{2} ≤ \frac{c_{18}}{2} .

Hence estimate (4.4) follows. According to (3.2), (4.1), (4.2), (4.4) and Lemma 3.3, [3, p. 245], and the definition of dual norm, we get the desired relation (4.5). □

5 Passage to the limit in (3.1) as k → ∞

The aim now is to pass to the limit in the process (3.1). By using estimates of Lemma 4.1 and standard compacity arguments, we deduce that there exists a subsequence (u_{k}, φ_{k}), not relabelled, such that, as k → ∞, we have

u_{k} → u weak in L^{2}(0, T, V),

u_{k} → u weak star L^{∞}(0, T, L^{2}(Ω)),

u_{k} → u weak in L^{2}(0, T, L^{2}(Γ_{N}^{u})),

\frac{∂u_{k}}{∂t} → \frac{∂u}{∂t} weak in L^{2}(0, T, V′),

φ_{k} → φ weak star in L^{∞}(0, T, H^{1}(Ω)).

On the other hand, since the space \{v ∈ L^{2}(0, T, V), \frac{∂v}{∂t} ∈ L^{2}(0, T, V′)\} is compactly embedded in L^{2}(Q_{T}), [2, p. 58], we can extract a subsequence from (u_{k}), not relabelled, such that

u_{k} → u strongly and a.e. in L^{2}(Q_{T}).

Moreover, we can assume that

θ_{k}(u_{k}) → θ(u) a.e. in Q_{T} and weak in L^{2}(0, T, H^{1}(Ω)).

Indeed, we have

∫_{0}^{t}(∫_{Ω} |θ_{k}(u_{k}) − θ(u)| dsdr
≤ ∫_{0}^{t}(∫_{Ω} |θ_{k}(u_{k}) − θ(u_{k})| ds)dr + ∫_{0}^{t}(∫_{Ω} |θ(u_{k}) − θ(u)| ds)dr
≤ csup_{|r|≤M}|θ_{k}(r) − θ(r)| + ∫_{0}^{t}(∫_{Ω} |θ(u_{k}) − θ(u)| ds)dr.

Now, arguing as in the proof of (3.11)–(3.13), we obtain

σ̃(u_{k})∇φ_{k} → σ(u)∇φ weak in L^{2}(Q_{T}),
σ̃(u_{k})φ_{k}∇φ_{k} → σ(u)φ∇φ weak in L^{2}(Q_{T}).

Moreover, using Aubin’s lemma (see [2, p. 7]), we obtain that u_{k} → u in C([0, T]; V′). Then u_{k}(0) → u(0) weak in V′. We consequently obtain u(x, 0) = \overline{u}(x, 0). Finally, we verify easily that the limit (u, φ) obtained is a solution of problem (1.1). This conclude to the proof of the main result. □

Remark. The question of uniqueness has been established in some special cases; see [4] and [8].

References

[1] A. Bamberger: Etude d’une équation doublement non linéaire. Jour. of Functional Analysis, Volume. 24, n 2, pp .148-155, 1977.

[2] J. L. Lions: Quelques méthodes de résolution de problèmes aux limites non linéaires, Dunod, Paris, 1969.

[3] P. Shi, M. Shillor, X. Xu: Existence of a solution of the stefan problem with joule’s heating. Jour. Differential Equations. Vol 105, No.2, october 1993.

[4] S.N. Antontsev, M. Chipot: The thermistor problem: existence, smoothness, uniqueness and blowup. SIAM Jour. Math.Anal. Vol 25, No. 4, pp. 1128-1156. July 1994.

[5] X. Xu: A strongly degenerate system involving an equation of parabolic type and equation of elliptic type, Commun. in Partial Differential Equations, 18, pp . 199-213, (1993).

[6] X. Xu: On the existence of bounded tempeature in the thermistor problem with degeneracy. Nonlinear Analysis,T.M.A 42, pp .199-213, (2000).

[7] X. Xu: Local and global existence of continuous temperature in electrical heating of conductors, Houston Journal of Mathematiques, vol 22, No.2, 1996.

[8] X. Xu: Existence and uniqueness for the nonstationary problem of the electrical heating of a conductor due to the joule-Thomson effect, Int. J. Math. Math. Sci. 16, pp.125-138 (1993).

Abderrahmane El Hachimi UFR Mathématiques Appliquées et Industrielles Faculté des Sciences B.P 20, El Jadida - Maroc e-mail adress: elhachimi@ucd.ac.ma Moulay Rachid Sidi Ammi UFR Mathématiques Appliquées et Industrielles Faculté des Sciences B.P 20, El Jadida - Maroc e-mail adress: sidiammi@hotmail.com