NONLINEAR KIRCHHOFF-CARRIER WAVE EQUATION IN A UNIT MEMBRANE WITH MIXED HOMOGENEOUS BOUNDARY CONDITIONS

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1 Electronic Journal of Differential Equations, Vol. 2525, No. 138, pp ISSN: URL: or ftp ejde.ath.txstate.edu login: ftp NONLINEAR KIRCHHOFF-CARRIER WAVE EQUATION IN A UNIT MEMBRANE WITH MIXED HOMOGENEOUS BOUNDARY CONDITIONS NGUYEN THANH LONG Abstract. In this paper we consider the nonlinear wave equation proble u tt B` u 2, ur 2 urr + 1 ur = fr, t, u, ur, < r < 1, < t < T, r li rurr, t <, r + u r1, t + hu1, t =, ur, = eu r, u tr, = eu 1 r. To this proble, we associate a linear recursive schee for which the existence of a local and unique weak solution is proved, in weighted Sobolev using standard copactness arguents. In the latter part, we give sufficient conditions for quadratic convergence to the solution of the original proble, for an autonoous right-hand side independent on u r and a coefficient function B of the for B = B u 2 = b + u 2 with b >. 1. Introduction In this paper, we consider the initial and boundary value proble u tt B u 2, u r 2 urr + 1 r u r = fr, t, u, u r, < x < 1, < t < T, li r + rur r, t <, u r 1, t + hu1, t =, ur, = u r, u t r, = u 1 r, 1.1 B, f, ũ, ũ 1 are given functions, u 2 = r ur, t 2 dr, u r 2 = r u rr, t 2 dr and h is a given positive constant. 2 Matheatics Subject Classification. 35L7, 35Q72. Key words and phrases. Nonlinear wave equation; Galerkin ethod; quadratic convergence; weighted Sobolev spaces. c 25 Texas State University - San Marcos. Subitted August 3, 24. Published Deceber 1, 25. 1

2 2 N. T. LONG EJDE-25/138 Many authors [6, 7, 15, 17, 18] have studied the proble v tt B 1 v 2, v 2 v = f 1 x, t, v, v t, v in Ω 1, T, v ν + hv = on Ω 1, T, or v = on Ω 1, T, vx, = v x, v t x, = v 1 x in Ω 1, Ω 1 is a bounded doain in R N with a sufficiently regular boundary Ω 1, N v 2 = v 2 x, tdx, v 2 = vx, t 2 dx = v x, t 2 dx, Ω 1 Ω 1 x i Ω 1 i=1 1.2 and ν is the outward unit noral vector on boundary Ω 1. With N = 1 and Ω 1 =, L the first equation in 1.2 has its origin in the nonlinear vibration of an elastic string c.f. Kirchhoff [7], for which the associated equation is ρhv tt P + Eh L v 2L y y, t 2 dy v xx =, v is the lateral deflection, x is the space coordinate, t is the tie, ρ is the ass density, h is the cross-section area, L is the length, E is Young s odulus, and P is the initial axial tension. Carrier [3] also established the odel L v tt = P + P 1 v 2 y, tdy v xx, P and P 1 are constants. In the case Ω 1 is an open unit ball of R N and the functions v, f 1, ṽ, ṽ 1 depend on x through r with r 2 = x 2 = N i=1 x2 i, we put vx, t = u x, t, f 1 x, t, v, v t, v = f 1 x, t, ṽ x = ũ x, ṽ 1 x = ũ 1 x, γ = N 1. Then B 1 v 2, v 2 v = B u 2 r, tr γ dr, u r r, t 2 r γ dr u rr + 1 r u r, Bξ, η = B 1 ω N ξ, ω N η and ω N is the area of the unit sphere in R N. Hence, we can rewrite proble 1.2 as u tt B u 2 r, tr γ dr, u r r, t 2 r γ dr u rr + 1 r u r = f 1 r, t in, 1, T, u r 1, t + hu1, t = on, T, or u1, t = on, T, ur, = ũ r, u t r, = ũ 1 r in, With N = 2, the first equation of 1.3 is the bi-diensional nonlinear wave equation describing nonlinear vibrations of the unit ebrane Ω 1 = {x, y : x 2 + y 2 < 1}. In the vibration process, the area of the unit ebrane and the tension at various points change in tie. The condition on the boundary Ω 1 describes elastic

3 EJDE-25/138 NONLINEAR KIRCHHOFF-CARRIER WAVE EQUATION 3 constraints, the constant h has a echanical signification. Boundary condition is satisfied autoatically if u is a classical solution of proble 1.1, for exaple, with u C 1 Ω, T C 2 Ω, T. This condition is also used in connection with Sobolev spaces with weight r see. [2, 16]. In the case of equation not involving the ter 1 r u r γ =, we have u tt B u 2 r, tdr, u r r, t 2 dr u rr = fr, t, u, u r, u t. 1.4 When f =, and B = B u rr, t 2 dr is a function depending only on u rr, t 2 dr, the Cauchy or ixed proble for 1.3 has been studied by any authors; see Ebihara, Medeiros and Miranda [5], Pohozaev [22] and the references therein. A survey of the results about the atheatical aspects of Kirchhoff odel can be found in Medeiros, Liaco and Menezes [2], [21]. Medeiros [19] studied proble 1.1 on a bounded open set Ω of R 3 with f = fu = bu 2 b > is a given constant. Hosoya and Yaada [6] considered proble 1.3 3,4-1.3 with f = fu = δ u α u δ > and α are given constants. In [9] the authors studied the existence and uniqueness of the solution of the equation u tt + λ 2 u B u 2 u + ε u t α 1 u t = F x, t, λ >, ε > and < α < 1 are given constants. In the case of the ter 1 r u r appearing in equation we have to eliinate the coefficient 1 r by using Sobolev spaces with appropriate weight see [11]. On the other hand, proble 1.1 with general nonlinear right-hand side fr, t, u, u r, u t given as a continuous function of five variables has not been studied copletely yet. In the present paper, we study proble 1.1 with soe fors of the right-hand side f. In the first part, we study proble 1.1 with the right-hand side fr, t, u, u r f C [, 1] R + R 2 satisfies the condition f r, f u, f u r in C [, 1] R + R 2. It is not necessary that f C 1 [, 1] R + R 2. First, we shall associate with equation a linear recurrent sequence which is bounded in a suitable function space. The existence of a local solution is proved by a standard copactness arguent. Note that the linearization ethod in this paper and in papers [2, 4, 14, 17, 18, 23] cannot be used in papers [5, 9, 12, 13, 15, 16, 19]. In the second part, we consider proble 1.1 corresponding to f = fr, u and Bη = b + η with given constant b >. We associate with equation a recurrent sequence u nonlinear defined by 2 u t 2 b + u r r, t 2 2 u rdr r r u r = fr, u 1 + u u 1 f u r, u 1 in, 1, T, with u satisfying The first ter u is chosen as u = ũ. If f C 2 [, 1] R, we prove that the sequence u converges quadratically. The results obtained here relatively are in part generalizations of those in [2, 4, 14, 17, 18, 23].

4 4 N. T. LONG EJDE-25/ Preliinary results, notation, function spaces Put Ω =, 1. We oit the definitions of the usual function spaces L p Ω, H Ω, W,p Ω. For any function v C Ω we define v as 1/2 v = rv 2 rdr and define the space V as the copletion of the space C Ω with respect to the nor. Siilarly, for any function v C 1 Ω we define v 1 as 1/2 v 1 = r[v 2 r + v r 2 ]dr and define the space V 1 as copletion of the space C 1 Ω with respect to the nor 1. Note that the nors and 1 can be defined, respectively, fro the inner products u, v = u, v + u, v = rurvrdr, r[urvr + u rv r]dr. Identifying V with its dual V we obtain the dense and continuous ebedding V 1 V V V 1. The inner product notation will be re-utilized to denote the duality pairing between V 1 and V 1. We then have the following leas, the proofs of which can be found in [2]: Lea 2.1. There exist constants K 1 > and K 2 > such that, for all v C 1 Ω and r Ω, i v 2 + v 2 1 v 2, ii v1 K 1 v 1, iii r vr K 2 v 1. Lea 2.2. The ebedding V 1 V is copact. Reark 2.3. In Lea 2.1, the constants K 1 and K 2 can be given explicitly as K 1 = and K 2 = We also note that li r + rvr = for all v V 1 see [1, Lea 5.4]. On the other hand, fro H 1 ε, 1 C [ε, 1], < ε < 1 and ε v H1 ε,1 v 1 for all v V 1, it follows that v [ε,1] C [ε, 1]. Fro both relations we deduce that rv C Ω for all v V 1. Now, we define the bilinear for au, v = hu1v1 + ru rv rdr, for u, v V 1, 2.1 h is a positive constant. Then for soe uniquely defined bounded linear operator A : V 1 V 1 we have au, v = Au, v for all u, v V 1. We then have the following lea. Lea 2.4. The syetric bilinear for a, defined by 2.1 is continuous on V 1 V 1 and coercive on V 1, i.e., i au, v C 1 u 1 v 1 ii av, v C v 2 1

5 EJDE-25/138 NONLINEAR KIRCHHOFF-CARRIER WAVE EQUATION 5 for all u, v V 1, C = 1 2 in{1, h} and C 1 = 1 + hk 2 1. The proof of Lea 2.4 is straightforward and we oit it. Lea 2.5. There exists an orthonoral Hilbert basis { w j } of the space V consisting of eigenfunctions w j corresponding to eigenvalues λ j such that i < λ 1 λ j + as j +, ii a w j, v = λ j w j, v for all v V 1 and j N. Note that fro ii it follows that { w j / λ j } is autoatically an orthonoral set in V 1 with respect to a, as inner product. The eigensolutions w j are indeed eigensolutions for the boundary value proble A w j 1 d r dr d w j r = λj w j, in Ω, dr li r + r d w j dr r < +, d w j dr 1 + h w j1 =. The proof of the above Lea can be found in [24, Theore 6.2.1] with V = V 1, H = V and a, as defined by 2.1. For functions v in C 2 Ω, we define 1/2, v 2 = r[v 2 r + v r 2 + Avr ]dr 2 and define the space V 2 as the copletion of C 2 Ω with respect to the nor 2. Note that V 2 is also a Hilbert space with respect to the scalar product u, v + u, v + Au, Av and that V 2 can be defined also as V 2 = {v V 1 : Av V }. We then have the following two leas whose proof of which can be found in [2]. Lea 2.6. The ebedding V 2 V 1 is copact. Lea 2.7. For all v V 2 we have i v L Ω 1 2 Av, ii v 3 2 Av, iii v 2 L Ω 2 v Av v. Also the following lea will be useful in Section 4. Lea 2.8. For all u V 1 and v V, the constant K 1 is given by Lea 2.1. u 2, v = 21 + K 2 1 u 2 1 v, 2.2 Proof. It suffices to prove that inequality 2.2 holds for u C 1 Ω and v C Ω. We have ur = u1 r u sds.

6 6 N. T. LONG EJDE-25/138 Hence, it follows fro Lea 2.1 that 2 u 2 r 2u u sds 2K 2 1 u r This iplies u 2, v = 2K 2 1 u 2 1 r ru 2 r vr dr r vr dr + 2 r1 r vr dr r r u s 2 ds. u s 2 ds. Note that the first integral herein can be estiated as 1/2 1/2 r vr dr rdr r vr 2 1 dr = 2 v. 2.3 Reversing the order of integration in the last integral of 2.3, we estiate that integral as = r1 r vr dr u s 2 ds s u s 2 ds r1 r vr dr s 1/2 s u s 2 ds r1 r 2 dr 1 2 u 2 v 1 2 u 2 1 v. 1/2 r vr 2 dr Fro the two estiates above, we obtain 2.2 and the lea is proved. For a Banach space X, we denote by X its nor, by X its dual space and by L p, T ; X,1 p the Banach space of all real easurable functions u :, T X such that T 1/p u L p,t ;X = ut p X dt < for 1 p <, Let u L,T ;X = ess sup <t<t ut X for p =. ut, u t = u t t = ut, u t = u tt t = üt, u r t = ut, u rr t denote respectively. ur, t, u r, t, t 2 u u r, t, t2 r, t, r 3. The general case 2 u r, t, r2 In this section, we consider initial and boundary value proble 1.1 with general right-hand side f = fr, t, u, u r. We ake the following assuptions: H1 ũ 1 V 1 and ũ V 2, H2 B C 1 R 2 + with Bξ, η b > for all ξ, η, H3 f C Ω R + R 2 and f/ r, f/ u, f/ u r C Ω R + R 2.

7 EJDE-25/138 NONLINEAR KIRCHHOFF-CARRIER WAVE EQUATION 7 With B and f satisfying assuptions H2 and H3, respectively, we introduce the following constants, for any M > and T > : K = K M, B = sup{bξ, η : ξ, η M 2 }, K 1 = K 1 M, B = sup { B + B ξ η ξ, η : ξ, η M 2}, K = K M, T, f = sup fr, t, u, v, r,t,u,v A K 1 = K 1 M, T, f = sup f + f + f r, t, u, v, r,t,u,v A r u v A = A M, T = {r, t, u, v : r 1, t T, u M 2 + 1/ 2, v M/ 2}. For each M > and T > we put { W M, T = v L, T ; V 2 : v L, T ; V 1 and v L 2, T ; V, } with v L,T ;V 2, v L,T ;V 1, v L 2,T ;V M, W 1 M, T = { v W M, T : v L, T ; V }. We shall choose as first initial ter u = ũ, suppose that 3.1 u 1 W 1 M, T, 3.2 and associate with proble 1.1 the following variational proble: Find u in W 1 M, T 1 so that ü t, v + b tau t, v = F t, v v V 1, u = ũ, u = ũ 1, b t = B u 1 t 2, u 1 t 2 = B Then, we have the following result. u 2 1r, trdr, u 1 r F r, t = f r, t, u 1 t, u 1 t. r, t 2 rdr, Theore 3.1. Let assuptions H1 H3 hold. Then there exist a constant M > depending on ũ, ũ 1, B, h and a constant T > depending on ũ, ũ 1, B, h, f such that, for u = ũ, there exists a linear recurrent sequence {u } W 1 M, T defined by Proof. The proof consists of several steps. Step 1: The Galerkin approxiation introduced by Lions [1]. Consider as in Lea 2.5 the basis w j = w j / λ j for V 1 and put u k t = k j=1 c k j tw j, 3.5

8 8 N. T. LONG EJDE-25/138 the coefficients c k j satisfy the syste of linear differential equations ü k t, w j + b ta u k t, w j = F t, w j, 1 j k, u k = ũ k, u k = ũ 1k, ũ k ũ strongly in V 2, ũ 1k ũ 1 strongly in V Suppose that u 1 satisfies 3.2. Then it is clear that syste 3.6 has a unique solution u k on an interval t T k T. The following estiates allows us to the take constant T k = T for all and k. Step 2: A priori estiates. Put S k t = X k t + Y k t + X k t = u k t 2 + b ta u k t, u k t, Y k t = a u k t, u k t + b t Au k t 2, A is defined by 2.1. Then it follows that S k t =S k ü k s 2 ds, 3.8 b s [ a u k s, u k s + Au k s 2 ] ds F s, u k s ds + 2 b s Au k s, ü k s ds + =S k + I I 5. a F s, u k s ds F s, ü k s ds We shall estiate step by step all integrals I 1,..., I 5. Integral I 1 : Using assuption H2, we obtain fro and that b t 2 B u 1 t 2 ξ, u 1 t 2 u 1 t, u 1 t + 2 B η 4M 2 K1. Cobining we obtain u 1 t 2, u 1 t 2 u 1 t, u 1 t I 1 4M 2 K1 b S k sds. 3.9 Integral I 2 : Since u 1 W 1 M, T, it follows fro Lea 2.7 that u 1 r, t M 2 + 1/ 2, u 1 r, t M/ 2, a.e. on Ω, T. 3.1 By the Cauchy-Schwarz inequality, it follows fro that I 2 2 F s u k s ds 2K X k sds.

9 EJDE-25/138 NONLINEAR KIRCHHOFF-CARRIER WAVE EQUATION 9 Integral I 3 : Using Lea 2.4, we have I 3 2C 1 F s 1 u k s 1 ds. On the other hand, fro and 3.1 we obtain F s K K2 1[ M ] 2. Then we deduce, fro that 2C1 [ 2 I 3 K M 2 2] t 1/2 K 1 C Y k sds. Integral I 4 : Using the inequality ab 3 4 a b2 a, b R, we get fro and that I 4 K 3 4 K 2 3 K 2 4b Au k s ü k s ds Au k s 2 ds S k sds Sk t. ü k s 2 ds Integral I 5 : We use again inequality ab 3 4 a b2 a, b R, we get fro and 3.8 that I 5 F s ük s ds 3 4 T K Sk t. Cobining the above estiates for I 1,..., I 5, we get S k t 3S k + C 1 M, T + C 2 M S k sds, 3.11 C 1 M, T = 45 4 T K2 + 9 T C 2 [ 2 1 K M 2 2 K 2C 1], C 2 M = K2 4b + 16M K1. Now, we need an estiate on the ter S k. We have S k = X k + Y k = ũ 1k 2 + aũ 1k, ũ 1k + B ũ 2 aũk, ũ k + Aũk 2. By eans of the convergence 3.7, we can deduce the existence of a constant M > independent of k and such that S k M 2 /

10 1 N. T. LONG EJDE-25/138 Note that, fro the assuption H3, we have li T + T Ki M, T, f =, i =, 1. Then, fro 3.12 we can always choose the constant T > such that M 2 /2 + C 1 M, T exp [ T C 2 M ] M 2, M 2 T b C K 1 + 2T K 1 exp [ 1 2 T K C 1 M 2 T b C K ] 1 < 1. It follows fro 3.11 and that for t T k for all t [, T k have u k {u ki S k t M 2 exp[ T C 2 M] + C 2 M S k sds T. By using Gronwall s lea we deduce fro here that S k t M 2 exp[ T C 2 M] exp[c 2 Mt] M 2 ]. So we can take constant T k W 1 M, T for all and k. We can extract fro {u k } such that u ki u in L, T ; V 2 weak, u ki u in L, T ; V 1 weak, ü ki ü in L 2, T ; V weak, 3.14 = T for all k and. Therefore, we } a subsequence u W M, T. Passing to the liit in 3.6, we have u satisfying 3.3 in L 2, T, weak. On the other hand, it follows fro and u W M, T that ü = b tau + F L, T ; V, hence u W 1 M, T and the proof of Theore 3.1 is coplete. Theore 3.2. Let assuptions H1-H3 hold. Then: i There exist constants M > and T > satisfying such that proble 1.1 has a unique weak solution u W 1 M, T. ii On the other hand, the linear recurrent sequence u defined by converges to the solution u of proble 1.1 strongly in the space W 1 T = { v L, T ; V 1 : v L, T ; V }. Furtherore, we have the estiate u u L,T ;V 1 + u u L,T ;V Ck T 1, k T = M 2 T b C K 1 + 2T K 1 [ 1 exp 2 T K C 1 M 2 T b C K ] 1 < 1, and C is a constant depending only on T, u, u 1 and k T. Proof. Existence of the solution. First, we note that W 1 T is a Banach space with respect to the nor see [1]: v W1T = v L,T ;V 1 + v L,T ;V.

11 EJDE-25/138 NONLINEAR KIRCHHOFF-CARRIER WAVE EQUATION 11 We shall prove that {u } is a Cauchy sequence in W 1 T. For this, set v = u +1 u. Then v satisfies the variational proble v t, w + b +1 ta v t, w + b +1 t b t Au t, w = F +1 t F t, w w V 1, v = v =. Taking w = v herein, after integrating in t, we get X t = b +1sa v s, v s ds b+1 s b s Au s, v s ds F +1 s F s, v s ds, X t = v t 2 + b +1 ta v t, v t. On the other hand, fro 3.1 2,4 and 3.2 we obtain It follows that b +1t 4M 2 K1, b +1 t b t 2 K 1 M v 1 t + 2 K 1 M v 1 t v t 2 + b C v t K 1 M v 1 t 1, F +1 t F t 2K 1 v 1 t 1. 4M 2 K1 C 1 v s 2 1ds + 8M K 1 v 1 s 1 Av s v s ds + 2 2K 1 v 1 s 1 v s ds 4M 2 K1 C 1 v s 2 1ds + 16M 2 K K 1 v 1 s 1 v s ds 8M 2 K1 + 2K 1 v 1 2 W 1T [ K C ] 1 M 2 K1 b C Using Gronwall s lea we deduce that v t 2 + b C v t 2 1 8M 2 K1 + 2K 1 v 1 2 W 1T exp {2T for t T. Hence v s 2 + b C v s 2 1 ds. v W1T k T v 1 W1T 1, [ 1 2 K C 1 M 2 K1 ]}, b C

12 12 N. T. LONG EJDE-25/138 k T = M 2 T b C K 1 + [ 1 2T K 1 exp 2 T K C 1 M 2 T b C K ] 1 < 1. Hence u +p u W1T u 1 u W1T 1 k T for all and p. It follows that {u } is a Cauchy sequence in W 1 T. Therefore, there exists u W 1 T such that u u strongly in W 1 T We also note that u W 1 M, T. Then fro the sequence {u } we can deduce a subsequence {u j } such that u j u in L, T ; V 2 weak, u j u in L, T ; V 1 weak, ü j ü in L 2, T ; V weak, with u W M, T. Noticing we have T b tau t B ut 2, ut 2 Aut, wt dt C 1 K u u L,T ;V 1 w L 1,T ;V 1 + 4C 1 M K 1 u 1 u L,T ;V 1 u L,T ;V 1 w L1,T ;V 1 k T 3.16 for all w L 1, T ; V 1. It follows fro that b tau B ut 2, ut 2 Au in L, T ; V 1 weak Siilarly F fr, t, u, u r L,T ;V 2K 1 u 1 u L,T ;V Hence, fro 3.15 and 3.18, we obtain F fr, t, u, u r strongly in L, T ; V Then, taking liits in 3.3 with = j +, there exists u W M, T satisfying üt, w + B ut 2, ut 2 a ut, w = fr, t, u, ur, w w V 1, u = ũ, u = ũ 1. On the other hand, fro 3.17 and we have ü = B u 2, u 2 Au + fr, t, u, ur L, T ; V. 3.2 Hence, u W 1 M, T and the proof of existence coplete. Uniqueness of the solution. Let u 1, u 2, be weak solutions of proble such that u 1 and u 2 are in W 1 M, T. Then w = u 1 u 2 satisfies the variational proble ẅt, v + b 1 ta wt, v + b1 t b 2 t Au 2 t, v = f 1 t f 2 t, v v V 1, w = ẇ =, bi t = B u i t 2, u i t 2, fi t = fr, t, u i, u i, i = 1, 2.

13 EJDE-25/138 NONLINEAR KIRCHHOFF-CARRIER WAVE EQUATION 13 Taking v = ẇ and integrating by parts, we obtain Put Then ẇt 2 + b 1 ta wt, wt = Xt = b 1 sa ws, ws ds b1 s b 2 s Au 2 s, ẇs ds f 1 s f 2 s, ẇs ds. Xt = ẇt 2 + b C wt [4 1 + C 1 M 2 K1 + ] 2K 1 b C b C Xsds for all t [, T ] follows. Using Gronwall s lea we deduce Xt =, i.e., u 1 = u 2 and the proof of Theore 3.2 is coplete. Reark 3.3. In the case of B 1 and f = ft, u, u t with f C 1 R + R 2 and ft,, = for all t, and with the hoogeneous Dirichlet boundary condition instead of 1.1 2, soe results have been obtained in [4]. In the case of f being in C 1 Ω R + R 2 and B 1 we have previously obtained soe results in [2]. We ephasize here that in the above, however, we do not need to assue that f is in C 1 Ω R + R A special case In this section, we consider initial boundary value proble 1.1 with an autonoous right-hand side independent of u r and an affine coefficient function B. Under these assuptions, we obtain stronger conclusion on the approach results in a quadratic convergence of the approxiation Theore 4.2. We ake the following assuptions: H4 Bη = b + η with b > a given constant. H5 f C 2 Ω R. With f satisfying assuption H5, for any M > we put K = K M, f = sup fr, u, r,u A f K 1 = K 1 M, f = sup + f r, u, r,u A r u 2 f K 2 = K 2 M, f = sup + 2 f r u u 2 r, u, r,u A A = A M = { r, u : r 1, u M 2 + 1/ 2 }. We shall choose as a constant in tie starting point u the initial data ũ. Assue u 1 W 1 M, T and consider the variational proble 3.3, b t = b + u t 2, F r, t = f r, t, u = fr, u 1 + u u 1 f u r, u 1, 4.1

14 14 N. T. LONG EJDE-25/138 with f r, t, u = fr, u 1 + u u 1 f u r, u 1. Then we have the following theore. Theore 4.1. Let H1, H4, and H5 hold. Then there exist constants M > and T > and the recurrent sequence {u } W 1 M, T defined by 3.3 and 4.1. Proof. The idea is the sae as in the proof of Theore 3.1. As there we define u k by , the functions b and F appearing in 3.5 are replaced by F k b k t = b + u k t 2, r, t = f r, t, u k = fr, u 1 + u k f u 1 u r, u 1, respectively. With S k, X k, Y k appearing in X k and Y k S k t =S k We can estiate S k defined by , the function b are replaced by b k, it follows that b k s [a u k s, u k s ] + Au k s 2 ds F k s, u k s ds + 2 b k s Au k s, ü k s ds + a F k s, u k s ds in a anner siilar to 3.11 as S k t 3S k + D M, T + D 1 M D = D M, T = 21 2 K + MK 1 2 F k s, ü k s ds. S k sds + D 2 S k s 2 ds, + 6C 1 T K M + 2, 1 + 3M 2 K 1 + K + MK M 2 /2 D 1 =D 1 M = C 1/C + 21K 2 1/2b C + 6C 1 2 4K M 2 /2K 2 b C 2, D 2 = 3 b 2 b C Fro convergence 3.7 we can deduce the existence of a constant M > independent of k and such that S k M 2 /6. Next, we can always choose a constant T >, so that D M, T M 2 /2 and 1 + D 1 M 2 expt D D 1 D 2 3M D 2 Then S k t 3 4 M 2 + D 1 M S k sds + D 2 S k s 2 ds. 4.3

15 EJDE-25/138 NONLINEAR KIRCHHOFF-CARRIER WAVE EQUATION 15 On the other hand, the function D 1 expd 1 t St = 4D 1, t T, 4.4 3M D 2 2 [expd 1 t 1] is the axial solution of the Volterra integral equation with non-decreasing kernel [8] Fro St = 3 4 M 2 + D 1 M Ssds + D 2 S 2 sds, t T. S k t St M 2, t T 4.5 follows for all k and. Hence u k W 1 M, T for all k and. Then, in a anner siilar to the proof of Theore 3.1, we can prove that the liit u W 1 M, T of the sequence {u k } when k + is the unique solution of variational proble 3.3 and 4.1. The proof of Theore 4.1 is coplete. The following result gives a quadratic convergence of the sequence {u } to a weak solution of proble 1.1 corresponding to f = fr, u and Bη = b + η. Theore 4.2. Let assuptions H1 and H4-H5, hold. Then i There exist constants M > and T > such that proble 1.1 corresponding to f = fr, u and Bη = b + η has a unique weak solution u W 1 M, T. ii On the other hand, the recurrent sequence {u } defined by 3.3 and 4.1 converges quadratically to the solution u strongly in the space W 1 T in the sense u u W1T C u 1 u 2 W 1T, C is a suitable constant. Furtherore, we have also the estiation u u W1T β 2 µ T 1 β for all, µ T = b C T K 2 T expt K1 b C T and β = 4Mµ T < 1. Note that the last condition is always satisfied by taking a suitable T >. Proof. First, we shall prove that {u } is a Cauchy sequence in W 1 T. For this, set v = u +1 u. Then v satisfies the variational proble v t, w + b +1 ta v t, w + b +1 t b t Au t, w = F +1 t F t, w, w V 1, v = v =, 4.6

16 16 N. T. LONG EJDE-25/138 with F +1 F = fr, u fr, u 1 + u +1 u f u r, u u u 1 f u r, u 1 f = v u r, u f 2 v2 1 u 2 r, λ, λ = u 1 + θv 1, < θ < 1, b +1 t b t = u +1 t 2 u t 2. Taking w = v in 4.6 and integrating in t, we get v t 2 + b +1 ta v t, v t = 2 2 u +1 s, u +1 s a v s, v s ds u+1 s 2 u s 2 Au s, v s ds f + 2 v u r, u, v s ds + = J J 4. v f u 2 r, λ, v s ds We can estiate herein the integrals J 1,..., J 4 step by step as J 1 2M 2 a v s, v s ds 2C 1 M 2 v s 2 1ds, J 2 4M 2 v s 1 v s ds, by 4.5, J 3 4K 1 v s v s ds 4K 1 v s 1 v s ds, J 4 4K 2 v 1s, 2 v s ds K K 2 1 v 1 s 2 1 v s ds, the last inequality follows fro 4.5 and Lea 2.8. Cobining the above estiates, we obtain v t 2 + b C v t 2 1 2C 1 M 2 v s 2 1ds + 22M 2 + K 1 + K K 2 1 v 1 s 2 1 v s ds. v s 1 v s ds Letting Z t = v s 2 + b C v s 2 1, the above inequality can be written as Z t K 1 T Z sds + K 2 T Z 2 1sds, 4.7 K 1 T = 2C 1M 2 b C + 2M 2 + K 1 b C K2 1K 2 2b C, K 2 T = 1 + K2 1K 2 2b C.

17 EJDE-25/138 NONLINEAR KIRCHHOFF-CARRIER WAVE EQUATION 17 Hence, we deduce fro 4.7 that µ T is the constant Fro 4.8, we obtain v W1T µ T v 1 2 W 1T, 4.8 µ T = b C T K 2 T expt K1 T b C. u u +p W1T β 2 µ T 1 β 4.9 for all and p β = 4M µt < 1. It follows that {u } is a Cauchy sequence in W 1 T. Then there exists u W 1 T such that u u strongly in W 1 T. Thus, and by applying a siilar arguent as used in the proof of Theore 3.2, u W 1 M, T is the unique weak solution of proble 1.1 corresponding to f = fr, u and Bη = b + η. Passing to the liit as p + for fixed, we obtain estiate 4.5 fro 4.9. This copletes the proof of Theore 4.2. Acknowledgents. The author wants to thank Professor Dung Le for this help, and the anonyous referees for their valuable suggestions. References [1] Adas, R. A.; Sobolev Spaces, Acadeic Press, NewYork, [2] Binh, D. T. T.; Dinh, A. P. N; Long, N. T.; Linear recursive schees associated with the nonlinear wave equation involving Bessel s operator, Math. Cop. Modelling, No. 5-6, [3] Carrier, G. F; On the nonlinear vibrations proble of elastic string, Quart. J. Appl. Math [4] Dinh, A. P. N; Long N. T.; Linear approxiation and asyptotic expansion associated to the nonlinear wave equation in one diension, Deonstratio Math , No. 1, [5] Ebihara, Y; Medeiros, L. A.; Milla, M. Minranda; Local solutions for a nonlinear degenerate hyperbolic equation, Nonlinear Anal [6] Hosoya, M.; Yaada, Y.; On soe nonlinear wave equation I: Local existence and regularity of solutions, J. Fac. Sci. Univ. Tokyo. Sect. IA, Math [7] Kirchhoff, G. R.; Vorlesungen über Matheatiche Physik: Mechanik, Teuber, Leipzig, 1876, Section [8] Lakshikantha, V; Leela, S.; Differential and Integral Inequalities, Vol.1. Acadeic Press, NewYork, [9] Lan, H. B.; Thanh, L. T.; Long, N. T.; Bang, N.T,; Cuong T.L., Minh, T.N.; On the nonlinear vibrations equation with a coefficient containing an integral, Zh. Vychisl. Mat. i Mat. Fiz , No. 9, ; translation in Coput. Math. Math. Phys , No. 9, [1] Lions, J. L.; Quelques éthodes de résolution des problè es aux liites non-linéaires, Dunod; Gauthier- Villars, Paris, [11] Long, N. T.; Dinh, A. P. N.; Periodic solutions of a nonlinear parabolic equation associated with the penetration of a agnetic field into a substance, Coputers Math. Appl , No. 1, [12] Long, N.T.; Dinh, A. P. N.; On the quasilinear wave equation: u tt u + fu, u t = associated with a ixed nonhoogeneous condition, Nonlinear Anal , No. 7, [13] Long, N. T.; Dinh, A. P. N.; A seilinear wave equation associated with a linear differential equation with Cauchy data, Nonlinear Anal , No. 8, [14] Long, N. T.; Die, T. N.; On the nonlinear wave equation u tt u xx = fx, t, u, u x, u t associated with the ixed hoogeneous conditions, Nonlinear Anal , No. 11,

18 18 N. T. LONG EJDE-25/138 [15] Long, N. T.; Thuyet, T. M.; On the existence, uniqueness of solution of the nonlinear vibrations equation, Deonstratio Math , No. 4, [16] Long, N. T.; Dinh, A. P. N.; Binh, D. T. T.; Mixed proble for soe seilinear wave equation involving Bessel s operator, Deonstratio Math , No. 1, [17] Long, N. T., Dinh, A. P. N.; Die, T. N.; Linear recursive schees and asyptotic expansion associated with the Kirchhoff-Carrier operator, J. Math. Anal. Appl , No. 1, [18] Long, N. T.; On the nonlinear wave equation u tt Bt, u x 2 u xx = fx, t, u, u x, u t associated with the ixed hoogeneous conditions, J. Math. Anal. Appl , No. 1, [19] Medeiros L. A.; On soe nonlinear perturbation of Kirchhoff-Carrier operator, Cop. Appl. Math [2] Medeiros, L.A.; Liaco, J.; Menezes, S. B.; Vibrations of elastic strings: Matheatical aspects, Part one, J. Coput. Anal. Appl. 4 22, No. 2, [21] Medeiros, L. A.; Liaco, J.; Menezes, S. B.; Vibrations of elastic strings: Matheatical aspects, Part two, J. Coput. Anal. Appl. 4 22, No. 3, [22] Pohozaev, S. I.; On a class of quasilinear hyperbolic equation, Math. USSR. Sb [23] Ortiz, E. L., Dinh, A. P. N.; Linear recursive schees associated with soe nonlinear partial differential equations in one diension and the Tau ethod, SIAM J. Math. Anal [24] Raviart, P. A.; Thoas, J. M., Introduction à l analyse nuérique des equations aux dérivées partielles, Masson, Paris, Nguyen Thanh Long Departent of Matheatics and Coputer Science, University of Natural Science, Vietna National University HoChiMinh City, 227 Nguyen Van Cu Str., Dist. 5, Hochiinh City, Vietna E-ail address: longnt@hcc.netna.vn longnt2@gail.co