On a nonlocal Cahn-Hilliard equation with a reaction term ∗ Stefano Melchionna† Elisabetta Rocca‡ January 8, 2015 5 1 0 2 Abstract n Weproveexistence,uniqueness,regularityandseparation propertiesforanonlocalCahn-Hilliard a equation with a reaction term. We deal here with the case of logarithmic potential and degenerate J mobility as well an uniformly lipschitz in u reaction term g(x,t,u). 7 ] P 1 Introduction A . h Our aim is to generalize existence, uniqueness, separation property and regularity results, proved by t a Gajewski, Zacharias [GZ] and Londen and Petzeltov`a [LP2] for the nonlocal Cahn-Hilliard equation, to m thenonlocalCahn-Hilliardequationwithreaction. Hence,weaimtostudythefollowinginitialboundary [ value problem: 1 ∂tu−∇·(µ∇v)=g(u) in Q, (1.1) v v =f′(u)+w in Q, (1.2) 1 4 w(x.t)= K(|x−y|)(1−2u(y,t))dy for (x,t)∈Q, (1.3) 5 ZΩ 1 n·µ∇v =0 on Γ, (1.4) 0 . u(x,0)=u (x), x∈Ω, (1.5) 1 0 0 where Q=Ω×(0,T), Ω⊂Rd is a bounded domain, Γ=∂Ω×(0,T), and n is the outer unit normal to 5 ∂Ω. The functions f and µ are definite by 1 : v f(u)=ulogu+(1−u)log(1−u), (1.6) i X 1 µ= =u(1−u). (1.7) r f′′(u) a The man novelty of the paper is that we can take into account in our analysis of the reaction term g in (1.1), which can be taken as a Lipschitz continuous function of the unknown u. Let us briefly recall here - for the readers’ convenience - the derivation of the nonlocal Cahn-Hilliard equation and the comparison with the local one. System (1.1)–(1.5) describes the evolution of a binary alloy with components A and B occupying a spatial domain Ω. We denote by u the local concentration of A. To describe phase separationin binary system one uses generally the standard local Cahn-Hilliard equation, which is derived (cf. [CH]) from a free energy functional of this form of the form τ2 E (u)= |∇u|2+F(u) dx. (1.8) CH 2 ZΩ(cid:18) (cid:19) ∗Acknowledgment. ThefinancialsupportoftheFP7-IDEAS-ERC-StG#256872(EntroPhase)isgratefullyacknowl- edgedbytheauthors. Thepresentpaperalsobenefits fromthesupportoftheGNAMPA(GruppoNazionaleperl’Analisi Matematica, laProbabilita`eleloroApplicazioni)ofINdAM(Istituto NazionalediAltaMatematica). †University of Vienna , Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. E-mail: mel- [email protected] ‡Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany. E-mail: [email protected] andDipartimentodiMatematica“F.Enriques”, Universit`adegliStudi diMilano,Milano I-20133, Italy. E-mail: [email protected] 1 Here F(u) denotes the Helmholtz free energy density of A. It is defined as F(u)=2K T u(1−u)+K Tf(u), (1.9) B c B where K is the Boltzmann’s constant, T is the system temperature, T is called critical temperature B c and f is defined as f(u)=ulnu+(1−u)ln(1−u). (1.10) Considering that the dynamics tends to minimize the energy E , Cahn obtained ([Ca]) the following CH equation for u: u +∇·J =0 (1.11) t where J is defined as J =−µ(u)∇v. (1.12) The function µ is named mobility and v denotes the first variation of E with respect to u: CH δE v = CH =F′(u)−τ2∆u, (1.13) δu known as chemical potential. For simplicity, in literature the mobility is often chosen constant although its physical (degenerate) relevant form is µ=au(1−u), a>0 (1.14) (see[Ca]),whereaisapositivefunctionpossiblydependingonuand∇useparatedfrom0(inliteraturea isoftenapositiveconstant). Equation(1.11)is,hence,a4thordernonlinearPDEknownasCahn-Hilliard equation: u +∇· µ(u)∇(F′(u)−τ2∆u) =0, (1.15) t which is usually coupled with the followin(cid:0)g boundary conditions:(cid:1) ∂u =0 on ∂Ω and µ(u)n·∇v =0 on ∂Ω. (1.16) ∂n This last condition ensures the mass conservation. Indeed, thanks to (1.16), we have: d u= u =− ∇·(µ(u)∇v)= µ(u)n·∇v =0. t dt ZΩ ZΩ ZΩ Z∂Ω Despite numerical results on the Cahn-Hilliard equation are in good agreementwith experiments, G. Giacomin and J. L. Lebowitz in [GL1] and [GL2] showed that Cahn-Hilliard equation cannot be derived frommicroscopicphenomena. ThismotivationledG.GiacominandJ.L.Lebowitztostudy theproblem of phase separation from the microscopic viewpoint using statistical mechanics. Then, performing the hydrodynamic limit they deduced a continuum model. By proceeding in this way they found a nonlocal version of the Cahn-Hilliard equation that is a second order nonlinear integrodifferential equation: u +∇·J =0 (1.17) t where J is defined as in (1.12), µ denotes the mobility term (defined as in (1.14)), and v = δE. Here the δu energy functional E is given by 1 E(u)= K(x−y)(u(x)−u(y))2dxdy+ f(u)+ku(1−u)dx. (1.18) 2 ZΩZΩ ZΩ This leads to v =f′(u)+w, where w =K∗(1−2u), (1.19) and where K is a symmetric positive convolution kernel, k(x) = K(x−y)dy and f is defined as in Ω (1.10). R Nonlocal Cahn-Hilliard equation is generally coupled with the boundary condition µ(u)n·∇v =0 on ∂Ω. (1.20) 2 Thus, the mass-conservation is still preserved. Notice that the nonlocal contribution 1 K(x − 2 Ω Ω y)(u(x)−u(y))2dxdy in (1.18), replacing the local one τ2 |∇u|2, better describes the long-range in- Ω 2 R R teractions between points in Ω. Moreover, let us note that the local term τ2 |∇u|2 could be formally R Ω 2 obtained from the nonlocal one (cf. [KRS]). R AddingareactiontermtotheCahn-Hilliardequationisusefulinseveralapplicationssuchasbiological models ([KS]), impainting algorithms([BEG]), polymers ([BO]). Cahn-Hilliardequationwith reactionis u +∇·J =g(u), (1.21) t where J =−µ∇v and v as in (1.13) or as in (1.19) and g(u)=g(x,t,u). ThemaindifficultiesinstudyingCahn-Hilliardequationwithreactionareduetothenon-conservation of the mass. Indeed, thanks to the boundary condition (1.20), we have d u= g 6=0. (1.22) dt ZΩ ZΩ Some analytical results on the local Cahn-Hilliard equation with reaction term are [CMZ], [Mi]. Ex- istence and uniqueness for nonlocal Cahn-Hilliard equation with constantmobility, polynomial potential and reaction term are proved in [DP]. Tothe bestofourknowledgenopreviousworksonthe nonlocalCahn-Hilliardequationwithreaction andwithsingularpotentialanddegeneratemobilityhavebeenproved. Furthermore,ourassumptionson the reaction term (see (G1)-(G3)) are more general then in [CMZ], [Mi] and [DP] and they are satisfied in every application we know (cf., e.g., [KS], [BEG], [BO]). Plan of the paper. In Section 2 we set notation, describe assumptions on data and state the main results. Existence and uniqueness are proved in Section 3. Regularity results are proved in Section 6. Section 7 is devoted to the proofs of the separation properties. Some remarks are stated in Section 8. Appendix (Section 9) contains example of convolution kernels and auxiliary theorems. 2 Assumptions on data and main results 2.1 Notation Set Ω⊂Rd, d∈N, a bounded domain with a sufficiently smooth boundary (e.g., of class C1,1). If X is a real Banachspace, X∗ will denote its dual. For every z ∈(H1(Ω))∗ we denote z¯= z, 1 . |Ω| Here h,i denotes the pairing of H1(Ω) and H1(Ω) ∗. Let us introduce also the space H1(Ω)D= {zE∈ 0 H1(Ω):z¯=0}. (cid:0) (cid:1) Set H1(0,T,X,X∗)={ z ∈L2(0,T,X): z ∈L2(0,T,X∗)}. t If z ∈ H1(0,T,X,X∗) the symbols z′, ∂ z, ∂z, and z will denote the partial derivative of z with t ∂t t respect to the t-variable (time). Let f ∈ C1(R), we use the symbol f′ to denote the derivative of the functionf. Finally,sety ∈H1([0,T]×Ω),we indicatethe partialderivativeofy withrespecttothe first variable (time) with the symbols ∂ y or ∂y and the partial derivate of y with respect to the x −variable t ∂t i with the symbol ∂ y. i Ifα:Rd →Randβ :Ω⊂Rd →Raremeasurablefunctionsα∗β willdenotetheconvolutionproduct definite by α∗β(x)= α(x−y)β(y)dy for x∈Rd. Ω R 2.2 Assumptions on data The given functions K, u and g will be assumed to fulfill the following conditions. 0 3 (K) The convolution kernel K :Rd →R satisfies the assumptions K(x)=K(−x) for a.a. x∈Rd, (K1) sup |K(x−y)|dy <+∞, (K2) x∈ΩZΩ ∀p∈[1,+∞] ∃r >0 such that kK∗ρk ≤r kρk , (K3) p W1,p(Ω) p Lp(Ω) ∃C >0 such that kK∗ρk ≤Ckρk ; (K4) W2,2(Ω) W1,2(Ω) (u0) The initial datum u is supposed to satisfy 0 u is measurable, (U01) 0 0≤u (x)≤1 for a.a. x∈Ω, (U02) 0 0<u¯ <1; (U03) 0 (G) The reaction term g :Ω×R+×[0,1]→R is such that g(x,t,s) is continuous, (G1) ∃L>0 such that |g(x,t,s )−g(x,t,s )|≤L|s −s |∀s ,s ∈[0,1], ∀x∈Ω, ∀t∈R+ (G2) 1 2 1 2 1 2 g(x,t,0)≥0≥g(x,t,1) ∀x∈Ω, ∀t∈R+. (G3) We remark that, as a consequence of (G1) for every T >0, there exist C >0 depending on T so that |g(x,t,s)|≤C ∀s∈[0,1],t∈[0,T],x∈Ω. (2.1) Furthermore, as a consequence of (G2), we have g is differentiable for a.a. s∈[0,1] and |∂ g(x,t,s)|≤L for a.a. (x,t,s)∈Ω×R+×[0,1], s where L as in (G2) (see [NZ]). Remark 1 Some examples of convolution kernels K which satisfy the above conditions (K1)-(K4) are given by Newton potentials: K(|x|)=k |x|2−d for d>2 d K(|x|)=−k ln|x| for d=2 (cid:26) 2 where k = cost > 0, gaussian kernel K(|x|) = Cexp(−|x|2/λ) and mollifiers (cf. Section 9.1 in the d Appendix). Remark 2 Examples of functions g which satisfy the conditions (G1)-(G3) are given by both classical reactions terms as g(u)=±(u3−u) and terms used in recent applications of the Cahn-Hilliard equations as g(x,t,u)=α(x,t)u(1−u) ([KS]), g(x,t,u)=λ(x)(h(x)−u) ([BEG]) or g(x,t,u)=−σ(x,t)u ([BO]) where λ,h,α and σ are continuous and positive functions, h<1. 2.3 Main results Before stating the main results of this work, let us introduce the definition of weak solution to system (1.1)-(1.5). Definition 3 Let u ,K,g be such that conditions (U01)-(U03), (K1)-(K4), (G1)-(G3) are satisfied. 0 Then, given T ∈(0,+∞), u is a weak solution to (1.1)-(1.5) on [0,T] if u∈H1(0,T,H1(Ω), H1(Ω) ∗), (2.2) 0≤u≤1 a.e.(cid:0)in Q, (cid:1) (2.3) 4 w=K∗(1−2u) a.e. in Q, w ∈C([0,T],W1,∞(Ω)), u(0)=u in L2(Ω), 0 andthefollowing variational formulation issatisfiedalmost everywherein (0,T)andfor everyψ ∈H1(Ω) hu ,ψi+(µ(u)∇w,∇ψ)+(∇u,∇ψ)=(g(u),ψ). (2.4) t Remark 4 As consequence of (2.2), u ∈ C([0,T],L2(Ω)). Hence, the initial condition (1.5) makes sense. Moreover, let us note that this notion of solution turns out to be particularly useful since it does not involve the potential f and so it can be stated for solutions u∈[0,1], not necessarily different from 0 and 1 (cf. also [FGR] for further comments on this point). Here we state our first result whose proof is given in Section 3. Theorem 5 Let (K1)-(K4), (U01)-(U03) and (G1)-(G3) be satisfied. Then there exists unique u∈H1(0,T,H1(Ω),(H1(Ω))∗)(֒→C([0,T],L2(Ω))) weak solution to (1.1) in the sense of Definition 3. Furthermore, if u i∈{1,2}, are two solutions to (1.1)-(1.4) in the sense of Definition 3 with initial i data u , i∈{1,2}, then, for every t∈[0,T], the following continuous dependence estimate: 0i ku −u k ≤exp(Ct)ku −u k (2.5) 1 2 L∞(0,t,L2(Ω)) 01 02 L2(Ω) holds true, where C >0 does not depend on t nor on u and u . 01 02 The proof is given in Section 3. Under additional assumptions on the initial data u and the function g we obtain more regularity on 0 u, as stated in the following result proved in Section 6. Theorem 6 Let the assumptions of Theorem 5 be satisfied. Let u be the weak solution to (1.1)-(1.5) in the sense of Definition 3. Moreover, assume that g and u satisfy: 0 ∃L>0 such that |g(x,t ,s)−g(x,t ,s)|≤L|t −t |, ∀t ,t ∈[0,T], ∀x∈Ω, ∀s∈[0,1], (G4) 1 1 1 2 1 2 u ∈H2(Ω), (2.6) 0 and n·(∇(u )+µ(u )∇K ∗(1−2u ))=0 on ∂Ω. (2.7) 0 0 0 Then u∈L∞(0,T,H2(Ω)). Remark 7 Since u∈L∞(0,T,H2(Ω))∩C([0,T],L2(Ω)), thanks to Lemma 32 in the Appendix, we have u∈C([0,T],Hs(Ω)) for every s<2 and hence u∈C([0,T],L∞(Ω)) if d≤3. If the initial data do not satisfy (2.6)-(2.7) the solution u is more regular only on the set [T ,T] for 0 any T >0. 0 Corollary 8 Letubesolutionto(1.1)-(1.5)inthesenseofDefinition3. LettheassumptionsofTheorem 5 be satisfied. Assume that g satisfies (G4). Then for every T ∈(0,T) u∈L∞(T ,T,H2(Ω)). 0 0 More regularity on v can be obtained under an additional assumption on the initial datum. 5 Theorem 9 Let the assumption of Theorem 5 be satisfied and let u such that 0 f′(u )∈L2(Ω). (2.8) 0 Then the weak solution u given by Theorem 5 fulfills v ∈L∞(0,T,L2(Ω)) ∇v ∈L2(0,T,L2(Ω)). (2.9) Remark 10 As a consequence of Theorem 9 the function v = f′(u)+w is well defined. Hence u 6= 0 and u6=1 a.e. in Ω×[0,T]. Furthermore u also satisfies the weak formulation given by Definition 3 with hu ,ψi+(µ(u)∇v,∇ψ)=(g(u),ψ), v =f′(u)+w, t instead of (2.4). Corollary 8 and Theorem 9 are proved in Section 6. In [LP2, Theorem 2.1] Londen and Petzeltov`a obtained the separation properties for the solution to (1.1)-(1.5) with g =0. We prove here the same results in the case g satisfies (G1)-(G3). Theorem 11 Let the assumptions of Theorem 6 be satisfied and d≤3. Then ∀T ∈(0,T) ∃k >0 such that k ≤u(x,t)≤1−k for a.a. x∈Ω,t∈(T ,T). (2.10) 0 0 Furthermore, if ∃k˜ >0 such that k˜ ≤u ≤1−k˜, (2.11) 0 then T =0. 0 Remark 12 If u do not satisfy (2.6) or (2.7), using Corollary 8 and applying Theorem 11 on the set 0 (t,T) where t>0 is small enough, we can anyway obtain (2.10). Theorem 11 is proved in Section 7. 3 Existence and uniqueness ThissectionisdevotedtotheproofofTheorem5. Wefirstproveuniquenessofsolutionsbydemonstrating estimate (2.5), then we prove existence of solutions by approximating our problem with a more regular problemP andthenpassingtothelimitasε→0viasuitablea-prioriestimatesandcompactnessresults. ε 4 Uniqueness We now prove the uniqueness of the solution. In the following formulas the symbol C denotes a positive constant depending on T, K, and g. It may vary even within the same line. Proof of (2.5). Let u and u be as in Theorem 5. Then i 0i h∂ u ,ψi=−(∇u +µ ∇w ,∇ψ)+(g(u ),ψ) ∀ψ ∈H1(Q), a.e. in (0,T), (4.1) t i i i i i where µ =µ(u )=u (1−u ) and w =K∗(1−2u ). Computing the difference of (4.1) with i=1 and i i i i i i i= 2, choosing ψ =u:=u −u and integrating on (0,t), t∈(0,T], we obtain 1 2 1 1 t ku(t)k2 − ku −u k2 = h∂ u,ui (4.2) 2 L2(Ω) 2 01 02 L2(Ω) t Z0 t =− |∇u|2 Z0 ZΩ t − (µ ∇w −µ ∇w )∇u 1 1 2 2 Z0 ZΩ t + (g(u )−g(u ))u. 1 2 Z0 ZΩ 6 Using the bounds onu ,u ,µ andµ (see (1.7) and(2.3)) andassumption(K3) we obtainthe following 1 2 1 2 estimates 1 1 (µ ∇w −µ ∇w )∇u ≤ |∇u|2+ |µ ∇w −µ ∇w |2 1 1 2 2 1 1 2 2 2 2 (cid:12)ZΩ (cid:12) ZΩ ZΩ (cid:12) (cid:12) and (cid:12) (cid:12) (cid:12) (cid:12) |µ ∇w −µ ∇w |2 ≤ |µ (∇w −∇w )|2+ |(µ −µ )∇w |2 1 1 2 2 1 1 2 1 2 2 ZΩ ZΩ ZΩ ≤Ck∇w −∇w k2 1 2 L2(Ω) + |(u −u )(1−u −u )∇w |2 1 2 1 2 2 ZΩ ≤Ck∇w −∇w k2 +Ck∇w k2 kuk2 1 2 L2(Ω) 2 L∞(Ω) L2(Ω) ≤Cr2kuk2 +Cr2 kuk2 ≤Ckuk2 2 L2(Ω) ∞ L2(Ω) L2(Ω) where r and r as in (K3). Furthermore, using (G2) we have 2 ∞ (g(u )−g(u ))u≤ Lu2 ≤Lkuk2 , 1 2 L2(Ω) ZΩ ZΩ where L as in (G2). So, thanks to (4.2), for every t∈(0,T), we obtain t ku(t)k2 ≤2ku −u k2 +C kuk2 . L2(Ω) 01 02 L2(Ω) L2(Ω) Z0 Using the Gronwall’s Lemma, we get (2.5), and so also uniqueness of solutions is proved. 5 Existence In order to show the existence of the solution to (1.1)–(1.5) we study an approximate problem P de- ε pending on a parameter ε. We prove the existence of the solution u to P and, finally, we obtain u as ε ε limit (for ε→0) of u in a proper functional space. ε 5.1 Approximate problem P ε We start extending the domain of the function g(x,t,s) to every s ∈ R since we cannot prove that the solution u to the approximateproblem satisfies the condition u ∈[0,1]for a.a. x∈Ω,t∈[0,T]. Let us ε ε define the function g1 :Ω×R+×R→R: g1(x,t,s)=g(x,t,0) ∀x∈Ω ∀t∈R+, s≤0 g1(x,t,s)=g(x,t,s) ∀x∈Ω ∀t∈R+, s∈[0,1] .  g1(x,t,s)=g(x,t,1) ∀x∈Ω ∀t∈R+, s≥1   We remark that g1 satisfies (G1)-(G3). Furthermore g1(x,t,s) ≤C ∀s∈R ∀(x,t)∈Q (5.1) where C as in (2.1) and (cid:12) (cid:12) (cid:12) (cid:12) g1(x,t,s )≥0≥g1(x,t,s ) ∀t∈R+, ∀x∈Ω, ∀s ≤0, ∀s ≥1. (5.2) 1 2 1 2 Let us consider the approximate problem P : find a solution u (we do not use the symbol u for ε ε simplicity of notation) to h∂ u,ψi+(µ ∇v,∇ψ)= g1(u),ψ ∀ψ ∈H1(Ω), a.e. in (0,T), (5.3) t ε (cid:0) (cid:1) 7 v =f′(u)+w a.e. in Q (5.4) ε w=K∗(1−2u) a.e. in Q, (5.5) n·µ ∇v =0 a.e. on Γ, (5.6) ε u(0,x)=u (x), for a.a. x∈Ω, (5.7) 0 where µ =max{µ+ε,ε} (5.8) ε and f is the solution to the following Cauchy-problem: ε 1 1 1 1 1 f′′ =(1+2a ) , f′( )=f′( ), and f ( )=f( ), (5.9) ε ε µ ε 2 2 ε 2 2 ε where a = (1+4ε)1/2−1. Thanks to (1.7) and (5.8), we have ε 2 ε for s<0 µ (s)= (s+a )(1+a −s) for s∈[0,1] . (5.10) ε  ε ε ε for s>0  Hence, µ is continuous. We remark that µ (s) is not decreasing for s ≤ 1/2 and not increasing for ε ε s≥1/2. This yields 1+4ε ε≤µ ≤µ (1/2)= . (5.11) ε ε 4 From (5.9) and (5.10) it follows 1+2aε for s<0 ε f′′(s)= 1+2aε for s∈[0,1] (5.12) ε  (s+aε)(1+aε−s)  1+2aε for s>0 ε and, in particular,  1+2a 0<f′′(s)≤ ε. (5.13) ε ε Furthermore f′′ satisfies the symmetry property ε 1 1 f′′ +s =f′′ −s ∀s∈R. (5.14) ε 2 ε 2 (cid:18) (cid:19) (cid:18) (cid:19) Thanks to (5.13), f′ is increasing and, thanks to f′(1/2) = f(1/2) = 0, f′(s) < 0 for s < 1/2 and ε ε ε f′(s)>0 for s>1/2. Using (5.12) we now obtain ε f′(s)<0 for s<0 ε  fε′(s)=ln 1+aεa+ε−ss for s∈[0,1] (5.15)  f′(s)>0 (cid:16) (cid:17) for s>1. ε Furthermore f′ satisfies  ε 1 1 f′ +s =−f′ −s ∀s∈R. (5.16) ε 2 ε 2 (cid:18) (cid:19) (cid:18) (cid:19) Since f′′ ≤ 1+2aε and f′(1/2)=0, we have f′(s)≤ 1+2aε(s−1/2) for s≥1/2. So, using (5.16), we get ε ε ε ε ε 1+2a |f′(s)|≤ ε |s−1/2| ∀s∈R. (5.17) ε ε As a consequence of (5.15) s= 1 minimizes f (s). From (5.16) we have 2 ε 1 1 f +s =f −s ∀s∈R. (5.18) ε ε 2 2 (cid:18) (cid:19) (cid:18) (cid:19) 8 Now, we show that 1 f (s)≥ s2−c ∀s∈R, (5.19) ε ε 2ε where c is a positive constant depending on ε. We start showing ε 1+a f (s)≥ ε (s−1/2)2−c′ ∀s∈R (5.20) ε 2ε ε where c′ is a positive constant depending on ε. We prove (5.20) for s > 1/2; the proof for s < 1/2 ε can be obtained using (5.18). As a consequence of (5.12) we have f′(s) = 1+2aε(s−1)+f′(1), s > 1. ε ε ε Furthermore f′(s) ≥ 0 for s > 1/2 as a consequence of (5.15). Hence f′(s) ≥ 1+2aεs− 1+2aε ∀s > 1/2 ε ε ε ε (the right term is negative for s∈[1/2,1]). From the last inequality follows by integration 1+2a 1+2a 1+2a 1 1+2a 1 f (s)−f (1/2)≥ εs2− εs− ε + ε ε ε 2ε ε 2ε 4 ε 2 1+2a 1+2a 1 1+2a 1 1+2a ≥ εs2−δs2− ε − ε + ε ∀δ >0. 2ε 2ε 4δ 2ε 4 ε We take into account 1+2aε > 1+aε, choose δ suitably and get (5.20). Hence, 2ε 2ε 1+a 1+a ε (s−1/2)2 = ε(s2−s−1/4) 2ε 2ε 1+a 1 ≥ ε((1−δ)s2−1/4− ) ∀δ >0. 2ε 8δ Choosing δ suitably small and using 1+aε > 1 we have (5.19). 2ε 2ε 5.2 Existence of the solution to the approximate problem The following lemma states the existence of a solution to (5.3)-(5.7) for a fixed ε>0 small enough. Lemma 13 Let ε< 1 (r as in (K3)). Let (K1)-(K3), (G2), (G1) and (5.1) be satisfied. Then there 2r2 2 exists u∈H1(0,T,H1(Ω),(H1(Ω))∗)∩L∞(0,T,L2(Ω)) solution to (5.3)-(5.7) such that µ1/2(u)|∇v| ≤C ε L2(0,T,L2(Ω)) (cid:13) (cid:13) (cid:13) (cid:13) where C is a positive constant depen(cid:13)ding on ε. (cid:13) Proof. The argument is based on a Faedo-Galerkin’s approximation scheme. We introduce the family {ei}i∈N of eigenfunctions of −∆+ Id : V → V∗ as a Galerkin base in V = H1(Ω). We define the orthogonalprojectorP :H =L2(Ω)→V =span({e }n )andu =P u . Wethenlookforfunctions n n i i=1 0n n 0 of the form n n u (t)= α (t)e and v (t)= β (t)e n k k n k k k=1 k=1 X X which solve the following approximating problem (u′ ,ψ)+(µ (u )∇v ,∇ψ)=(g1,ψ) ∀ψ ∈V (5.21) n ε n n n n v =P (K ∗(1−2u )+f′(u )) n n n ε n g1 =P g1(u ) n n n u (0)=u . (5.22) n 0n(cid:0) (cid:1) This approximating problem is equivalent to solve a Cauchy problem for a system of ODEs in the n unknowns (α ). As a consequence of (5.8), (G1), (G2) and (5.9), for every ψ ∈ V , the func- i n tions (m(u )∇v ,∇ψ) and (g ,ψ) are locally Lipschitz with respect to the variables α uniformly in n n n i 9 t. Hence there exists T ∈ R such that system (5.21) has an unique solution α ,...,α ,β ,...,β ∈ n + 1 n 1 n C1([0,T );R). n We now want to prove a-priori estimates for u uniformly in n. Henceforth we shall denote by C a n positiveconstantwhichdependonε,butitisindependentofnandt. ThevaluesofC maypossiblyvary even within the same line. We choose ψ =v as test function and get n (u′ ,v )+(µ (u )∇v ,∇v )=(g1,v ). n n ε n n n n n Thus, (u′ ,v )=(u′ ,f′(u ))+(u′ ,K∗(1−2u )) n n n ε n n n d = f (u )+ K(x−y)u (x)(1−u (y)) . ε n n n dt (cid:18)ZΩ ZΩZΩ (cid:19) From this follows by integration on (0,t): t f (u )+ K(x−y)u (x)(1−u (y)) (t)+ µ (u )|∇v |2 (5.23) ε n n n ε n n (cid:18)ZΩ ZΩZΩ (cid:19) Z0 ZΩ t = (g1,v )+ f (u )+ K(x−y)u (x)(1−u (y)) (0). n n ε n n n Z0 (cid:18)ZΩ ZΩZΩ (cid:19) Thanks to (5.17) we have |f′(s)|≤C|s|+C. Due to (5.1), we have ε (g1,f′(u ))≤C+Cku k2 . (5.24) n ε n n H Using (5.19) and (K3), we obtain, for δ >0 to be announced, K(x−y)u (x)(1−u (y)) (5.25) n n ZΩZΩ 1 + f (u )≥ u 2−c +(K∗(1−u ),u ) ε n n ε n n H 2ε ZΩ ZΩ 1 ≥ ku k2 −c −r ku k k1−u k 2ε n H ε 2 n H n H 1 ≥ −r ku k2 −C −r |Ω|ku k 2ε 2 n H ε 2 n H (cid:18) (cid:19) 1 ≥ −r −δ ku k2 −C , 2ε 2 n H δ,ε (cid:18) (cid:19) where C denotes a constant depending on both ε and δ. Since 1 > r , we choose δ such that δ,ε 2ε 2 1 −r −δ =C >0. From (5.1) and (K3) follows 2ε 2 (cid:0) (cid:1) (g1,K∗(1−2u ))≤CkK∗(1−2u )k (5.26) n n n H ≤C+Dku k ≤C+Dku k2 . n H n H Using (5.23), (5.24), (5.25) and (5.26) we get t t ku (t)k2 + µ (u )|∇v |2 ≤C+D ku k2 . (5.27) n H ε n n n H Z0 ZΩ Z0 We now use Gronwall’s Lemma and get the estimates ku k ≤C (5.28) n L∞(0,T,H) and µ1/2(u )|∇v | ≤C. (5.29) ε n n L2(0,T,H) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 10