International Mathematical Forum, Vol. 0, 205, no. 0, 477-486 HIKARI Lt, www.m-hikari.com http://x.oi.org/0.2988/imf.205.5757 Existence an Uniqueness of Solution for Caginalp Hyperbolic Phase Fiel System with Polynomial Growth Potential Mayeul Evrar Isseret Goyau, Fièle Moukamba, Daniel Moukoko an Franck Davhys Reval Langa Faculté es Sciences et Techniques Université Marien NGouabi, B.P. 69, Brazzaville, Congo Copyright c 205 Mayeul Evrar Isseret Goyau et al. This article is istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. Abstract Our aim in this article is to stuy the existence an the uniqueness of solution for Caginalp hyperbolic phase-fiel system, with initial conitions, homogenous Dirichlet bounary conitions an polynomial growth potential in boune an smooth omain. Mathematics Subject Classification: Primary: 35D05, 35D0, 35D20, 35D25, 35D30, Seconary: 46F30 Keywors: Caginalp hyperbolic phase-fiel system, polynomial growth potential, Dirichlet bounary conitions, phase space Corresponing author
478 Mayeul Evrar Isseret Goyau et al. Introuction We consier the folllowing Caginalp hyperbolic phase-fiel system ɛ 2 u 2 2 α 2 + u + α u + fu = α,. α with homogenous Dirichlet bounary conitions α = u u,.2 u = α = 0,.3 an initial conitions u t=0 = u 0, u t=0 = u, α t=0 = α 0, α t=0 = α,.4 where ɛ > 0 is a relaxation parameter. u = ux, t express a phase-fiel or orer parameter. α = αx, t is the thermal isplacement or primitive of relative temperature θ, with αx, t = t 0 θx, ττ + α 0. is a boune an smooth omain, of class C 2 in R n n 3, an the smooth bounary, of classe C 2 of. Hypothesis of potential f. f is of class C 2,.5 f0 = 0,.6 c 0 F s fss + c, c 0 0, c 0, s R,.7 with F s = s 0 fττ, f s c 2 s 2p +, c 2 > 0, p > 0, s R,.8 f s c 3, c 3 0, s R..9 We will precise restrictions on p when n = 3 about certain stapes, when being in an obligation. Such stuies have alreay been carrie out by authors; in the case of parabolichyperbolic systems see [], [2], [4], [5], [6], [9]. We can also mention the recent work of Daniel Moukoko, for example [7] et [8] in which this system was the subject of a stuy with regular potential fs = s 3 s, an also those of Doumbé Bongola brice Lanry, [3] in which the parabolic-hyperbolic system governe by a potential polynomial growth is stuie.
Caginalp hyperbolic phase fiel system 479 Notations. *.,. enotes the scalar prouct on L 2, an. associate norm. *.,. X enotes the scalar prouct on X, et. X associate norm. * is a measure of. 2 A priori estimates Multiplying. an.2 respectively by u we get the following respective equations t t + ɛ u 2 + 2 F u, α 2 H whose sum is t + α 2 α an, an then integrating over, 2 = 2 α, u, + 2 α 2 + 2 α 2 = 2u, α 2 u + ɛ u 2 + 2 F u, + α 2 H + α 2 = 2u, α, from which we euce t + ɛ u 2 + 2 F u + c 0, + α 2 H + α 2 2 u, α. Now, apply Holer an Young inequalities. Then we get, where, α 2., 2 + 2 α 2 + 2 α 2 2 + 2 α 2 + 2 α 2 t E 2 + α 2 + 2 α 2 C, 2.2 E = + ɛ u 2 + 2 F u + c 0, + α 2 H + α 2. Thanks to the estimate.7, we obtain The estimates 2.2 an 2.3 imply F u + c 0, 0. 2.3 t E + u 2 + α 2 + α 2 H k E.
480 Mayeul Evrar Isseret Goyau et al. Applying the Gronwall s lemma, we fin + ɛ u 2 + α 2 H + α 2 + for all, t [0, T ]. Multiplying. by u + ɛ u t 2 2 H t from which we euce the inequality 0 u τ 2 + α τ 2 + α τ 2 H τ E 0e kt, 2.4 an integrating over. We get the equation 2 H = 2f u u, u + 2 α, u, u 2 t H 2 + ɛ u 2 H 2 H 2 f u u u α x + 2, u. 2.5 Thanks to the estimate.8, we have f u u u x c u 2p + u u x. 2.6 On the other han, if n = 2 an p > 0, one fins, through the Höler s inequality, u 2p u u x u 2p u L 22p+ L 22p+ u. The continuous injection H L 22p+ implies u 2p u u x c u 2p u H H 2 u H. 2.7 If n = 3, in this case p we fin, through Holer s inequality for p = u 2 u u x u 2 L u 6 L 6 u, which implies u 2 u u x c u 2 H u H 2 u H. 2.8 So starting from 2.6, 2.7 an 2.8 we euce f u u u x c u q u H H 2 + u H. 2.9 Inserting 2.9 into 2.5. We get + ɛ u t 2 2 H 2 H c u q u H H 2 + u H + 2 α H u H.
Caginalp hyperbolic phase fiel system 48 which implies t 2 + ɛ u 2 H + 2 u 2 H C u 2q H u 2 H 2 + + α 2 H. Since u L 0, T ; H0, we euce the estimate + ɛ u t 2 2 H + 2 u 2 H C u 2 H + α 2 2 H + C, 2.0 with C > 0. Equation 2. gives the estimate α 2 H + α t 2 + α 2 + 2 α 2 H C + C 2 u 2 H, 2. with C > 0 an C 2 > 0. Aing 2.0 an 2., with u L 0, T ; H0, we get the estimate where 0 t E 3 + 2 u 2 H + α 2 + α 2 H k 3 E 3 + C, E 3 = 2 + ɛ u 2 H + α 2 H + α 2. Appling the Gronwall s lemma with t [0, T ]. We get t E 3 + 2 u τ 2 H + α τ 2 + α τ 2 H τ E 3 0e k3t + C, 2.2 Multiplying. by 2 u 2, We get t u 2 + 2ɛ 2 u 2 2 = 2 u, 2 u 2 2fu, 2 u 2 + 2 α, 2 u 2. We euce inequality t u 2 + 2ɛ 2 u 2 2 2 u, 2 u 2 + 2 fu 2 u x + 2 α 2, 2 u. 2.3 2 With the assumption.8, we fin the following estimate u u fu c 2 s 2p s + s = C u 2p+ + u, 0 0 which implies fu 2 u 2 x C u 2p + u 2 u 2 x.
482 Mayeul Evrar Isseret Goyau et al. We euce the estimate fu 2 u 2 x C u q u H H + 2 u 2. Inserting above estimate into 2.3, we fin the following estimate t u 2 + 2ɛ 2 u 2 2 2 u H 2 2 u 2 + C u q H u H + 2 u 2 + 2 α 2 u 2. Appling Holer an Young inequalities, we get t u 2 + ɛ 2 u 2 2 6 ɛ u 2 H 2 + C ɛ Multiplying.2 by α α 2 H + α t 2 2 H u 2 H + + 6 ɛ α 2. 2.4 an integrating over, we get the following estimate + α 2 H + 2 α 2 H 2 2 2 H. 2.5 Multiplying.2 by 2 α, an integrating over, we get 2 α t 2 + α 2 H + 2 2 α 2 2 = 2 α, 2 α 2 + 2 u, 2 α + 2 u 2, 2 α 2. Appling Holer an Young inequalities, we fin the estimate α t 2 + α 2 H + 2 α 2 2 C α 2 H + C 2 2 + C 3 u 2. 2.6 In this stuy, we have three main results; existence theorem, uniqueness theorem, existence theorem with more regularity. 3 Existence an Uniqueness of solution. Theorem 3. Existence. We assume u 0, u, α 0, α H0 L2 H0 L 2 an F u 0 < +, then the problem.-.4 possesses at least one solution u, α such that u, α L 0, T ; H0, u L 0, T ; L 2 L 2 0, T ; L 2, α L 0, T ; L 2 L 2 0, T ; H0, for all T > 0. The prove of this theorem is base on 2.4 an the stanar Galerkin scheme see [7].
Caginalp hyperbolic phase fiel system 483 Theorem 3.2 Uniqueness. Assume the hypothesis of Theorem 3. verifie, then the problem.2-.4 possesses a unique solution u, α, such that u, α L 0, T ; H0, u L 0, T ; L 2 L 2 0, T ; L 2, α L 0, T ; L 2 L 2 0, T ; H0, for all T > 0, with p an n = 3. Proof. Let u, α an u 2, α 2 two solutions of the problem.-.4, with u 0, u, α 0, α, u2 0, u2, α2 0, α2 H 0 L2 H0 L2 their respective initial ata. Let u = u u 2 an α = α α 2. Then u, α satisfies ɛ 2 u 2 + u u + fu fu 2 = α, 3. 2 α 2 + α α α = u u, 3.2 u = α = 0, u t=0 = u 0 = u 0 u 2 0, u t=0 = u = u u 2, α t=0 = α 0 = α 0 α 2 0, α t=0 = α = α α 2. Integrate over the prouct of 3. by u. We fin + ɛ u t 2 2 2 fu fu 2 u x + 2 α, u.3.3 Lagrange theorem gives an estimate fu fu 2 = f δ 0 u + δ 0 u 2 u avec 0 δ 0. The hypothesis.8 allows to write, when n = 3 et p, inequalities f δ 0 u + δ 0 u 2 c 2 δ 0 u + δ 0 u 2 2p + c 2 2 δ0 u 2 2p + δ 0 2 u 2 2p +. We euce the estimate f δ 0 u + δ 0 u 2 C u 2p + u 2 2p +. Insert the above estimate into 3.3. We get + ɛ u t 2 +2 u 2 C u 2p + u 2 2p + H H u +2 α u, then we have t + ɛ u 2 + u 2 C + 2 α 2. 3.4
484 Mayeul Evrar Isseret Goyau et al. u, α verifies equation 2. which implies α 2 H + α t 2 + α 2 + α 2 H C u 2 H 2. 3.5 A together 3.4 an γ 3.5 with γ > 0, such that - 2γ > 0, we obtain the estimate t E 2 + C4 u 2 with k 2 > 0, inepenent of ɛ, where E 2 = Apply the Gronwall s lemma. We get E 2 + t 0 for all t [0, T ]. C 4 u τ 2 + γ α 2 + γ α 2 H k 2 E 2 + ɛ u 2 + γ α 2 H + γ α 2. + γ α τ 2 + γ α τ 2 H τ E 2 0e k2t, We euce the continuous epenence of the solution relative to the initial conitions, hence the uniqueness of the solution. The existence an uniqueness of the solution of problem.2 -.4 being proven in a larger space, we will seek a solution with more regularity. Theorem 3.3. We assume that u, α is solution of problem.2-.4, an in that p an n = 3, with u 0, u, α 0, α H0 H2 H0 H 0 H 2 H0 then, the problem -4 has a unique solution u, α such that u, α L 0, T ; H0 H2, u L 0, T ; H0 L2 0, T ; H0 α, L 0, T ; H0 L2 0, T ; H0 H 2 an 2 u, 2 α L 2 0, T ; L 2, for all T > 0. 2 2 Proof. By Theorems 3. an 3.2, the problem.2-.4 has a unique solution u, α such that u, α L 0, T ; H0 u, L 0, T ; L 2 L 2 0, T ; L 2, α L 0, T ; L 2 L 2 0, T ; H0, for all T > 0. Thanks to those hypothesis, one can affirm : * 2.2 implies u L 0, T ; H 0 H2 an u L 0, T ; H 0 L2 0, T ; H 0. * Since α L 0, T ; L 2 L 2 0, T ; H0 an u L 0, T ; H0 H2, then 2.4 implies 2 u L 2 0, T ; L 2. 2
Caginalp hyperbolic phase fiel system 485 * Since u L 0, T ; H0 u an L 0, T ; H0 L2 0, T ; H0,2.5 implies α L 0, T ; H 2 H0 α, L 0, T ; H0 L2 0, T ; H0 H2. * Since u L 0, T ; H0 u, L 0, T ; L 2 an α L 0, T ; H 2 H0, 2.6 implies 2 α L 2 0, T ; L 2. 2 References [] Haim Brezis, Analyse Fonctionnelle, Theorie et Application, Masson, Paris, 983. [2] Laurence Cherfils, Stefania Gatti an Alain Miranville, Corrigenum to Exitence an globalsolutions to the Caginalp phase-fiels system with ynamic bounary conitions an singular potentials, J. Math. Anal. Appl., 348 2008, no. 2, 029-030. http://x.oi.org/0.06/j.jmaa.2008.07.058 [3] Doumbé Bongola brice Lanry, Etue e moèles e champ e phases e type Caginalp. Thèse soutenue à la Faculté es Sciences Fonamentales et Appliquées e Poitiers, le 30 mai 203. [4] Alain Miranville an Ramon Quintanilla, Some generalizations of the Caginalp phase-fiel system, Applicable Analysis, 88 2009, no. 6, 877-894. http://x.oi.org/0.080/000368090304282 [5] Alain Miranville an Ramon Quintanilla, A generalizations of the Caginalp phase-fiel system base on the Cattaneo law, Nonlinear Analysis: Theory, Methos an Applications, 7 2009, 2278-2290. http://x.oi.org/0.06/j.na.2009.0.06 [6] A. Miranville an R. Quintanilla, A Caginalp phase-fiel system with nonlinear coupling, Nonlinear Analysis: Real Worl Appl., 200, 2849-286. http://x.oi.org/0.06/j.nonrwa.2009.0.008 [7] Moukoko Daniel, Well-poseness an long time behavior of a hyperbolic Caginalp system, Journal of Applie Analysis an Computation, 4 204, no. 2, 5-96. [8] Daniel Moukoko, Etue e moèles hyperboliques e champ e phases e Caginalp, These unique, Faculté es Sciences et Techniques, Université Marien NGOUABI, 23 janvier 205. [9] Daniel Moukoko, Fiele Moukamba, Langa Franck Davhys Reval, Global Attractor for Caginalp Hyperbolic Fiel-phase System with Sin-
486 Mayeul Evrar Isseret Goyau et al. gular Potential, Journal of Mathematics Research, 7 205, 65-77. http://x.oi.org/0.5539/jmr.v7n3p65 Receive: August, 205; Publishe: September, 205