We establish the existence of a solution in a certain sense to a strongly degenerate problem consisting in a coupled nonlinear
parabolic-elliptic system. The diffusion term in the parabolic equation is of the form − div a(x, t, u, ∇u), where a is an operator
of the Leray–Lions type. Moreover, the second equation is nonuniformly elliptic.Sur l’existence de solutions pour un système fortement dégénéré. On montre l’existence d’une solution dans un certain sens
d’un problème fortement dégénéré constitué par un système non-linéaire de deux équations aux dérivées partielles couplées du
type parabolique-elliptique, le terme de diffusion de l’équation parabolique étant de la forme − div a(x, t, u, ∇u), où a est un
opérateur du type de Leray–Lions. En outre, la seconde équation de ce système est non-uniformement elliptique.Ministerio de Ciencia y Tecnología BFM2003-01187Junta de Andalucía FQM-31

González Montesinos, María Teresa

Ortegón Gallego, Francisco

idUS. Depósito de Investigación Universidad de Sevilla

Partial Differential EquationsOn the existence of solutions for a strongly degenerate systemMaría Teresa González Montesinos a, Francisco Ortegón Gallego ba Departamento de Matemáticas, Universidad de Cádiz, Facultad de Ciencias Empresariales, 11002 Cádiz, Spainb Departamento de Matemáticas, Universidad de Cádiz, CASEM, Campus del Río San Pedro, 11510 Puerto Real, SpainAbstractWe establish the existence of a solution in a certain sense to a strongly degenerate problem consisting in a coupled nonlinear parabolic-elliptic system. The diffusion term in the parabolic equation is of the form − div a(x, t, u, ∇u), where a is an operator of the Leray–Lions type. Moreover, the second equation is nonuniformly elliptic. Sur l’existence de solutions pour un système fortement dégénéré. On montre l’existence d’une solution dans un certain sens d’un problème fortement dégénéré constitué par un système non-linéaire de deux équations aux dérivées partielles couplées du type parabolique-elliptique, le terme de diffusion de l’équation parabolique étant de la forme − div a(x, t, u, ∇u), où a est un opérateur du type de Leray–Lions. En outre, la seconde équation de ce système est non-uniformement elliptique.Version française abrégéeOn étudie un problème aux limites d’évolution constitué par deux équations aux dérivées partielles non-linéaires dutype parabolique-elliptique. Ce problème peut être considéré comme une généralisation du problème du thermisteur[1–5,8–10]. Le résultat énoncé dans cette Note est une version plus générale d’un résultat de Xu [8].Soient N  2, T > 0, et Ω ⊂ RN un ensemble ouvert, borné et suffisament régulier, on pose ΩT = Ω × (0, T ) etΓT = ∂Ω × (0, T ). Étant données les fonctions u0 :Ω → R et ϕ0 :ΓT → R, on considère le problème⎧⎪⎨⎪⎩∂u∂t− diva(x, t, u,∇u) = σ(u)|∇ϕ|2 dans ΩT ,div(σ (u)∇ϕ) = 0 dans ΩT ,u(·,0) = u0 dans Ω, u = 0, ϕ = ϕ0 sur ΓT .(1)Dorénavant, on prend p  2 et on pose p′ = pp−1 . Les hypothèses sur les données sont (H.1)–(H.5) (voir plus bas).On observe donc que l’équation elliptique peut être dégénérée sur l’ensemble {u = +∞}. En effet, l’hypo-thèse (H.5) permet de considérer le cas σ(s) → 0 quand s → +∞ (ce qui se passe dans les plupart des cas d’intérêtpratique). Par conséquent, on ne peut pas s’attendre à obtenir d’estimations à priori pour ∇ϕ, et donc ϕ n’appar-tient pas à aucun espace de Sobolev. Ceci veut dire que l’hypothèse (H.5) empêche la recherche de solutions faiblesde (1). Néanmoins, si l’on considère la expression Φ = σ(u)∇ϕ comme une seule fonction, alors on peut montrerque Φ ∈ L2(ΩT )N , et ceci va nous permettre donner une nouvelle formulation du système (1) pour lequel on montrel’existence d’une solution : il s’agît de la solution de capacité (voir la Définition 2.1 ci-après), une notion introduitepar Xu [8] dans l’étude d’une version modifiée du problème du thermisteur.Le résultat principal de cette Note est le suivant :Théorème 0.1. Sous les hypothèses (H.1)–(H.6), le problème (1) admet au moins une solution de capacité.1. Introduction and statement of the problemWe prove an existence result to a coupled nonlinear parabolic-elliptic system, which may be regarded as a general-ized version of the well-known thermistor problem arising in electromagnetism [1–5,8–10]. Also, this Note extends aprevious result due to Xu [8].For N  2 and T > 0, let Ω ⊂ RN be an open, bounded and smooth enough set, and write ΩT = Ω × (0, T ) andΓT = ∂Ω × (0, T ). Given the functions u0 :Ω → R and ϕ0 :ΓT → R, we consider the system⎧⎪⎨⎪⎩∂u∂t− diva(x, t, u,∇u) = σ(u)|∇ϕ|2 dans ΩT ,div(σ (u)∇ϕ) = 0 dans ΩT ,u(·,0) = u0 dans Ω, u = 0, ϕ = ϕ0 sur ΓT .(1)For p  2, let p′ = pp−1 and suppose the following hypotheses on data:(H.1) a :ΩT ×R×RN → RN is a Carathéodory function.(H.2) There exists α > 0 such that, for all s ∈ R and ξ, η ∈ RN , and almost everywhere in ΩT , [a(x, t, s, ξ) −a(x, t, s, η)](ξ − η) α|ξ − η|p .(H.3) a(x, t, s,0) = 0.(H.4) There exist a nonnegative function b ∈ Lp′(ΩT ) and a constant β > 0 such that, for all s ∈ R and ξ ∈ RN , andfor almost every (x, t) ∈ ΩT , |a(x, t, s, ξ)| β[b(x, t)+ |s|p−1 + |ξ |p−1].(H.5) σ ∈ C(R) and 0 < σ(s) σ¯ , for all s ∈ R.(H.6) ϕ0 ∈ L2(0, T ;H 1(Ω))∩L∞(ΩT ) and u0 ∈ L2(Ω).Our interest is focused on the search of solutions to system (1) bearing in mind, apart from the form of the diffusionterm in the parabolic equation of (1), that the hypothesis on σ prevents from the search of weak solutions. Indeed,since σ(s) may converge to zero as |s| → ∞, the elliptic equation of (1) becomes degenerate when u is infinite.Thus, no a priori estimates for ∇ϕ will be available and so, ϕ might not belong to a Sobolev space. However, takingthe function Φ = σ(u)∇ϕ instead of ϕ, it is possible to prove that Φ ∈ L2(ΩT )N , which enable us to give a newformulation of (1). Its solution is called capacity solution. The notion of capacity solution was first introduced by Xuin the analysis of a modified version of the thermistor problem [7–10].From this point on, we use the following notation: Lp(X) stands for Lp(0, T ;X), X being a Banach space.We also introduce the following spaces together with their respective dual spaces: H = H 10 (Ω), H ′ = H−1(Ω),V = W 1,p0 (Ω), V ′ = W−1,p′(Ω),H= L2(H),H′ = L2(H ′), V = Lp(V ) and V ′ = Lp′(V ′). Also, we write a(u,∇u)instead of a(x, t, u,∇u). Finally, when passing to the limit, the subsequences will be denoted in the same way as theirrespective original sequences, except where explicitly noted.2. Notion of capacity solution and main resultDefinition 2.1. A triplet (u,ϕ,Φ) is called a capacity solution to problem (1) if the following conditions are fulfilled:(C.1) u ∈ W = {v ∈ V | ∂v∂t∈ V ′}, ϕ ∈ L∞(ΩT ), Φ ∈ L∞(L2(Ω))N .(C.2) dudt − diva(u,∇u) = div(ϕΦ) in V ′, divΦ = 0 in H′.(C.3) For every S ∈ C10(R), S(u)ϕ − S(0)ϕ0 ∈H and S(u)Φ = σ(u)[∇(S(u)ϕ)− ϕ∇S(u)].(C.4) u(·,0) = u0.The nonuniform character of the elliptic equation in (1) is one of the main difficulties in the resolution of thissystem. Consider the new hypothesis (H.5′), namely,σ ∈ C(R) and 0 < σ1  σ(s) σ2, for all s ∈ R. (H.5′)We have the following result [6]:Theorem 2.2. Under hypotheses (H.1)–(H.6), with (H.5′) instead of (H.5), system (1) admits a weak solution (u,ϕ),that is, u ∈ V , dudt ∈ V ′, ϕ − ϕ0 ∈ L∞(H)∩L∞(ΩT ), u(·,0) = u0 in Ω andt∫0〈dudt, φ〉+t∫0∫Ωa(u,∇u)∇φ =t∫0∫Ωσ(u)|∇ϕ|2φ, ∀φ ∈ V ∩L∞(ΩT ), ∀t ∈ [0, T ], (2)∫Ωσ(u)∇ϕ∇ξ = 0, ∀ξ ∈H, a.e. t ∈ (0, T ). (3)Furthermore, the pair (u,ϕ) satisfies the following properties:‖ϕ‖L∞(ΩT )  ‖ϕ0‖L∞(ΩT ), ess supt∈[0,T ]∫Ωσ(u)|∇ϕ|2  C(σ1, σ2, ϕ0), ‖u‖H  C(u0, ϕ0, T , σ1, σ2, α); (4)〈σ(u)|∇ϕ|2, ξ 〉H′,H = −∫ΩTσ(u)ϕ∇ϕ∇ξ, for all ξ ∈H. (5)The main result of this Note now follows:Theorem 2.3. Under hypotheses (H.1)–(H.6), there exists a capacity solution to problem (1).3. Proof of Theorem 2.3Let n ∈ N and put σn(s) = σ(s)+ 1/n. The approximate problem of (1) is given by∂un∂t− diva(un,∇un) = div(σn(un)ϕn∇ϕn)in ΩT , (6)div(σn(un)∇ϕn)= 0 in ΩT , (7)un = 0, ϕn = ϕ0 on ΓT , un(·,0) = u0 in Ω. (8)By (H.5), 1/n  σn(s)  σ¯ + 1 = σˆ . According to Theorem 2.2, problem (6)–(8) admits a weak solution (un,ϕn)satisfying (2)–(5), where (div(σn(un)ϕn∇ϕn)) ⊂H′ ↪→ V ′ is bounded. Also, the sequences (un), (a(un,∇un)) and( dundt ) are bounded in V , Lp′(ΩT )N and V ′, respectively.We can then deduce the existence of functions ϕ ∈ L∞(ΩT ), Φ ∈ L∞(L2(Ω)N) and u ∈ V such that ϕn ⇀ ϕweakly– in L∞(ΩT ), σn(un)∇ϕn ⇀Φ weakly– in L∞(L2(ΩT )N) and un ⇀ u weakly in V . Furthermore, as (un)is relatively compact in Lp(ΩT ), we may assume thatun −→ u strongly in Lp(ΩT ) and a.e. in ΩT , (9)and thus, dundt ⇀dudt weakly in V ′. Therefore, the triplet (u,ϕ,Φ) satisfies condition (C.1) of Definition 2.1. Ow-ing to (H.5) and (9), σn(un) ⇀ σ(u) weakly– in L∞(ΩT ) and almost everywhere in ΩT . Finally, there existsΥ ∈ Lp′(ΩT )N such that a(un,∇un)⇀ Υ weakly in Lp′(ΩT )N .3.1. Strong convergence of (∇un)Since (1) is nonlinear, (9) does not lead to Υ = a(u,∇u). To do so, it is enough to show that ∇un → ∇u almosteverywhere in ΩT . In fact, this is the goal of the next result [6]:Lemma 3.1. Assume (H.1)–(H.6) and let (un,ϕn) be a weak solution of system (6)–(8). Then (un) is relatively compactin Lq(W 1,q0 (Ω)) for all q ∈ [1,p).3.2. L1-convergence of (ϕn)In order to show the strong convergence of (ϕn) in L1(QT ), up to a suitable subsequence, we need some usefulresults. First, we have the following lemma, which is a modified version of that due to Xu [8]:Lemma 3.2. Let (un) be a bounded sequence in V and relatively compact in Lp(ΩT ). Then there exists a subsequence(un(k)) ⊂ (un) such that, for every ε > 0, there exist a constant value M = M(ε) > 0 and a function ψ ∈ L1(W 1,1(Ω))such that 0ψ  1, ‖ψ − 1‖L1(ΩT ) + ‖∇ψ‖L1(ΩT )  ε and |u|, |un(k)|M in {ψ > 0} for all k  1.On the other hand, we can show the following intermediate results:Lemma 3.3. For every S ∈ C10(R), S(un)ϕn ⇀ S(u)ϕ weakly- in L∞(ΩT ).Lemma 3.4. For every S ∈ C10(R) such that 0 S  1,lim supn→∞∫ΩTσn(un)∣∣∇(S(un)ϕn − S(u)ϕ)∣∣2 C‖S′‖∞(1 + ‖S′‖∞).Now we are ready to state the strong convergence of some subsequence (ϕn(k)) in L1(ΩT ).Lemma 3.5. There exists a subsequence (ϕn(k)) ⊂ (ϕn) such that ϕn(k) → ϕ strongly in L1(ΩT ).Proof of Lemma 3.5. The sequence (un) satisfies the assumptions of Lemma 3.2, so that, for every ε > 0, there existM > 0 and ψ ∈ L1(W 1,1(Ω)) as in Lemma 3.2. Furthermore, σn(k)(un(k)) CM in {ψ > 0} for all k  1.Choose (Sm) ⊂ C10(R) so that 0 Sm  1, Sm = 1 in [−M,M] and ‖S′m‖∞ C/m for all m 1 and some C > 0.Then we can write∫ΩT|ϕn(k) − ϕ| =∫ΩT∣∣Sm(un(k))ϕn(k) − Sm(u)ϕ∣∣ψ +∫ΩT|ϕn(k) − ϕ|(1 −ψ).Taking into account Lemma 3.4 and properties of Sm, it can be shown thatlim supk→∞∫ΩT|ϕn(k) − ϕ|Cε +KM[C‖S′m‖∞(1 + ‖S′m‖∞)]1/2 Cε +KMC1/2[Cm(1 + Cm)]1/2,where KM = CC−1/2M (measΩ)1/2T 1/2. Making ε → 0 and m → ∞ yields the result. The convergences deduced in the previous sections lead to (C.2) of Definition 2.1. Finally, in order to obtain (C.3),it is enough to make k → ∞ in the equalityS(un(k))σn(k)(un(k))∇ϕn(k) = σn(k)(un(k))[∇(S(un(k))ϕn(k))− ϕn(k)∇S(un(k))].This ends the proof of Theorem 2.3.Remark 1. The uniqueness of capacity solutions to problem (1) has not been analyzed yet and it is a very complextask. All known uniqueness results for the thermistor problem rely on restrictive hypotheses on data, for instanceϕ0 ∈ L∞(0, T ;W 1,∞(Ω)). However, in that setting there is no need to search for capacity solutions, because theregularity of the solutions leads us to the context of weak solutions.Remark 2. The hypothesis p  2 is used in two points: firstly, in the inclusion L2(0, T ;H−1(Ω)) ↪→ Lp′(0, T ;W−1,p′(Ω)), which yields div (σ (w)ϕ∇ϕ) ∈ Lp′(0, T ;W−1,p′(Ω)), for every w ∈ Lp(Q). Secondly, along the proofof the strong convergence in L1 of a subsequence (ϕn(k)) (see [6] for details).AcknowledgementsThe authors wish to thank the referee for her/his useful suggestions and corrections on the previous version of thispaper which have improved its presentation.This work was partially supported by Ministerio de Ciencia y Tecnología under grant BFM2003-01187 with theparticipation of FEDER fund and by Consejería de Educación y Ciencia de la Junta de Andalucía group FQM-315.References[1] S.N. Antontsev, M. Chipot, The thermistor problem: existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal. 25 (4) (1994) 1128–1156.[2] M.T. González Montesinos, Estudio matemático de algunos problemas no lineales del electromagnetismo relacionados con el problema deltermistor, PhD Thesis, Universidad de Cádiz, 2002.[3] M.T. González Montesinos, F. Ortegón Gallego, The evolution thermistor problem with degenerate thermal conductivity, Comm. Pure Appl.Anal. 1 (3) (2002) 313–325.[4] M.T. González Montesinos, F. Ortegón Gallego, On certain non-uniformly and singular non-uniformly elliptic systems, Nonlinear Anal. 54(2003) 1193–1204.[5] M.T. González Montesinos, F. Ortegón Gallego, Algunos resultados sobre el problema del termistor, Bol. Soc. Esp. Mat. Apl. 31 (2005)109–138.[6] M.T. González Montesinos, F. Ortegón Gallego, Existence of a capacity solution to a coupled nonlinear parabolic-elliptic system. Commun.Pure Appl. Anal., in press.[7] X. Xu, A degenerate Stefan-like problem with Joule’s heating, SIAM J. Math. Anal. 23 (1992) 1417–1438.[8] X. Xu, A strongly degenerate system involving an equation for parabolic type and an equation of elliptic type, Comm. Partial DifferentialEquations 18 (1993) 199–213.[9] X. Xu, The thermistor problem with conductivity vanishing for large temperature, Proc. Roy. Soc. Edinburgh Sect. A 127 (1994) 1–21.[10] X. Xu, On the existence of bounded temperature in the thermistor problem with degeneracy, Nonlinear Anal. 42 (2000) 199–213.

Partial Differential Equations: On the existence of solutions for a strongly degenerate system

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Partial Differential Equations: On the existence of solutions for a strongly degenerate system

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