We prove the existence of solutions of the viscous Cahn-Hilliard equation in whole domain when the nonlinear term in the second order diffusion grows as uq for the critical case when N >= 3. Our results improve the ones in [9, 12]

Bui, L.T.T.

Dao, A. N

Díaz Díaz, Jesús Ildefonso

We prove the existence of solutions of the viscous Cahn-Hilliard equation in whole domain when the nonlinear term in the second order diffusion grows as uq for the critical case when N >= 3. Our results improve the ones in [9, 12].DGISPI (Spain)Vietnam National University, HCM cityUniversidad Complutense de MadridDepto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

Docta Complutense

Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 176, pp. 1–8.ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.eduCRITICAL CASE FOR THE VISCOUS CAHN-HILLIARDEQUATIONLE TRONG THANH BUI, ANH NGUYEN DAO, JESÚS ILDEFONSO DÍAZAbstract. We prove the existence of solutions of the viscous Cahn-Hilliardequation in whole domain when the nonlinear term in the second order diffusiongrows as uq for the critical case when N ≥ 3. Our results improve the ones in[9, 12].1. IntroductionIn this article, we study the initial-value problemut = ∆[ϕ(u)− α∆u+ βut] in RN × (0, T ) := Q,u(x, 0) = u0 in RN × {0} ,(1.1)where the nonlinearity ϕ satisfies the following assumptions:(H1) ϕ ∈W 1,∞loc (R), ϕ(0) = 0, and ϕ(s)s ≥ 0, for any s ∈ R.(H2) There exists K > 0 such that|ϕ(u)| ≤ K(|u|+ |u|q), (1.2)for some q ∈ (1,∞) if N = 1, 2; or q ∈(1, N+2N−2]if N ≥ 3.(H3) There exists s0 > 0 such that ϕ′(s) ≥ 0, if |s| ≥ s0.Forward-backward parabolic equations arise in a variety of applications, such asedge detection in image processing [25], aggregation models in population dynamics[24], and stratified turbulent shear flow [1], theory of phase transitions [4, 5, 21],control theory in [11], etc. A different well-known equation of this type is thePerona-Malik equation,wt = div( ∇w1 + |∇w|2), (1.3)which is parabolic if |∇w| < 1 and backward parabolic if |∇w| > 1. Similarly, theequationut = ∆( u1 + u2)(1.4)is parabolic if |u| < 1 and backward parabolic if |u| > 1. Observe that in one spacedimension the above equations are formally related setting u = wx. A different2010 Mathematics Subject Classification. 35B25, 35K55, 35R25, 28A33, 35D99.Key words and phrases. Forward-backward parabolic equations; singular limits;pseudo-parabolic regularization; Cahn-Hilliard regularization; viscous Cahn-Hilliard equation.c©2017 Texas State University.Submitted April 30, 2016. Published July 11, 2017.12 L. T. T. BUI, A. N. DAO, J. I. DÍAZ EJDE-2017/176well-known equation of application in theory of phase transitions isut = ∆ϕ(u) (1.5)where the famous choice of nonlinearity ϕ(u) = u3 − u.Clearly, forward-backward parabolic equations lead to ill-posed problems. Oftena higher order term is added to the right-hand side to regularize the equation. Twomain classes of additional terms are encountered in the mathematical literature,which, e.g. in case of equation (1.4), (1.5), reduce to:(i) ε∆[ψ(u)]t, with ψ′ > 0, leading to third-order pseudo-parabolic equations(ε > 0 being a small parameter; for example, see [2, 8, 14, 20, 23, 26, 27, 32, 33, 34]);(ii) −ε∆2u, leading to fourth-order Cahn-Hilliard type equations (for example,see [3, 4, 28, 31] and references therein).Remarkably, when ψ(u) = u either of the above regularizations can be regardedas a particular case of the viscous Cahn-Hilliard equation,νut = ∆[ϕ(u)− α∆u+ βut] (α, β, ν > 0) , (1.6)choosing either α = ε or β = ε; here ϕ(u) = u3 − u or ϕ(u) = u1+u2 for equation(1.5), whereas in general it involves a non-monotonic function.Equation (1.6) has been derived by several authors using different physical con-siderations (in particular, see [16, 18, 22]). It is worth mentioning the wide litera-ture concerning both the relationship between the viscous Cahn-Hilliard equationand phase field models, and generalized versions of the equation suggested in [16](and references therein). Besides, the existence results were obtained under suit-able nonlinearity ϕ in bounded smooth domain of RN (see [9, 10, 13]). Moreover,in the latter reference authors give us the rigorous proof of convergence to solu-tions of either the Cahn-Hilliard equation, or of the Allen-Cahn equation, or of theSobolev equation, depending on the choice of the parameter α, β. Recently, in [12]the authors gave the analysis of equation (1.6) in RN under some assumptions onthe growth of nonlinearity ϕ satisfying (H2), but not including the critical caseq = N+2N−2 .In light of the above considerations, by using some sharp a priori estimates for asuitable auxiliary approximation problem, we will prove the existence of solutionsof problem (1.1) for a class of nonlinear functions ϕ satisfying the growth condition(H2) including the critical case q = N+2N−2 . Thus, our existence results enhance apart of the ones of Dlotko, et al. [12]. Our existence theorem is as follows:Theorem 1.1. Let u0 ∈ H1(RN ), and q = N+2N−2 . Let ϕ satisfy (H1)–(H3). Then,there exists a weak solution of problem (1.1).Remark 1.2. Note that we do not assume the boundedness on ϕ′, see [9, (1.1)].Thus, our results also improve the ones of Bui, et al. [9].Before proving Theorem 1.1, we give a definition of weak solutions of (1.1).Definition 1.3. Let α, β > 0, and let u0 ∈ H1(RN ). By a weak solution of problem(1.1) we mean any function u ∈ C([0, T ];H2(RN )) ∩ C1([0, T ];L2(RN )) such thatϕ(u) ∈ C([0, T ];L2(RN )), andut = ∆v in Qu = u0 in RN × {0}(1.7)EJDE-2017/176 VISCOUS CAHN-HILLIARD EQUATION 3in the sense of distribution. Here v ∈ C([0, T ];H2(RN ) ∩ H1(RN )) and for everyt ∈ [0, T ] the function v(·, t) is the unique solution of the elliptic problem−β∆v(·, t) + v(·, t) = ϕ(u)(·, t)− α∆u(·, t) in RN ,lim|x|→∞v(x, t) = 0. (1.8)The function v is called a chemical potential.2. Proof of Theorem 1.1We first mention the description of our method. We start by considering theexistence of weak solutions of the viscous Cahn-Hilliard problem with Dirichletboundary conditions in the ball Bn, which has center at the origin and radiusn ≥ 1:ut = ∆[ϕn(u)− α∆u+ βut] in Bn × (0, T ) =: Qnu = ∆u = 0 on ∂Bn × (0, T )u = u0n = u0φn in Bn × {0} ,(2.1)where φn(x) = φ(x/n), and φ ∈ C∞(RN ) such that φ(x) = 1 if |x| < 1/2, andφ(x) = 0 if |x| > 1. And ϕn is just a truncated function of ϕ as in [9]:ϕn(u) =ϕ(u), if |u| ≤ n,ϕ(n) + (u− n), if u > n,ϕ(−n) + (u+ n), if u < n.Secondly, we establish a priori estimates for those solutions of problem (2.1) beingindependent of n. Finally, we shall pass to the limit as n→∞ (in a suitable way)to get a desired result.It is not difficult to verify that ϕn is a globally Lipschitz function, and ϕn(u)u ≥0. A well-posed result for problem (2.1) is proved in [9, Theorem 2.1]. Thus, thereexists a unique weak solution un of problem (2.1) in Bn × (0, T ). Remind thatvn = ϕn(un)− α∆un + βunt.Then, multiplying both sides of this equation with ∂tun and integrating over Bn ×(0, t) yields∫BnΦn(un)(x, t) dx+α2∫Bn|∇un|2(x, t) dx+ β∫ t0∫Bnu2nt dx ds=∫ t0∫Bnvn∂tun dx ds+∫BnΦn(u0n)(x) dx+α2∫Bn|∇u0n|2 dx, for t ∈ (0, T ),with Φn(u) =∫ u0ϕn(s)ds. Note that ∆vn = unt. Then, we obtain∫ΩΦn(un)(x, t) dx+α2∫Ω|∇un|2(x, t) dx+ β∫ t0∫Ωu2nt dx ds+∫ t0∫Ω|∇vn|2 dx ds=∫ΩΦn(u0n)(x) dx+α2∫Ω|∇u0n|2 dx(2.2)4 L. T. T. BUI, A. N. DAO, J. I. DÍAZ EJDE-2017/176Since ϕn(s)s ≥ 0 and assumption (H2), we have0 ≤ Φn(u) =∫ u0ϕn(s)ds ≤K2u2 +K(N − 2)2Nu2NN−2 ,so there is a positive constant C = C(K,N) such that∫BnΦn(u0n)dx ≤ C(∫Bnu20dx+∫Bnu2NN−20 dx). (2.3)From Sobolev’s embedding theorem, we obtain‖u0n‖L2NN−2 (Bn)≤ C(N)‖∇u0n‖L2(Bn). (2.4)A combination of (2.4) and (2.3) implies that∫BnΦ(u0n) dx is bounded by aconstant depending only on ‖u0‖H1(RN ). Therefore, there is a positive constantC = C(N, ‖u0‖H1(RN )) such that∫BnΦn(un)(x, t) dx+α2∫Bn|∇un|2(x, t) dx+ β∫ t0∫Bnu2nt dx ds+∫ t0∫Bn|∇vn|2 dx ds ≤ C.(2.5)Next, using un as a test function to the first equation of (2.1) yields12∫Bnu2n(x, t) dx+β2∫Bn|∇un|2(x, t) dx+ α∫ t0∫Bn(∆un)2 dx ds+∫ t0∫Bnϕ′n(u)|∇un|2dx ds≤∫Bnu20n(x, t) dx+β2∫Bn|∇u0n|2(x, t) dxUsing (H3) yields12∫Bnu2n(x, t) dx+β2∫Bn|∇un|2(x, t) dx+ α∫ t0∫Bn(∆un)2 dx ds≤∫Bn×(0,t)∩{|un|≤s0}−ϕ′n(u)|∇un|2dx ds+∫Bnu20n(x, t) dx+β2∫Bn|∇u0n|2(x, t) dxBy (H1), there is a positive constant C0 such that |ϕ′n(s)| < C0, for any |s| ≤ s0.Then,12∫Bnu2n(x, t) dx+β2∫Bn|∇un|2(x, t) dx+ α∫ t0∫Bn(∆un)2 dx ds≤ C0∫Bn×(0,t)∩{|un|≤s0}|∇un|2dx ds+∫Bnu20n(x, t) dx+β2∫Bn|∇u0n|2(x, t) dxBy (2.5),∫Bn|∇un|2dx is bounded by a constant depending only on ‖u0‖H1(RN ).This fact and the last inequality imply that there is a positive constant, still denotedby C = C(‖u0‖H1(RN )) such that12∫Bnu2n(x, t) dx+β2∫Bn|∇un|2(x, t) dx+ α∫ t0∫Bn(∆un)2 dx ds ≤ C. (2.6)EJDE-2017/176 VISCOUS CAHN-HILLIARD EQUATION 5Next, we show that ‖ϕn(un)‖L2(Bn×(0,T )) is uniformly bounded for any n ≥ 1.By (H2), it suffices to show that un ∈ L2(N+2)N−2 (Bn×(0, T )) is bounded by a constantnot depending on n. Indeed, from Sobolev’s embedding theorem we have for N ≥ 3,‖un(·, t)‖L2NN−2 (Bn)≤ C1(N)‖∇un(·, t)‖L2(Bn).From (2.6) or (2.5), there is a positive constant C = C(‖u0‖H1(RN ))such that‖un(·, t)‖L2NN−2 (Bn)≤ C. (2.7)Thanks to Gagliardo-Nirenberg inequality, we obtain‖un(·, t)‖L2NN−4 (Bn)≤ C2(N)‖∇un(·, t)‖L2NN−2 (Bn)‖un(·, t)‖L2NN−2 (Bn). (2.8)Combining (2.7) and (2.8) yields‖un(·, t)‖L2NN−4 (Bn)≤ C ′2(N)‖∇un(·, t)‖L2NN−2 (Bn). (2.9)Using Sobolev’s embedding theorem again yields‖∇un(·, t)‖L2NN−2 (Bn)≤ C3(N)‖D2un(·, t)‖L2(Bn).By the boundary condition, we can use the integration by parts formula to get‖D2un(·, t)‖2L2(Bn) = ‖∆un(·, t)‖2L2(Bn). Thus,‖∇un(·, t)‖2L2NN−2 (Bn)≤ C3(N)‖∆un(·, t)‖2L2(Bn). (2.10)By (2.10) and (2.9), there exists a constant C > 0 not depending on n such that‖un(·, t)‖2L2NN−4 (Bn)≤ C‖∆un(·, t)‖2L2(Bn). (2.11)Now, it follows from the interpolation theorem that‖un(·, t)‖L2(N+2)N−2 (Bn)≤ ‖un(·, t)‖θL2NN−4 (Bn)‖un(·, t)‖1−θL2NN−2 (Bn),with θ = N−2N+2 .By (2.7), from the last inequality, we obtain‖un(·, t)‖2(N+2)N−2L2(N+2)N−2 (Bn)≤ C‖un(·, t)‖2L2NN−4 (Bn)(2.12)A combination of (2.12), (2.11), and (2.6) yields∫ T0‖un(·, t)‖2(N+2)N−2L2(N+2)N−2 (Bn)dt ≤ C∫ T0‖∆un(·, t)‖2L2(Bn)dt ≤ C(T,N, u0). (2.13)Therefore, we obtain the above claim.It remains to pass to the limit as n→∞ in the equation satisfied by un. Thanksto the uniform estimates in (2.2), (2.5), (2.6), and (2.13), we can mimic the proofof [9, Theorem 2.4] to getun∗⇀ u in L∞((0, T );H1(RN )) , (2.14)unt ⇀ ut in L2(Q) , (2.15)∆un ⇀ ∆u in L2(Q), (2.16)un → u a.e. in Q, (2.17)up to a subsequence.6 L. T. T. BUI, A. N. DAO, J. I. DÍAZ EJDE-2017/176Next, we prove that ϕn(un) converges weakly to ϕ(u) in L2(Q). In fact, weobserve that ϕn(un)→ ϕ(u) as n→∞ a.e. in Q by (2.17). Moreover, the sequence{ϕn(un)}n≥1 is uniformly bounded in L2(Bn× (0, T )) for any n ≥ 1. Thus, there isa subsequence (still denoted by {ϕn(un)}n≥1) such that ϕn(un) converges weaklyto ϕ(u) in L2(Q), see [17, Theorem 13.44].Now, it suffices to show that u is a weak solution of (1.1). We write the equationsatisfied by un in the weak sense:For any ψ ∈ C1([0, T ]; C2c (RN )) such that ψ(., T ) = 0, we have∫Q(ψ)−unψt dx ds−∫supp(ψ)u0n(x)ψ(x, 0) dx=∫Q(ψ)(ϕ(un)∆ψ − α∆un∆ψ + βun∆ψt) dx ds,for any n ≥ 1 such that supp(ψ) ⊂ Bn, and Q(ψ) = supp(ψ) × (0, T ). Passing tothe limit as n→∞ in the above equation yields∫Q(ψ)−uψt dx ds−∫supp(ψ)u0(x)ψ(x, 0)dx =∫Q(ψ)(ϕ(u)∆ψ − α∆u∆ψ + βu∆ψt) dx ds.Or, u is a weak solution of problem (1.1). This completes the proof.Acknowledgments. The research leading to the present results has received fund-ing from research grant of Vietnam National University, HCM city, project number:C2016-18-24. J. I. 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Valadier, A Course on Young Measures, Rend. Ist. Mat. Univ. Trieste, 26 (1994), suppl.,349-394 (1995).Le Trong Thanh BuiFaculty of Mathematics and Computer Science, University of Science, Vietnam Na-tional University, Ho Chi Minh city, 227 Nguyen Van Cu, D. 5, Ho Chi Minh city, Viet-namE-mail address: bltthanh@hcmus.edu.vn8 L. T. T. BUI, A. N. DAO, J. I. DÍAZ EJDE-2017/176Anh Nguyen Dao (corresponding author)Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton DucThang University, VietnamE-mail address: daonguyenanh@tdt.edu.vnJesús Ildefonso D́ıazInstituto de Matematica Interdisciplinar, Universidad Complutense de Madrid, 28040Madrid, SpainE-mail address: ildefonso.diaz@mat.ucm.es

English

Critical case for the viscous Cahn-Hilliard equation

We prove the existence of solutions of the viscous Cahn-Hilliard equation
 in whole domain when the nonlinear term in the second order  diffusion grows
 as $u^q$ for the critical case when $N\geq 3$. Our results improve the ones
 in [9,12]

Critical case for the viscous Cahn-Hilliard equation

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