Electronic Journal of Differential Equations, Vol. 2006(2006), No. 77, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (loin: ftp) STRONG GLOBAL ATTRACTOR FOR A QUASILINEAR NONLOCAL WAVE EQUATION ON PERIKLES G. PAPADOPOULOS, NIKOLAOS M. STAVRAKAKIS Abstract. We study the lon time behavior of solutions to the nonlocal quasilinear dissipative wave equation u tt φ(x) u(t) 2 u + δu t + u a u = 0, in, t 0, with initial conditions u(x, 0) = u 0 (x) and u t(x, 0) = u 1 (x). We consider the case N 3, δ > 0, and (φ(x)) 1 a positive function in L N/2 ( ) L ( ). The existence of a lobal attractor is proved in the stron topoloy of the space D 1,2 ( ) L 2 (RN ). 1. Introduction Our aim in this work is to study the quasilinear hyperbolic initial-value problem u tt φ(x) u(t) 2 u + δu t + u a u = 0, x, t 0, (1.1) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x, (1.2) with initial conditions u 0, u 1 in appropriate function spaces, N 3, and δ > 0. Throuhout the paper we assume that the functions φ, : R satisfy the condition (G1) φ(x) > 0, for all x and (φ(x)) 1 := (x) L N/2 ( ) L ( ). For the modellin process we refer the reader to some of our earlier papers [11, 13] or to the oriinal paper by Kirchhoff in 1883 [8]. There he proposed the so called Kirchhoff strin model in the study of oscillations of stretched strins and plates. In bounded domains there is a vast literature concernin the attractors of semilinear waves equations. We refer the reader to the monoraphs [3, 14]. Also in the paper [4], the existence of lobal attractor in a weak topoloy is discussed for a eneral dissipative wave equation. Ono [9], for δ 0, has proved lobal existence, decay estimates, asymptotic stability and blow up results for a deenerate non-linear wave equation of Kirchhoff type with a stron dissipation. On the other hand, it seems that very few results are achieved for the unbounded domain case. In our previous work [11], we proved lobal existence and blow-up results for an equation of Kirchhoff type in all of. Also, in [13] we proved the existence of 2000 Mathematics Subject Classification. 35A07, 35B30, 35B40, 35B45, 35L15, 35L70, 35L80. Key words and phrases. Quasilinear hyperbolic equations; Kirchhoff strins; lobal attractor; unbounded domains; eneralized Sobolev spaces; weihted L p spaces. c 2006 Texas State University - San Marcos. Submitted May 10, 2006. Published Juy 12, 2006. 1
2 P. G. PAPADOPOULOS, N. M. STAVRAKAKIS EJDE-2006/77 compact invariant sets for the same equation. Recently, in [12] we studied the stability of the oriin for the eneralized equation of Kirchhoff strins on, usin central manifold theory. Also, Karahalios and Stavrakakis [5], [7] proved existence of lobal attractors and estimated their dimension for a semilinear dissipative wave equation on. The presentation of this paper is follows: In Section 2, we discuss the space settin of the problem and the necessary embeddins for constructin the evolution triple. In Section 3, we prove existence of an absorbin set for our problem in the enery space X 0. Finally in Section 4, we prove that there exists a lobal attractor A in the stron topoloy of the enery space X 1 := D 1,2 ( ) L 2 ( ), so extendin some earlier results of us on the asymptotic behavior of the problem (see [13]). Notation. We denote by B R the open ball of with center 0 and radius R. Sometimes for simplicity we use the symbols C 0, D 1,2, L p, 1 p, for the spaces C 0 ( ), D 1,2 ( ), L p ( ), respectively; p for the norm L p ( ), where in case of p = 2 we may omit the index. The symbol := is used for definitions. 2. Space Settin. Formulation of the Problem As it is already shown in the paper [11], the space settin for the initial conditions and the solutions of problem (1.1)-(1.2) is the product space X 0 := D(A) D 1,2 ( ), N 3. Also the space X 1 := D 1,2 ( ) L 2 ( ), with the associated norm e 1 (u(t)) := u 2 D + u 1,2 t 2 L is introduced, where the space L 2 ( ) is defined to be the closure 2 of C0 ( ) functions with respect to the inner product (u, v) L 2 ( ) := uvdx. (2.1) It is clear that L 2 ( ) is a separable Hilbert space and the embeddin X 0 X 1 is compact. The homoeneous Sobolev space D 1,2 ( ) is defined, as the closure of C0 ( ) functions with respect to the followin enery norm u 2 D := 1,2 u 2 dx. It is known that D 1,2 ( ) = { u L 2N N 2 ( ) : u (L 2 ( )) N} and D 1,2 ( ) is embedded continuously in L 2N N 2 ( ), that is, there exists k > 0 such that u 2N k u D 1,2. (2.2) N 2 The space D(A) is oin to be introduced and studied later in this section. The followin eneralized version of Poincaré s inequality is oin to be frequently used u 2 dx α u 2 dx, (2.3) for all u C0 and L N/2, where α := k 2 1 N/2 (see [1, Lemma 2.1]). It is shown that D 1,2 ( ) is a separable Hilbert space. Moreover, the followin compact embeddin is useful.
EJDE-2006/77 STRONG GLOBAL ATTRACTOR 3 Lemma 2.1. Let L N/2 ( ) L ( ). Then the embeddin D 1,2 L 2 is 2N compact. Also, let L 2N pn+2p ( ). Then the followin continuous embeddin D 1,2 ( ) L p ( ) is valid, for all 1 p 2N/(N 2). For the proof of the above lemma, we refer to [6, Lemma 2.1]. To study the properties of the operator φ, we consider the equation φ(x) u(x) = η(x), x, (2.4) without boundary conditions. Since for every u, v C0 ( ) we have ( φ u, v) L 2 = u v dx, (2.5) we may consider (2.4) as an operator equation of the form A 0 u = η, A 0 : D(A 0 ) L 2 ( ) L 2 ( ), η L 2 ( ). (2.6) The operator A 0 = φ is a symmetric, stronly monotone operator on L 2 ( ). Hence, the theorem of Friedrichs is applicable. The enery scalar product iven by (2.5) is (u, v) E = u vdx and the enery space X E is the completion of D(A 0 ) with respect to (u, v) E. It is obvious that the enery space is the homoeneous Sobolev space D 1,2 ( ). The enery extension A E = φ of A 0, φ : D 1,2 ( ) D 1,2 ( ), (2.7) is defined to be the duality mappin of D 1,2 ( ). We define D(A) to be the set of all solutions of equation (2.4), for arbitrary η L 2 ( ). Usin the theorem of Friedrichs we have that the extension A of A 0 is the restriction of the enery extension A E to the set D(A). The operator A = φ is self-adjoint and therefore raph-closed. Its domain D(A), is a Hilbert space with respect to the raph scalar product (u, v) D(A) = (u, v) L 2 + (Au, Av) L 2, for all u, v D(A). The norm induced by the scalar product is u D(A) = { u 2 dx + φ u 2 dx which is equivalent to the norm Au L 2 = { φ u 2 dx } 1/2. So we have established the evolution quartet } 1/2, D(A) D 1,2 ( ) L 2 ( ) D 1,2 ( ), (2.8) where all the embeddins are dense and compact. Finally, the definition of weak solutions for the problem (1.1) (1.2) is iven. Definition 2.2. A weak solution of (1.1)-(1.2) is a function u such that the followin three conditions are satisfied: (i) u L 2 [0, T ; D(A)], u t L 2 [0, T ; D 1,2 ( )], u tt L 2 [0, T ; L 2 ( )],
4 P. G. PAPADOPOULOS, N. M. STAVRAKAKIS EJDE-2006/77 (ii) for all v C0 ([0, T ] ( )), satisfies the eneralized formula T T ) (u tt (τ), v(τ)) L 2 dτ + ( u(t) 2 u(τ) v(τ)dx dτ 0 0 +δ T 0 (u t (τ), v(τ)) L 2 dτ + T 0 ( u(τ) a u(τ), v(τ)) L 2 dτ = 0, (2.9) (iii) u satisfies the initial conditions u(x, 0) = u 0 (x), u 0 D(A), u t (x, 0) = u 1 (x), u 1 D 1,2 ( ). 3. Existence of an Absorbin Set. In this section we prove existence of an absorbin set for our problem (1.1)-(1.2) in the enery space X 0. First, we ive existence and uniqueness results for the problem (1.1)-(1.2) usin the space settin established previously. Theorem 3.1 (Local Existence). Consider that (u 0, u 1 ) D(A) D 1,2 and satisfy the nondeeneracy condition u 0 > 0. (3.1) Then there exists T = T ( u 0 D(A), u 1 ) > 0 such that the problem (1.1)-(1.2) admits a unique local weak solution u satisfyin u C(0, T ; D(A)) and u t C(0, T ; D 1,2 ). Moreover, at least one of the followin two statements holds: (i) T = +, (ii) e(u(t)) := u(t) 2 D(A) + u t(t) 2 D 1,2, as t T. For the proof of the above theorem, we refer to [11, Theorem 3.2]. Remark 3.2. The nondeeneracy condition (3.1) is imposed by the method which is used even for the proof of existence of local solutions of the problem (1.1)-(1.2). For more details we refer to the proof of Theorem 3.2 in [11]. Also we must notice that this condition is necessary even in the case of bounded domains (e.., see [9] and [10]). Lemma 3.3. Assume that a 0, N 3. If the initial data (u 0, u 1 ) D(A) D 1,2 and satisfy the condition u 0 > 0, (3.2) then u(t) > 0, for all t 0. (3.3) Proof. Let u(t) be a unique solution of the problem (1.1)-(1.2) in the sense of Theorem 3.1 on [0, T ). Multiplyin (1.1) by 2 u t (in the sense of the inner product in the space L 2 ) and interatin it over, we have d dt u t(t) 2 + u(t) 2 d dt u(t) 2 D(A) +2 u t (t) 2 + 2( u a u, u t (t)) = 0 Since u 0 > 0, we see that u(t) > 0 near t = 0. Let T := sup{t [0, + ) : u(s) > 0 for 0 s < t}, (3.4) then T > 0 and u(t) > 0 for 0 t < T. By contradiction we may prove that T = +.
EJDE-2006/77 STRONG GLOBAL ATTRACTOR 5 Theorem 3.4 (Absorbin Set). Assume that 0 a < 2/(N 2), N 3, M 0 := 1 2 u 0 2 > 0, (u 0, u 1 ) D(A) D 1,2 and δ 4 > 4α 1/2 R 2 c 2 3, (3.5) where c 3 := (max{1, M0 1 })1/2 and R a iven positive constant. Then the ball B 0 := B X0 (0, R ), for any R > R, is an absorbin set in the enery space X 0, where R 2 := 2k 2R 2(a+1) ( δ δ 4 4R2 c 2 ) 3 1. α Proof. Given the constants T > 0, R > 0, we introduce the two parameter space of solutions X T,R := {u C(0, T ; D(A)) : u t C(0, T ; D 1,2 ), e(u) R 2, t [0, T ]}, where e(u) := u t 2 D + u 2 1,2 D(A). The set X T,R is a complete metric space under the distance d(u, v) := sup 0 t T e(u(t) v(t)). Followin [9] we introduce the notation T 0 := sup{t [0, ) : u(s) 2 > M 0, 0 s t}. Condition 1 2 u 0 2 = M 0 > 0 implies T 0 > 0 and u(t) 2 > M 0 > 0, for all t [0, T 0 ]. Next, we set v = u t + εu for sufficiently small ε. Then, for calculation needs, equation (1.1) is rewritten as v t + (δ ε)v + ( φ(x) u 2 ε(δ ε))u + f(u) = 0. (3.6) Multiplyin equation (3.6) by Av = ( ϕ )v = v = (u t + εu), and interatin over, we obtain (usin Hölder inequality with p 1 = 1 N, q 1 = N 2 2N, r 1 = 1 2 ) 1 d { u 2 2 dt D 1,2 u 2 D(A) + ε(δ ε) v 2 D1,2 + u 2 } D 2 1,2 + (δ ε) v 2 D + 1,2 ε u 2 D 1,2 u 2 D(A) + ε2 (δ ε) u 2 D (3.7) 1,2 ( d ) dt u 2 D u 2 1,2 D(A) + u a L Na u L N 2 2N v. We observe that θ(t) := u 2 D 1,2 u 2 D(A) + ε(δ ε) v 2 D1,2 + u 2 D 2 1,2 (3.8) M 0 u 2 D(A) + u t 2 D 1,2 c 2 3 e(u). Also ( d ) ( dt u 2 D u 2 1,2 D(A) = 2 uu t ϕ dx ) u 2 D(A) 2 ( u 2 1/2 ( D(A)) ut 2 1/2 u 2 L) 2 D(A) 2α 1/2 R 2 e(u) 2α 1/2 R 2 c 2 3θ(t). (3.9) Applyin Youn s inequality in the last term of (3.7) and usin relations (3.8), (3.9) and the estimates u a L Na Ra and u L 2N N 2 u D(A) R, (3.10)
6 P. G. PAPADOPOULOS, N. M. STAVRAKAKIS EJDE-2006/77 inequality (3.7) becomes (for suitably small ε) d dt θ(t) + C θ(t) C(R), (3.11) δ ( ) where C = 1 2 δ/4 4α 1/2 R 2 c 2 3 > 0 and C(R) = R 2(a+1). An application of Gronwall s inequality in the relation (3.11) ives θ(t) θ(0)e C t + 1 e C t C C(R). (3.12) δ Followin the reasonin developed by K. Ono (see [9]), the nondeeneracy condition u 0 > 0 and the relation (3.3), imply that u(s) > M 0 > 0, 0 s t, t [0, + ). Now, lettin t, in the relation (3.12) conclude that R2(a+1) lim sup θ(t) := R t δc. 2 (3.13) So, the ball B 0 := B X0 (0, R ), for any R > R, is an absorbin set for the associated semiroup S(t) in the enery space of solutions X 0. Corollary 3.5 (Global Existence). The unique local solution the problem (1.1)- (1.2) defined by Theorem 3.1 exists lobally in time. Proof. Combinin inequality (3.13) and the aruments developed in the proof of [11, Theorem 3.2], we conclude that the solution of the problem (1.1)-(1.2) exists lobally in time. 4. Stron Global Attractor in the space X 1 In this section we study the problem (1.1)-(1.2) from a dynamical system point of view. We need the followin results. Theorem 4.1. Assume that 0 a 4/(N 2), where N 3. D(A) D 1,2 and satisfy the nondeeneracy condition If (u 0, u 1 ) u 0 > 0, (4.1) then there exists T > 0 such that the problem (1.1)-(1.2) admits local weak solutions u satisfyin u C(0, T ; D 1,2 ) and u t C(0, T ; L 2 ). (4.2) Proof. The proof follows the lines of [11, Theorem 3.2], so we just sketch the proof. The compactness of the embeddin X 0 X 1 implies e 1 (u(t)) e(u(t)), where the associated norms are e 1 (u(t)) := u 2 D 1,2 + u t 2 L 2 and e(u(t)) := u 2 D(A) + u t 2 D 1,2. Then, for some positive constant R an a priori bound can be found of the form e 1 (u(t)) e(u(t)) R 2. Hence the solutions u of the problem (1.1)-(1.2) satisfy u L (0, T ; D 1,2 ), u t L (0, T ; L 2 ). Finally, the continuity properties (4.2), are proved followin ideas from [14, Sections II.3 and II.4]. Next, the stron continuity of the semiroup S(t) is proved in the space X 1.
EJDE-2006/77 STRONG GLOBAL ATTRACTOR 7 Lemma 4.2. The mappin S(t) : X 1 X 1 is continuous, for all t R. Proof. Let u, v two solutions of the problem (1.1)-(1.2) such that u tt φ(x) u 2 u + δu t = u a u, v tt φ(x) v 2 v + δv t = v a v. Let w = u v. So, we have w tt φ u 2 w + δw t = φ{ u 2 v 2 } v ( u a u v a v) w(0) = 0, w t (0) = 0. Multiplyin the previous equation by 2w t and interatin over, we et w t w tt dx 2 u 2 ww t dx + 2δ wt 2 dx R N = { u 2 v 2 } vw t dx 2 ( u a u v a v))w t dx. Hence d dt e (w) + 2δ w t 2 L 2 (4.3) = ( d dt u 2 ) w 2 + 2{ u 2 v 2 }( v, w t ) 2( u a u v a v, w t ) L 2 I 1 (t) + I 2 (t) + I 3 (t). (4.4) So d dt e (w) I 1 (t) + I 2 (t) + I 3 (t), (4.5) where e (w) = w t 2 L + C 2 u w 2 D and C 1,2 u = u 2 D. To estimate the above 1,2 interals, more smoothness of the solutions u, v is needed. Theorem 3.1 uarantees the uniqueness of local solutions in the space X 0, if the initial conditions (u 0, u 1 ) X 0. To improve these results, it is assumed that (u 0, u 1 ) X 1. Then, applyin aain Theorem 3.1, it could be proved the existence of a local solution (u, u t ) in X 1. Furthermore, we may obtain I 1 (t) = (2 uu t φ(x)(x)dx) w 2 2( u 2 D(A) )1/2 ( u t 2 L 2 )1/2 w 2 (4.6) 2R k( u t 2 D 1,2)1/2 w 2 2R 2 k w 2 C 2 e (w), where C 2 = 2R 2 k. Also, the followin estimation is valid I 3 (t) I 3 (t) α 1 ( u 2 v 2 ) (u v) w t L 2 α 1 2R 2 w D 1,2 w t L 2 C A ( C u 2C u w 2 D 1,2 + 1 2 w t 2 L 2 ) C A C B e (w), (4.7)
8 P. G. PAPADOPOULOS, N. M. STAVRAKAKIS EJDE-2006/77 where we have used Youn s inequality and C A = 2α 1 R, 2 C B = max( 1 2, 1 2C u ). Hence, ( ) I 2 (t) ( u + v )( (u v) ) vw t dx 2R w D 1,2( v 2 D(A) )1/2 ( w t 2 L 2 )1/2 2R 2 w D 1,2( w t 2 L 2 )1/2 2R ( 2 C u w 2 D 2C + 1 1,2 u 2 w t 2 L ) C ΓC 2 B e (w), where C Γ = 2R. 2 Finally, usin relations (4.6)-(4.8), estimation (4.5) becomes (4.8) d dt e (w) (C 2 + C A C B + C Γ C B )e (w) C e (w), (4.9) where C = C 2 + C A C B + C Γ C B and the proof is completed. Remark 4.3 (Continuity in X 1 ). It is important to state that the operator S(t) : X 0 X 0 associated to the problem (1.1)-(1.2) is weakly continuous in the space X 0, but it is stronly continuous in the space X 1. Therefore, we will study problem (1.1)-(1.2) as a dynamical system in the space X 1 := D 1,2 ( ) L 2 ( ). Remark 4.4 (Uniqueness in X 1 ). Assumin that the initial data are from the space X 1, relation (4.9) uarantees the uniqueness of the solutions for the problem (1.1)-(1.2). Indeed, if û a = (u 0, u 1 ), û b = (u 0, u 1), from inequality (4.9) take S(t)û a S(t)û b X1 C( û a X1, û b X1 ) û a û b X1. (4.10) Remark 4.5. Accordin to Theorem 3.4 we have that the ball B 0 := B X0 (0, R ) is an absorbin set in the space X 0, so and in X 1 by the compact embeddin. So, we obtain the followin theorem. Theorem 4.6. The dynamical system iven by the semiroup (S t ) t 0, possesses an invariant set, which attracts all bounded sets of X 1, denoted by A = t 0 s t S s B 0 X 1. The above set is also compact, so it is lobal attractor for the stron topoloy of X 1. Proof. First, we have that operators (S t ) t 0 form a semiroup on X 1 and that S t : X 1 X 1 is continuous, for all t R (Lemma 4.2). Also, we have that the ball B 0, is an absorbin set in X 1 (Remark 4.5). Our oal is to prove that the functional invariant set A is compact for the stron topoloy of X 1. So, we must show that for a point w 1 A, the sequence S(t j )u j 0 converes stronly to w 1 in X 1. Here, we have that (u j 0 ) j N and (t j ) j N, are two sequences such that (u j 0 ) is bounded in X 1, t j oes to +, as j oes to + and S(t j )u j 0 converes weakly to w 1 in the space X 1, as j oes to + (for more details we refer to [2] and [3]). We fix T > 0 and note that the sequence S(t j T )u j 0 is bounded in X 1 thanks to the existence of an absorbin set in X 1. Hence from this sequence we may extract a subsequence j 1 such that, for some v 1 X 1, S(t j1 T )u j1 0 v 1, as j 1. (4.11) Introducin the notation u j1 (t) := S(t j1 + t T )u j1 0, (4.12)
EJDE-2006/77 STRONG GLOBAL ATTRACTOR 9 we deduce from (4.11) that u j1 (t) S(t)v 1, as j 1, (4.13) since S(t) is weakly continuous on X 1. Usin the enery type estimate (3.12) and the fact that the sequence θ(u j1 (0)) = θ(s(t j1 T )u j1 0 ) is bounded by a constant, let say C, we obtain lim sup θ(s(t j 1 )u j1 0 ) j Ce C T + 1 e C T C(R). (4.14) 1 C δ Applyin the invariance of the set A, for v 1 (t) = S(t)v 1, we et θ(w 1 ) = θ(s(t )v 1 ) e C T θ(v 1 ) + 1 e C T C Subtractin by parts relations (4.14) and (4.15) we et C(R). (4.15) δ lim j 1 sup θ(s(t j 1 )u j1 0 ) θ(w 1) + e C T (C θ(v 1 )). (4.16) Since T is chosen arbitrarily, for T = 0 we have lim j 1 sup θ(s(t j 1 )u j1 0 ) θ(w 1). (4.17) On the other hand, since S(t j1 )u j1 0 converes weakly to w 1 in X 1, we have that lim inf j1 θ(s(t j1 0 ) θ(w 1). So we et lim θ(s(t j)u j 0 ) = θ(w 1). (4.18) j Usin aain the fact that S(t)A = A and that θ(t) is weakly continuous, we obtain lim S(t j)u j 0 j 2 X 1 = w 1 2 X 1. (4.19) Therefore, S(t j )u j 0 converes stronly to w 1 in the space X 1 as j. Thus, we obtain that A is a lobal attractor in the stron topoloy of X 1 (see also [14]). Acknowledments. This work was supported by the Pythaoras project 68/831 from the EPEAEK proram on Basic Research from the Ministry of Education, Hellenic Republic (75% by European Funds and 25% by National Funds). References [1] K. J. Brown and N. M. Stavrakakis; Global Bifurcation Results for a Semilinear Elliptic Equation o n all of, Duke Math. J., 85, (1996), 77-94. [2] J. M. Ghidalia, A Note on the Stron towards Attractors of Damped Forced KdV Equations, J. Differential Equations, 110, (1994), 356-359. [3] J. K. Hale, Asymptotic Behaviour of Dissipative Systems, Mathematical Surveys and Monoraphs 25, AMS, Providence, R.I., 1988. [4] J. K. Hale and N M Stavrakakis, Compact Attractors for Weak Dynamical Systems, Applicable Analysis, Vol 26, (1998), 271-287. [5] N. I. Karachalios and N. M. Stavrakakis, Existence of Global Attractors for semilinear Dissipative Wave Equations on, J. Differential Equations, 157, (1999), 183-205. [6] N. I. Karachalios and N M Stavrakakis; Global Existence and Blow-Up Results for Some Nonlinear Wave Equations on, Adv. Diff. Equations, Vol 6, (2001), 155-174. [7] N. I. Karachalios and N. M. Stavrakakis; Estimates on the Dimension of a Global Attractor for a Semilinear Dissipative Wave Equation on, Discrete and Continuous Dynamical Systems, Vol 8, (2002), 939-951. [8] G. Kirchhoff, Vorlesunen Über Mechanik, Teubner, Leipzi, 1883.
10 P. G. PAPADOPOULOS, N. M. STAVRAKAKIS EJDE-2006/77 [9] K. Ono, On Global Existence, Asymptotic Stability and Blowin - Up of Solutions for some Deenerate Non-Linear Wave Equations of Kirchhoff Type with a Stron Dissipation, Math. Methods Appl. Sci., 20 (1997), 151-177. [10] K. Ono, Global Existence and Decay Properties of Solutions for Some Mildly Deenerate Nonlinear Dissipative Kirchhoff Strins, Funkcial. Ekvac., 40, (1997), 255-270. [11] P. G. Papadopoulos and N. M. Stavrakakis, Global Existence and Blow-Up Results for an Equation of Kirchhoff Type on, Topoloical Methods in Nonlinear Analysis, 17, (2001), 91-109. [12] P. G. Papadopoulos and N. M. Stavrakakis, Central Manifold Theory for the Generalized Equation of Kirchhoff Strins on, Nonlinear Analysis TMA, 61, (2005), 1343-1362. [13] P. G. Papadopoulos and N. M. Stavrakakis, Compact Invariant Sets for Some Quasilinear Nonlocal Kirchhoff Strins on, submitted. [14] R. Temam, Infinite- Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sc., 68, (2nd Edition), Spriner-Verla, 1997. Perikles G. Papadopoulos Department of Mathematics, National Technical University, Zorafou Campus, 157 80 Athens, Greece E-mail address: perispap@math.ntua.r Nikolaos M. Stavrakakis Department of Mathematics, National Technical University, Zorafou Campus, 157 80 Athens, Greece E-mail address: nikolas@central.ntua.r