Existence of weak solutions for the thermistor problem with degeneracy

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1 22-Fez conference on Partial ifferential Equations, Electronic Journal of ifferential Equations, Conference 9, 22, pp or 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 u t θ(u) = σ(u) ϕ 2 in (, T ), div(σ(u) ϕ) = in (, T ), u = u on (, T ), θ(u) n β(x, t)(u u) = on Γu N (, T ), ϕ = ϕ on (, T ), ϕ n = on Γϕ N (, T ), u(x, ) = u(x, ) in, (1.1) where is a regular open bounded subset of R N, N 1, with smooth boundary and T a positive real. Here, and Γϕ are two nonempty open subsets of with smooth boundaries, = Γu, Γϕ N = Γϕ and n is the outward normal derivative to. While θ, σ, β, u, and ϕ are known functions of their arguments. Mathematics Subject Classifications: 35K2, 35K35, 35K5, 35K6. Key words: Thermistor, degeneracy, existence, regularization, theorem of Leray Schauder, monotonicity-compacity method. c 22 Southwest Texas State University. Published ecember 28,

2 128 Existence of weak solutions 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],[], [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() =. (H2) σ is a real positive continuous function. (H3) β is a continuous function from [, [ to [, [. (H) u W 1, ( (, T )) and ϕ L (, T, W 1, ()) with u(x, t) M a.e. in (, T ), where M is positive constant. In [6] Xu obtained existence of weak solutions of (1.1) under (H2) (H) and the hypothesis (H1 ) θ is an increasing C 1 -function from R to R, with θ() =. The result of Xu states that for each M > u L ( (,T )), there exists a δ > such that if ϕ L ( (,T )) < δ one can find a weak solution (u, ϕ) with u L ( (,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 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 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 = (, T ). efine the space V = {v H 1 (), v = u on Γ u } with inner product ((u, v)) = Σ N u v i=1 x i x i ds = u v ds, and, the duality bracket between V and V. Also define the space W (, T ) = {v L 2 (, T, V ) : v t L2 (, T, V )}.

3 Abderrahmane El Hachimi & Moulay Rachid Sidi Ammi 129 efinition 2.1 By a weak solution of (1.1), we mean a pair of function (u, ϕ) such that u L 2 (, T, V ), u t L2 (, T, V ), ϕ L 2 (, T, H 1 ()), and for a.e. t u t, v θ(u) θ(u) v ds β(u u)v ds Γ u n v ds = σ(u)ϕ ϕ v ds σ(u)ϕ ϕ Γ ϕ n v ds, for v V C1 (), σ(u)ϕ ψ ds = σ(u) ϕ n ψ ds, for ψ H1 (). The main result of this section is the following. Theorem 2.2 Under hypotheses (H1) (H), Problem (1.1) has a weak solution (u, ϕ) such that u L 2 (, T, V ) L 2 (, T, H 1 ()) L 2 (, T, W s,2 ()), s : < s < 1, u t L2 (, T, V ), θ(u) L 2 (, T, H 1 ()) and ϕ L 2 (, T, H 1 ()). Moreover, u(x, t) M a.e. in Q T. Remark. If (u, ϕ) is a solution of problem (1.1), then u W (, T ) which is compactly embedded in L 2 (, T, L 2 ()). 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 1 k θ k, θ k() = and θ k θ in C loc (R), and from σ we introduce the truncated function σ(m) if s > M, σ(s) = σ(s) if s M, σ() if s <. Now, we define the regularized-truncated problem (3.1) associated with (1.1): u k t θ k(u k ) = σ(u k ) ϕ k 2 in Q T, div( σ(u k ) ϕ k ) = in Q T, θ k (u k ) n u k = u on (, T ), β(x, t)(u k u) = on (, T ), ϕ k = ϕ on (, T ), (3.1) ϕ k n = on Γϕ N (, T ), u k (x, ) = u(x, ) in.

4 13 Existence of weak solutions Lemma 3.1 Let u L 2 (, T, H 1 ()) and ϕ L 2 (, T, H 1 ()). Then in the distributional sense. div( σ(u)ϕ ϕ) = σ(u) ϕ 2 For the proof of this lemma see ([6], p. 25). Remarks. 1) By efinition 2.1, (u k, ϕ k ) is a solution of (3.1) if and only u k t, v ((θ k(u k ), v)) β(u k u)v ds θ k(u) u Γ u n v ds = σ(u k )ϕ k ϕ k v ds σ(u k )ϕ ϕ (3.2) n v ds, for all v V C 1 (), and σ(u k ) ϕ k ψ ds = σ(u k ) ϕ n ψ ds, for all ψ H1 (). (3.3) 2) The boundary integral in the right term of (3.3) makes sense since the operator trace from H 1 () to the boundary space L 2 ( ) is linear and compact. In fact, one can show that for each ϕ L (, T, H 1 ()), the restriction of ϕ to (, T ) belongs to the space L 2 (, T, L 2 ( )). For the rest of this paper, we shall denote by c i different constants depending only on and the data but not on k. Theorem 3.2 Under Hypotheses (H1) (H), there exists at least a weak solution (u k, ϕ k ) of (3.1), such that u k W (, T ), ϕ k L (, T, H 1 ()), u k (x, ) = u(x, ) a.e. in, and satisfying (3.2) (3.3). Moreover, 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 u k,w, v θ t k(w) u k,w v ds β(u k,w u)v ds θ k(u) u Γ u n v ds = σ(w)ϕ k,w ϕ k,w v ds σ(w)ϕ ϕ Γ ϕ n v ds, for allv V C1 (). (3.) (It is easy to show that such a solution exists and is unique.) Let S k : W (, T ) W (, T ) be the operator defined by S k (w) = ϕ k,w where ϕ k,w is the unique solution of σ(w) ϕ k,w ψ ds = σ(w) ϕ n ψ ds, for ψ H1 (). (3.5)

5 Abderrahmane El Hachimi & Moulay Rachid Sidi Ammi 131 By [3, Lemma 2.2], we have ϕ k,w L () 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.) 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.) (3.5). Then, we have the following a-priori estimates ϕ k,w L2 (,T,H 1 ()) c 1, (3.6) u k,w L 2 (,T,V ) c 2, (3.7) u k,w L 2 (,T,L 2 ( )) c 3, (3.8) u k,w L t 2 (,T,V ) c, (3.9) where the positive constants c i (i = 1... ) are not depending on w, nor on k. Proof. (i) Taking ψ = ϕ k,w ϕ in (2.11) and using the properties of σ, the conditions of ϕ and young s inequality, we deduce that σ(w) ϕ k,w 2 c 5 ϕ 2, where c 5 is a positive constant. Which obviously implies (3.6). (ii) Taking v = u k,w in (3.), it follows that u k,w t =, u k,w θ k(w) u k,w 2 ds σ(w)ϕ k,w ϕ k,w u k,w ds θ k(u) u n u k,w ds. β(u k,w u)u k,w ds σ(w)ϕ ϕ n u k,w ds So, we get 1 2 t u k,w 2 L 2 () θ k(w) u k,w 2 ds β u k,w 2 ds = βu k,w u ds σ(w)ϕ k,w ϕ k,w u k,w ds σ(w)ϕ ϕ n u k,w ds θ k(u) u n u ds.

6 132 Existence of weak solutions Using a lemma3.3 in [3, p. 25] and the hypotheses on θ k and β, and applying Young s inequality, there exist positive constants c i, (i = ) such that 1 2 t u k,w 2 L 2 () c 6 u k,w 2 ds c 7 u k,w 2 ds βu k,w u ds σ(w)ϕ k,w ϕ k,w u k,w ds c 8 c 6 2 c 7 σ(w)ϕ ϕ n u k,w ds ϕ k,w u k,w ds θ k(u) u n u ds βu k,w u u k,w 2 ds c 9 u k,w 2 ds c 7 θ k(u) u n u ds σ(w)ϕ ϕ n u k,w ds ϕ k,w 2 ds c 1 u k,w 2 ds c 11 ϕ n 2 ds u 2 ds c 12. Then we have 1 2 t u k,w 2 L 2 () c 6 u k,w 2 ds c 7 u k,w 2 ds 2 c 13 c 7 u k,w 2 ds c 7 u k,w 2 ds Therefore, c 13 c 7 c 13 c t u k,w 2 L 2 () c 6 2 u k,w 2 ds c 7 u k,w 2 ds c 7 2 u k,w 2 ds c 7 2 u k,w 2 ds u k,w 2 ds. u k,w 2 ds c 13 c 7 u 2 ds c 1. (3.1) Integrating (3.1) on (, T ), we conclude to the estimates (3.7) and (3.8). (iii) According to (3.), (3.6), (3.7), (3.8) and a lemma3.3 in [3, p. 25], we obtain u k,w L2 (,T,V t ) c. Hence Lemma 3.3 is proved.

7 Abderrahmane El Hachimi & Moulay Rachid Sidi Ammi 133 Now, we define the space { v W (, T ), v L2 (,T,V ) c W := 2, v t L 2 (,T,V ) c, v(x, t) M a.e. in Q T, v() = u(x, ) in Note that W is a non empty convex set, and by Lemma 3.3, the operator U k : W W 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 (, T ). Then, using the weak compactness of W we shall conclude that U k has a fixed point w in the set W. To prove the weak continuity, assume that (w j ) j is a sequence in W satisfying w j w weakly in W (, T ) and let (u k,wj, ϕ k,wj ) the corresponding sequence of solutions of (3.) (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 (, T, V ), w j w t t weak in L 2 (, T, V ), u k,wj u k weak in L 2 (, T, V ), u k,wj u k weak in L 2 (, T, L 2 ()), u k,wj u k weak in L 2 (, T, V ), t t ϕ k,wj ϕ k weak in L 2 (, T, H 1 ()). 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 σ is continuous and bounded, then, thanks to the dominated convergence theorem of Lebesgue, it follows that By the trace theorem, we derive that σ(w j ) σ(w) strongly in L 2 (Q T ). σ(w j ) ϕ k,w j n σ(w) ϕ k n weak in L2 ( ). 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 }. σ(w j ) ϕ k,wj α 1 weak in L 2 (Q T ), (3.11) θ k(w j ) u k,wj α 2 weak in L 2 (Q T ), (3.12) σ(w j )ϕ k,wj ϕ k,wj α 3 weak in L 2 (Q T ). (3.13) Now, as ϕ k,wj ϕ k weak in L 2 (Q T ), we have σ(w j ) ϕ k,wj σ(w) ϕ k in (Q T ). Consequently, α 1 = σ(w) ϕ k.

8 13 Existence of weak solutions In a similar way, we obtain α 2 = θ k(w) u k, α 3 = σ(w)ϕ k ϕ k. Then, passing to the limit as j in the relations (3.) and (3.5) satisfied by (u k,wj, ϕ k,wj ), we deduce immediately that u k = U k (w) and ϕ k = S k (w). By the unique solvability of (3.), all the sequence u k,wj converges to u k = U k (w) weakly in W (, T ). This completes the proof. Remark By theorem 3.2, we have u k M. Hence σ(u k ) = σ(u k ). 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.1 Let (u k, ϕ k ) be a solution of (3.1). Under the hypotheses (H1) (H), there exist constants c i (i = ) such that, for any k 1, the following estimates hold u k (t) 2 L 2 () u(x, ) 2 L 2 () c 15, (.1) u k 2 L 2 (,T,V ) u(x, ) 2 L 2 () c 15, (.2) 1 k u k 2 L 2 (,T,V ) c 16 2k ( u(x, ) 2 L 2 () c 17), (.3) θ k (u k ) 2 L 2 (,T,H 1 ()) c 18, (.) u k t L 2 (,T,V ) c 19. (.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. 25], we obtain 1 2 t u k(t) 2 L 2 () c 2 u k 2 ds c 21 u k 2 ds c 22 ϕ k u k ds c 23 u k u ds c 2 u k ϕ n ds c 25 u u ds. n The same arguments as in the proof of Lemma 3.3 lead to t u k(t) 2 L 2 () u k 2 ds u k 2 ds c 26. (.6)

9 Abderrahmane El Hachimi & Moulay Rachid Sidi Ammi 135 Integrating (.6) on (, t) yiel ds u k (t) 2 L 2 () Q t u k 2 ds t u k 2 ds u(x, ) 2 L 2 () c 15. Hence (.1) and (.2) are satisfied. On the other hand, by (3.2) we have u k t, u k ((θ k (u k ), u k )) β u k 2 = σ(u k )ϕ k ϕ k u k ds βu k u ds σ(u k )ϕ ϕ n u k ds θ k(u) u Γ u n u ds. Therefore, arguing exactly as above, we deduce that T T u k (t) 2 L 2 () 2 ((θ k (u k ), u k ))dt u k 2 ds c 16 ( u(x, ) 2 L 2 () c 17). Furthermore, T and < 1 k θ k (u k). Then ((θ k (u k ), u k ))dt = T ( ) θ k(u k ) u k 2 ds dt 1 k u k 2 L 2 (,T,H 1 ()) c 16 2 ( u(x, ) 2 L 2 () c 17). Which gives estimate (.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 u k t, θ k(u k ) θ k (u k ) 2 H 1 () β(u k u)θ k (u k ) ds = σ(u k )ϕ k ϕ k θ k (u k ) ds σ(u k )ϕ ϕ n θ k(u k ) ds. θ k(u) u n θ k(u k ) ds Straightforward calculations and Bamberger s lemma [8, p. 8] give d { dt ( u k (x,.) ) } θ k (r)dr ds 1 2 θ k(u k ) 2 H 1 () c 27. (.7) Integrating (.7) from to T yiel ds ( u k (x,t ) ) θ k (r)dr ds 1 2 θ k(u k ) 2 L 2 (,T,H 1 ()) c 18 2.

10 136 Existence of weak solutions Hence estimate (.) follows. According to (3.2), (.1), (.2), (.) and Lemma 3.3, [3, p. 25], and the definition of dual norm, we get the desired relation (.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.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 (, T, V ), u k u u k u weak star L (, T, L 2 ()), weak in L 2 (, T, L 2 ()), u k u weak in L 2 (, T, V ), t t ϕ k ϕ weak star in L (, T, H 1 ()). On the other hand, since the space {v L 2 (, T, V ), v t L2 (, 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 Moreover, we can assume that u k u strongly and a.e. in L 2 (Q T ). θ k (u k ) θ(u) a.e. in Q T and weak in L 2 (, T, H 1 ()). Indeed, we have t ( θ k (u k ) θ(u) dsdr t ( θ k (u k ) θ(u k ) ds)dr csup r M θ k (r) θ(r) t ( t ( θ(u k ) θ(u) ds)dr θ(u k ) θ(u) ds)dr. Now, arguing as in the proof of (3.11) (3.13), we obtain σ(u k ) ϕ k σ(u) ϕ weak in L 2 (Q T ), σ(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([, T ]; V ). Then u k () u() weak in V. We consequently obtain u(x, ) = u(x, ). 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.

11 Abderrahmane El Hachimi & Moulay Rachid Sidi Ammi 137 Remark. The question of uniqueness has been established in some special cases; see [] and [8]. References [1] A. Bamberger: Etude d une équation doublement non linéaire. Jour. of Functional Analysis, Volume. 2, n 2, pp , [2] J. L. Lions: Quelques méthodes de résolution de problèmes aux limites non linéaires, unod, Paris, [3] P. Shi, M. Shillor, X. Xu: Existence of a solution of the stefan problem with joule s heating. Jour. ifferential Equations. Vol 15, No.2, october [] S.N. Antontsev, M. Chipot: The thermistor problem: existence, smoothness, uniqueness and blowup. SIAM Jour. Math.Anal. Vol 25, No., pp July 199. [5] X. Xu: A strongly degenerate system involving an equation of parabolic type and equation of elliptic type, Commun. in Partial ifferential Equations, 18, pp , (1993). [6] X. Xu: On the existence of bounded tempeature in the thermistor problem with degeneracy. Nonlinear Analysis,T.M.A 2, pp , (2). [7] X. Xu: Local and global existence of continuous temperature in electrical heating of conductors, Houston Journal of Mathematiques, vol 22, No.2, [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 (1993). Abderrahmane El Hachimi UFR Mathématiques Appliquées et Industrielles Faculté des Sciences B.P 2, El Jadida - Maroc adress: elhachimi@ucd.ac.ma Moulay Rachid Sidi Ammi UFR Mathématiques Appliquées et Industrielles Faculté des Sciences B.P 2, El Jadida - Maroc adress: sidiammi@hotmail.com