Electronic Journal of Qualitative Theory of Differential Equations 27, No. 9, 1-24; http://www.math.u-szege.hu/ejqte/ Global solutions an exponential ecay for a nonlinear couple system of beam equations of Kirchhoff type with memory in a omain with moving bounary M. L. Santos an U. R. Soares Departamento e Matemática, Universiae Feeral o Pará Campus Universitario o Guamá, Rua Augusto Corrêa 1, CEP 6675-11, Pará, Brazil. Abstract In this paper we prove the exponential ecay in the case n > 2, as time goes to infinity, of regular solutions for a nonlinear couple system of beam equations of Kirchhoff type with memory an weak amping u tt 2 u M( u 2 L 2 ( t v 2 L 2 ( t u t g 1 (t s u(ss αu t h(u v = in v tt 2 v M( u 2 L 2 ( t v 2 L 2 ( t v t g 2 (t s v(ss αv t h(u v = in ˆQ, ˆQ in a non cylinrical omain of R n1 (n 1 uner suitable hypothesis on the scalar functions M, h, g 1 an g 2, an where α is a positive constant. We show that such issipation is strong enough to prouce uniform rate of ecay. Besies, the coupling is nonlinear which brings up some aitional ifficulties, which plays the problem interesting. We establish existence an uniqueness of regular solutions for any n 1. 2 Mathematics Subject Classification: 35B35, 35L7 Keywors: Moving bounary, global solution, exponential ecay.. 1. INTRODUCTION Let be an open boune omain of R n containing the origin an having C 2 bounary. Let γ : [, [ R be a continuously ifferentiable function. See hypothesis (1.15-(1.17 on γ. Let EJQTDE, 27 No. 9, p. 1
us consier the family of subomains { t } t< of R n given by t = T(, T : y x = γ(ty whose bounaries are enote by Γ t an ˆQ the non cylinrical omain of R n1 with lateral bounary ˆQ = ˆ = t< t< t {t} Γ t {t}. Let us consier the Hilbert space L 2 ( enowe with the inner prouct (u,v = u(xv(xx an corresponing norm u 2 L 2 ( = (u,u. We also consier the Sobolev space H 1 ( enowe with the scalar prouct (u,v H 1 ( = (u,v ( u, v. We efine the subspace of H 1 (, enote by H 1(, as the closure of C ( in the strong topology of H 1 (. By H 1 ( we enote the ual space of H 1 (. This space enowe with the norm inuce by the scalar prouct is, owing to the Poincaré inequality a Hilbert space. We efine for all 1 p < an if p = ((u,v H 1 ( = ( u, v u 2 L 2 ( C u 2 L 2 (, u p L p ( = u(x p x, u L ( = supess u(x. x In this work we stuy the existence of strong solutions as well the exponential ecay of the energy of the nonlinear couple system of beam equations of Kirchhoff type with memory given EJQTDE, 27 No. 9, p. 2
by u tt 2 u M( u 2 L 2 ( t v 2 L 2 ( t u t g 1 (t s u(ss αu t h(u v = in ˆQ, (1.1 v tt 2 v M( u 2 L 2 ( t v 2 L 2 ( t v t g 2 (t s v(ss αv t h(u v = in ˆQ, (1.2 u = v = u ν = v ν = on ˆ, (1.3 (u(x,,v(x, = (u (x,v (x, (u t (x,,v t (x, = (u 1 (x,v 1 (x in, (1.4 where ν = ν(σ,t is the unit normal at (σ,t ˆΣ irecte towars the exterior of ˆQ. If we enote by η the outer normal to the bounary Γ of, we have, using a parametrization of Γ ν(σ,t = 1 r (η(ξ, γ 2 (tξ η(ξ, ξ = σ γ(t (1.5 where r = (1 γ 2 (t ξ η(ξ 2 1 2. In fact, fix (σ,t ˆΣ. Let ϕ = be a parametrization of a part U of Γ, U containing ξ = σ γ(t. The parametrization of a part V of ˆΣ is ψ(σ,t = ϕ( σ γ(t = ϕ(ξ =. We have ψ(σ,t = 1 γ(t ( ϕ(ξ, γ (tξ ϕ(ξ. From this an observing that η(ξ = ϕ(ξ/ ϕ(ξ, the formula (1.5 follows. Let ν(, t be the x-component of the unit normal ν(,, ν 1. Then by the relation (1.5, one has ν(σ,t = η( σ. (1.6 γ(t In this paper we eal with nonlinear couple system of beam equations of Kirchhoff type with memory over a non cylinrical omain. We show the existence an uniqueness of strong solutions to the initial bounary value problem (1.1-(1.4. The metho we use to prove the result of existence an uniqueness is base on the transformation of our problem into another initial bounary value problem efine over a cylinrical omain whose sections are not timeepenent. This is one using a suitable change of variable. Then we show the existence an uniqueness for this new problem. Our existence result on non cylinrical omain will follow using the inverse transformation. That is, using the iffeomorphism τ : ˆQ Q efine by τ : ˆQ Q, (x,t t (y,t = ( x,t (1.7 γ(t EJQTDE, 27 No. 9, p. 3
an τ 1 : Q ˆQ efine by τ 1 (y,t = (x,t = (γ(ty,t. (1.8 Denoting by φ an ϕ the functions φ(y,t = u τ 1 (y,t = u(γ(ty,t, ϕ(y,t = v τ 1 (y,t = v(γ(ty,t (1.9 the initial bounary value problem (1.1-(1.4 becomes φ tt γ 4 2 φ γ 2 M(γ n 2 ( φ 2 L 2 ( ϕ 2 L 2 ( φ t g 1 (t sγ 2 (s φ(ss αφ t A(tφ a 1 t φ a 2 φ h(φ ϕ = in Q, (1.1 ϕ tt γ 4 2 ϕ γ 2 M(γ n 2 ( φ 2 L 2 ( ϕ 2 L 2 ( ϕ t g 2 (t sγ 2 (s ϕ(ss αϕ t A(tϕ a 1 t ϕ a 2 ϕ h(φ ϕ = in Q, (1.11 φ Γ = ϕ Γ = φ ν Γ = ϕ ν Γ =, (1.12 (φ,ϕ t= = (φ,ϕ (φ t,ϕ t t= = (φ 1,ϕ 1 in, (1.13 where an n n A(tφ = yi (a ij yj φ, A(tϕ = yi (a ij yj ϕ i,j=1 i,j=1 a ij (y,t = (γ γ 1 2 y i y j (i,j = 1,...,n, a 1 (y,t = 2γ γ 1 y, a 2 (y,t = γ 2 y(γ γ γ (αγ (n 1γ. (1.14 To show the existence of a strong solution we will use the following hypotheses: γ n > 2, γ if n 2, (1.15 γ L (,, inf γ(t = γ >, (1.16 t< γ W 2, (, W 2,1 (,. (1.17 Note that the assumption (1.15 means that ˆQ is ecreasing if n > 2 an increasing if n 2 in the sense that when t > t an n > 2 then the projection of t on the subspace t = contain the projection of t on the same subspace an contrary in the case n 2. The above metho EJQTDE, 27 No. 9, p. 4
was introuce by Dal Passo an Ughi [21] to stuy certain class of parabolic equations in non cylinrical omain. We assume that h C 1 (R satisfies h(ss, s R. Aitionally, we suppose that h is superlinear, that is with the following growth conitions h(ss H(s, H(z = z h(ss, s R, h(t h(s C(1 t ρ 1 s ρ 1 t s, t,s R, for some C > an ρ 1 such that (n 2ρ n. Concerning the function M C 1 ([, [, we assume that where M(τ = τ M(τ m, M(ττ M(τ, τ, (1.18 M(ss an where λ 1 is the first eigenvalue of the spectral Dirichlet problem < m < λ 1 γ 2 L (1.19 We recall also the classical inequality 2 w = λ 1 w in, w = w = on Γ. ν w L 2 ( λ 1 w L 2 (. (1.2 Unlike the existing papers on stability for hyperbolic equations in non cylinrical omain, we o not use the penalty metho introuce by J. L. Lions [16], but work irectly in our non cylinrical omain ˆQ. To see the issipative properties of the system we have to construct a suitable functional whose erivative is negative an is equivalent to the first orer energy. This functional is obtaine using the multiplicative technique following Komornik [8] or Rivera [2]. We only obtaine the exponential ecay of solution for our problem for the case n > 2. The main ifficulty in obtaining the ecay for n 2 is ue to the geometry of the non cylinrical omain because it affects substantially the problem, since we work irectly in ˆQ. Therefore the case n 2 is an important open problem. From the physics point of view, the system (1.1-(1.4 escribes the transverse eflection of a streche viscoelastic beam fixe in a moving bounary evice. The viscoelasticity property of the material is characterize by the memory terms t g 1 (t s u(ss, t g 2 (t s v(ss. EJQTDE, 27 No. 9, p. 5
The uniform stabilization of plates equations with linear or nonlinear bounary feeback was investigate by several authors, see for example [7, 9, 1, 11, 13, 22] among others. In a fixe omain, it is well-known, the relaxation function g ecays to zero implies that the energy of the system also ecays to zero, see [2, 12, 19, 23]. But in a moving omain the transverse eflection u(x,t an v(x,t of a beam which charges its configuration at each instant of time, increasing its eformation an hence increasing its tension. Moreover, the horizontal movement of the bounary yiels nonlinear terms involving erivatives in the space variable. To control these nonlinearities, we a in the system a frictional amping, characterize by u t an v t. This term will play an important role in the issipative nature of the problem. A quite complete iscussion in the moelling of transverse eflection an transverse vibrations, respectively, of purely for the nonlinear beam equation an elastic membranes can be foun in J. Ferreira et al. [6], J. Límaco et al. [17] an L. A. Meeiros et al. [18]. This moel was propose by Woinowsky [24] for the case of cylinrical omain, without the terms u an t g 1(t s u(ss but with the term M( u 2 u. See also Eisley [5] an Burgreen [1] for physics justification an backgroun of the moel. We use the stanar notations which can be foun in Lion s book [15, 16]. In the sequel by C (sometimes C 1,C 2,... we enote various positive constants which o not epen on t or on the initial ata. This paper is organize as follows. In section 2 we prove a basic result on the existence, regularity an uniqueness of regular solutions. We use Galerkin approximation, Aubin-Lions theorem, energy metho introuce by Lions [16] an some technical ieas to show existence regularity an uniqueness of regular solution for the problem (1.1-(1.4. Finally, in section 3, we establish a result on the exponential ecay of the regular solution to the problem (1.1-(1.4. We use the technique of the multipliers introuce by Komornik [8], Lions [16] an Rivera [2] couple with some technical lemmas an some technical ieas. 2. EXISTENCE AND REGULARITY OF GLOBAL SOLUTIONS In this section we shall stuy the existence an regularity of solutions for the system (1.1-(1.4. For this we assume that the kernels g i : R R is in C 1 (,, an satisfy where g i, g i, γ 2 1 g i (sγ 2 (ss = β i >, i = 1,2, (2.1 γ 1 = sup γ(t. t< EJQTDE, 27 No. 9, p. 6
To simplify our analysis, we efine the binary operator g Φ(t t γ(t = g(t sγ 2 (s Φ(t Φ(s 2 sx. With this notation we have the following lemma. Lemma 2.1 For Φ C 1 (,T : H 2( an g C1 (R we have t g(t sγ 2 (s Φ(s Φ t sx = 1 g(t 2 γ 2 Φ 2 x 1 ( 2 g Φ γ 1 [ g Φ t 2 t γ ( g(s γ 2 (s s an t g(t sγ 2 (s Φ(s Φ t sx = 1 g(t 2 γ 2 Φ 2 x 1 ( 2 g Φ γ 1 [ g Φ t 2 t γ ( g(s γ 2 (s s ] Φ 2 x, ] Φ 2 x. an g Φ(t γ(t. The well- The proof of this lemma follows by ifferentiating the terms g Φ(t γ(t poseness of system (1.1-(1.13 is given by the following theorem. Theorem 2.1 Let us take (φ,ϕ (H 2( H4 ( 2, (φ 1,ϕ 1 (H 2(2 an let us suppose that assumptions (1.15-(1.2 an (2.1 hol. Then there exists a unique solution (φ,ϕ of the problem (1.1-(1.13 satisfying φ,ϕ L (, : H( 2 H 4 (, φ t,ϕ t L (, : H 1 (, φ tt,ϕ tt L (, : L 2 (. Proof. Let us enote by B the operator Bw = 2 w, D(B = H 2 ( H 4 (. It is well know that B is a positive self ajoint operator in the Hilbert space L 2 ( for which there exist sequences {w n } n N an {λ n } n N of eigenfunctions an eigenvalues of B such that the set of linear combinations of {w n } n N is ense in D(B an λ 1 < λ 2... λ n as n. Let us enote by m m φ m = (φ,w j w j, ϕ m = (ϕ,w j w j, j=1 j=1 m φ m 1 = m (φ 1,w j w j ϕ m 1 = (ϕ 1,w j w j. j=1 j=1 EJQTDE, 27 No. 9, p. 7
Note that for any {(φ,φ 1,(ϕ,ϕ 1 } (D(B H 2(2, we have φ m φ strong in D(B, ϕ m ϕ strong in D(B, φ m 1 φ 1 strong in H 2( an ϕm 1 ϕ 1 strong in H 2(. Let us enote by V m the space generate by w 1,...,w m. Stanar results on orinary ifferential equations imply the existence of a local solution (φ m,ϕ m of the form m (φ m (t,ϕ m (t = (g jm (t,f jm (tw j, j=1 to the system φ m tt w jy α φ m t w jy γ 4 2 φ m w j y γ 2 M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( φ m w j y t g 1 (t sγ 2 (s φ m (s w j sy A(tφ m w j y a 1 φ m t w j y a 2 φ m w j y h(φ ϕw j y =, (2.2 (j = 1,...,m, ϕ m tt w jy α ϕ m t w jy γ 4 2 ϕ m w j y γ 2 M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( ϕ m w j y t g 2 (t sγ 2 (s ϕ m (s w j sy A(tϕ m w j y a 1 ϕ m t w jy a 2 ϕ m w j y h(φ ϕw j y =, (2.3 (j = 1,...,m, (φ m (y,,ϕ m (y, = (φ m,ϕ m, (φ m t (y,,ϕ m (y, = (φ m 1,ϕ m 1. (2.4 The extension of the solution to the whole interval [, is a consequence of the first estimate which we are going to prove below. A Priori estimate I Multiplying the equations (2.2 by g jm (t an (2.3 by f jm (t, summing up the prouct result EJQTDE, 27 No. 9, p. 8
in j = 1,2,...,m an using the Lemma 2.1 we get 1 2 t m 1 (t,φm,ϕ m α φ m t 2 L 2 ( A(tφ m φ m t y a 1 φ m t φm t y a 2 φ m φ m t y α ϕ m t 2 L 2 ( A(tϕ m ϕ m t y a 1 ϕ m t ϕ m t y (n [ 2γ 2γ n1 γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( M(γn 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( ] M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( = 1 g 1 (t 2 γ 2 ( φm 2 L 2 ( 1 2 g 1 φm γ 1 g 2 (t 2 γ 2 ( ϕm 2 L 2 ( 1 2 g 2 ϕm γ 4 γ γ 5 φm 2 L 2 ( 4 γ γ 5 ϕm 2 L 2 (, where m 1 (t,φ m,ϕ m = φ m t 2 L 2 ( ( 1 γ 2 (t t g 1 (sγ 2 (ss φ m 2 L 2 ( γ n M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( γ 4 φ m 2 L 2 ( g 1 φm ϕ m t 2 L γ 2 ( ( 1 t γ 2 (t g 2 (sγ 2 (ss ϕ m 2 L 2 ( γ 4 ϕ m 2 L 2 ( g 2 ϕm 2 H(φ ϕy. γ From (1.16-(1.17 an (1.19-(1.2 it follows that γ n M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( γ 4 ( φ m 2 L 2 ( ϕm 2 L 2 ( m 1 γ 2 ( φ m 2 L 2 ( ϕm 2 L 2 (. (2.5 where m 1 = ( λ 1 m γ 2. Taking into account (1.16, (1.17, (2.1 an the last equality we obtain 1 2 t m 1 (t,φm,ϕ m α φ m t 2 L 2 ( α ϕm t 2 L 2 ( C( γ γ m 1 (t,φm,ϕ m. (2.6 Integrating the inequality (2.6, using Gronwall s Lemma an taking account (1.17 we get m 1 (t,φ m,ϕ m A priori estimate II t φ m s (s 2 L 2 ( ϕm s (s 2 L 2 ( s C, m N, t [,T]. (2.7 Now, if we multiply the equations (2.2 by λ j g jm an (2.3 by λ j f jm an summing up in EJQTDE, 27 No. 9, p. 9
j = 1,...,m we get 1 2 t φm t 2 L 2 ( α φm t 2 L 2 ( γ 4 2 t φm 2 L 2 ( γ 2 2 M(γn 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( t φm 2 L 2 ( t g 1 (t sγ 2 (s φ m (s φ m t sy A(tφ m φ m t y a 1 φ m t φ m t y a 2 φ m φ m t y h(φ m ϕ m φ m t y =, 1 2 t ϕm t 2 L 2 ( α ϕm t 2 L 2 ( γ 4 2 t ϕm 2 L 2 ( γ 2 2 M(γn 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( t ϕm 2 L 2 ( t g 2 (t sγ 2 (s ϕ m (s ϕ m t sy A(tϕ m ϕ m t y a 1 ϕ m t ϕm t y a 2 ϕ m ϕ m t y h(φ m ϕ m ϕ m t y =. Summing the last two equalities an using the lemma 2.1 we obtain where 1 2 t m 2 (t α( φm t 2 L 2 ( ϕm t 2 L 2 ( = 1 g 1 (t 2 γ 2 ( φm 2 L 2 ( 1 2 g 1 φm 2γ γ γ 5 ( φm 2 L 2 ( ϕm 2 L 2 ( (γ 2 M(γ n 2 (( φ m 2 L t 2 ( ϕm 2 L 2 ( ( φ m 2 L 2 ( ϕm 2 L 2 ( A(tφ m φ m t y a 1 φ m t φ m t y a 2 φ m φ m t y 1 g 2 (t 2 γ 2 ( ϕm 2 L 2 ( 1 2 g 2 ϕm A(tϕ m ϕ m t γ y a 1 ϕ m t ϕm t y a 2 ϕ m ϕ m t y h(φ m ϕ m φ m t y h(φ m ϕ m ϕ m t y m 2 (t = φm t 2 L 2 ( ( 1 γ 2 (t t g 1 (sγ 2 (ss φ m 2 L 2 ( γ 4 ( φ m 2 L 2 ( ϕm 2 L 2 ( g 1 φm ϕ m t γ 2 L 2 ( γ 2 M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( ( φm 2 L 2 ( ϕm 2 L 2 ( ( 1 t γ 2 (t g 2 (sγ 2 (ss ϕ m 2 L 2 ( g 2 ϕm γ. EJQTDE, 27 No. 9, p. 1
From (1.16-(1.17 an (1.19-(1.2, we have γ 4 ( φ m 2 L 2 ( ϕm 2 L 2 ( γ 2 M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( ( φm 2 L 2 ( ϕm 2 L 2 ( m 1 γ 2 ( φ m 2 L 2 ( ϕm 2 L 2 (. Using similar arguments as (2.7, the hypothesis for the function h an observing the above inequality we obtain m 2 (t t A priori estimate III ( φ m s 2 L 2 ( ϕm s 2 L 2 (s C, m N, t [,T]. (2.8 It easy to see from (2.2-(2.3 an of the growth hypothesis for the function h together with the Sobolev s imbeing that φ m tt ( 2 L 2 ( ϕm tt ( 2 L 2 ( C, m N. (2.9 Differentiating the equations (2.2-(2.3 with respect to the time, we obtain φ m tttw j y α φ m tt w j y γ 4 2 φ m t w j y t (γ 2 M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( φm w j y 4γ γ 5 2 φ m w j y g ( γ 2 φ m ( w jy t g 1(t sγ 2 (s φ m (s w j sy t (A(tφm w j y t (a 1 φ m t w jy h (φ m ϕ m (φ m t ϕ m t w jy t (a 2 φ m w j y = (2.1 ϕ m tttw j y α ϕ m tt w j y γ 4 2 ϕ m t w j y t (γ 2 M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( ϕm w j y 4γ γ 5 2 ϕ m w j y g 2( γ 2 ϕ m ( w jy t g (t sγ 2 (s ϕ m (s w j sy t (A(tϕm w j y t (a 1 ϕ m t w j y h (φ m ϕ m (φ m t ϕ m t w jy t (a 2 ϕ m w j y =. (2.11 EJQTDE, 27 No. 9, p. 11
Multiplying the equations (2.1-(2.11 by g jm (t an f jm (t, respectively, an summing up the prouct result in j = 1,2,...,m we have 1 φ m tt 2 y α φ m tt 2 y γ 4 φ m t 2 y 2 t 2 t t (γ 2 M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( φm φ m tt y 4γ γ 5 2 φ m φ m tt y g 1( γ 2 φ m φ m tt y ( t g 1 (t sγ 2 (s φ m (s φ m tt sy t (A(tφm φ m tt y t (a 1 φ m t φ m tt y h (φ m ϕ m (φ m t ϕ m t φ m tt y t (a 2 φ m φ m tt y = (2.12 1 ϕ m tt 2 y α ϕ m tt 2 y γ 4 ϕ m t 2 y 2 t 2 t t (γ 2 M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( ϕm ϕ m tt y 4γ γ 5 2 ϕ m ϕ m tt y g 2( γ 2 ϕ m ( ϕm tt y t g (t sγ 2 (s ϕ m (s ϕ m tt sy t (A(tϕm ϕ m tt y t (a 1 ϕ m t ϕm tt y h (φ m ϕ m (φ m t ϕ m t ϕm tt y t (a 2 ϕ m ϕ m tt y =. (2.13 Let us take p n = 2n n 2. From the growth conition of the function h an from the Sobolev imbeing we obtain h (φ m ϕ m φ m t φ m tt y C (1 2 φ m ϕ m ρ 1 φ m t φ m tt y [ C [ C ] 1 (1 2 φ m ϕ m ρ 1 n y (1 (φ m ϕ m 2 y ] ρ 1 2 [ n φ m t pn y Taking into account the estimate (2.7, we conclue that h (φ m ϕ m φ m t φ m tt y [ φ m t 2 y ] 1 [ pn ]1 φ m tt 2 y [ 2 φ m tt 2 y ]1 2 ]1 2. [ ]1 [ ]1 C φ m t 2 2 y φ m tt 2 2 y { } C φ m t 2 y φ m tt 2 y. (2.14 EJQTDE, 27 No. 9, p. 12
Similarly we get { h (φ m ϕ m ϕ m t φm tt y C ϕ m t 2 y { h (φ m ϕ m φ m t ϕ m tt y C φ m t 2 y { h (φ m ϕ m ϕ m t ϕm tt y C ϕ m t 2 y } φ m tt 2 y, (2.15 } ϕ m tt 2 y, (2.16 } ϕ m tt 2 y. (2.17 Summing the equations (2.12 an (2.13, substituting the inequalities (2.14-(2.17 an using similar arguments as (2.7-(2.8 we obtain, after some calculations an taking into account the lemma 2.1 an hypothesis on M where 1 2 t m 3 (t α( φ m tt (t 2 L 2 ( ϕm tt (t 2 L 2 ( C( γ γ ( φ m 2 L 2 ( φm t 2 L 2 ( C( γ γ ( ϕ m 2 L 2 ( ϕm t 2 L 2 ( C( φ m t 2 L 2 ( ϕm t 2 L 2 ( C( γ γ m 3 (t, m 3 (t = φm tt 2 L 2 ( ϕm tt 2 L 2 ( g 1 φ t γ g 2 ϕ t γ γ 2 M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( ( φm t 2 L 2 ( ϕm t 2 L 2 ( ( t t g 1 (sγ 2 (ss φ t 2 L 2 ( ( g 2 (sγ 2 (ss ϕ t 2 L 2 (. Using Gronwall s lemma an relations (1.17, (2.7-(2.8 we get m 3 (t α t A priori estimate IV ( φ m ss(s 2 L 2 ( ϕm ss(s 2 L 2 (s C, t [,T], m N. (2.18 Multiplying the equations (2.2-(2.3 by g jm (t an f jm (t, respectively, summing up the prouct result in j = 1,2,...,m an using the hypothesis on h we euce 1 2 t m 4 (t φ m t 2 L 2 ( ϕm t 2 L 2 ( γ 4 ( φ 2 L 2 ( ϕ 2 L 2 ( γ 2 M(γ n 2 ( φ m 2 L 2 ( ϕ 2 L 2 ( ( φm 2 2 ϕ 2 L 2 ( t g 1 (sγ 2 (s φ m (sφ m (tsy t g 2 (sγ 2 (s ϕ m (sϕ m (tsy C( γ γ ( φ m 2 L 2 ( ϕm 2 L 2 ( φm 2 L 2 ( ϕ m 2 L 2 ( φm t 2 L 2 ( ϕm t 2 L 2 (, (2.19 EJQTDE, 27 No. 9, p. 13
where m 4 (t = 2 (φ m φ m t ϕ m ϕ m t y α( φm 2 L 2 ( ϕm 2 L 2 (. Choosing k > 2 α, we obtain k m 1 (t m 4 (t (k 2 α ( φm t 2 L 2 ( ϕm t 2 L 2 ( φm 2 L 2 ( ϕm 2 L 2 ( (k 2 α γ 4 ( φ m 2 L 2 ( ϕm 2 L 2 ( γ n M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( >. (2.2 Now, multiplying the equations (2.2 an (2.3 by kg jm (t an kf jm (t, respectively, summing up the prouct result an combining with (2.19, we get, taking into account (2.5 1 2 t (k m 1 (t m 4 (t (kα 1( φm t 2 L 2 ( ϕm t 2 L 2 ( (γ 4 ( φ m 2 L 2 ( ϕm 2 L 2 ( C( γ γ (k m 1 (t m 4 (t. (2.21 From (2.21, using the Gronwall s Lemma we obtain the following estimate, taking into account (2.5 an (1.17 t k m 1 (t m 4 (t ( φ m s (s 2 L 2 ( ϕm s (s 2 L 2 ( φm (s 2 L 2 ( ϕ m (s 2 L 2 ( s C( φ 1 2 L 2 ( ϕ 1 2 L 2 ( φ 2 L 2 ( ϕ 2 L 2 (.(2.22 From estimates (2.7, (2.8, (2.18 an (2.22 it s follows that (φ m,ϕ m converge strong to (φ,ϕ L 2 (, : H 2 (. Moreover, since M C1 [, [ an φ m, ϕ m are boune in L (, : L 2 ( L 2 (, : L 2 (, we have for any t > t C M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( M(γn 2 ( φ 2 L 2 ( ϕ 2 L 2 ( t ( φ m φ 2 H 1 ( ϕm ϕ 2 H 1 (, where C is a positive constant inepenent of m an t, so that M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( ( φm,w j M(γ n 2 ( φ 2 L 2 ( ϕ 2 L 2 ( ( φ,w j an M(γ n 2 ( φ m 2 L 2 ( ϕm 2 L 2 ( ( ϕm,w j M(γ n 2 ( φ 2 L 2 ( ϕ 2 L 2 ( ( ϕ,w j. Using similar arguments as above we conclue that h(φ m ϕ m h(φ ϕ in L 2 (,T;L 2 (. EJQTDE, 27 No. 9, p. 14
Letting m in the equations (2.2-(2.3 we conclue that (φ, ϕ satisfies (1.1-(1.11 in L (, : L 2 (. Therefore we have that φ,ϕ L (, : H 2 ( H4 (, φ t,ϕ t L (, : H 1 (, φ tt,ϕ tt L (, : L 2 (. To prove the uniqueness of solutions of problem (1.1, (1.11, (1.12 an (1.13 we use the metho of the energy introuce by Lions [16], couple with Gronwall s inequality an the hypotheses introuce in the paper about the functions g i, h,m an the obtaine estimates. To show the existence in non cylinrical omain, we return to our original problem in the non cylinrical omain by using the change of variable given in (1.8 by (y,t = τ(x,t, (x,t ˆQ. Let (φ,ϕ be the solution obtaine from Theorem 2.1 an (u,v efine by (1.9, then (u,v belong to the class u,v L (, : H( 2 t H 4 ( t, (2.23 u t,v t L (, : H 1 ( t, (2.24 u tt,v tt L (, : L 2 ( t. (2.25 Denoting by u(x,t = φ(y,t = (φ τ(x,t, v(x,t = ϕ(y,t = (ϕ τ(x,t then from (1.9 it is easy to see that (u,v satisfies the equations (1.1-(1.2 in L (, : L 2 ( t. If (u 1,v 1, (u 2,v 2 are two solutions obtaine through the iffeomorphism τ given by (1.7, then (φ 1,ϕ 1, (φ 2,ϕ 2 are the solution to (1.1-(1.11. By uniqueness result of Theorem 2.1, we have (φ 1,ϕ 1 = (φ 2,ϕ 2, so (u 1,v 1 = (u 2,v 2. Therefore, we have the following result. Theorem 2.2 Let us take (u,v (H 2( H 4 ( 2, (u 1,v 1 (H 2( 2 an let us suppose that assumptions (1.15-(1.17, (1.19-(1.2 an (2.1 hol. Then there exists a unique solution (u, v of the problem (1.1-(1.4 satisfying (2.23-(2.25 an the equations (1.1-(1.2 in L (, : L 2 ( t. EJQTDE, 27 No. 9, p. 15
3. EXPONENTIAL DECAY In this section we show that the solution of system (1.1-(1.4 ecays exponentially. To this en we will assume that the memory g i satisfies: ( 1 g i (t C 1g i (t (3.1 g i (ss = β i >, i = 1,2 (3.2 for all t, with positive constant C 1. Aitionally, we assume that the function γ( satisfies the conitions γ, t, n > 2, (3.3 < max t< γ (t 1, (3.4 where = iam(. The conition (3.4 (see also (1.5 imply that our omain is time like in the sense that ν < ν where ν an ν enote the t-component an x-component of the outer unit normal of ˆ. Remark: It is important to observe that to prove the main Theorem of this section, that is, Theorem 3.1 as well the Lemmas 3.4 an 3.5 we use the following substantial hypothesis: M(s m >, s [, [. (3.5 This because we worke irectly in our omain with moving bounary, where the geometry of our omain influence irectly in the problem, what generate several technical ifficulties in limiting some terms in Lemma 3.5 an consequent to prove Theorem 3.1. To facilitate our calculations we introuce the following notation (g u(t = t t g(t s u(t u(s 2 sx. First of all we will prove the following three lemmas that will be use in the sequel. Lemma 3.1 Let F(, be the smooth function efine in t [, [. Then γ F(x,tx = F(x,tx F(x,t(x νγ t, (3.6 t t t t γ Γ t where ν is the x-component of the unit normal exterior ν. EJQTDE, 27 No. 9, p. 16
Proof. We have by a change variable x = γ(ty, y F(x,tx = F(γ(ty,tγ n (ty t t t = ( F t (γ(ty,tγn (ty n γ i=1 γ x i( F t (γ(ty,tγn (ty n γ (tγ n 1 (tf(γ(ty,ty. If we return at the variable x, we get F γ F(x,tx = (x,tx t t t t γ t x F(x,tx n γ γ t F(x,tx. Integrating by parts in the last equality we obtain the formula (3.6. Lemma 3.2 Let v H 2 ( H 1 (. Then for all i = 1,...,n we have v y i = η i v η (3.7 Proof. We consier r C 2 (, R n such that r = ν on Γ. (3.8 (It is possible to choose such a fiel r( if we consier that the bounary Γ is sufficiently smooth. Let θ D(Γ an ϕ H m ( with m > max( n 2,2 such that ϕ Γ = θ. Since D(Γ H m 1 2(Γ, such function ϕ exists an we have 2 (vr j ϕy = y i y j Note that is regular, we also obtain 2 (vr j ϕy = y j y i It follows that which implies (3.7. Γ Γ v η j (vr j ϕγ = θη i y j Γ η Γ η j (vr j ϕγ = Γ y i = θ v Γ. y i θ v v Γ = θ(η i Γ y i Γ y i Γ θ D(Γ (i,j = 1,...,n. θηj 2 Γ v y i Γ EJQTDE, 27 No. 9, p. 17
From the Lemma 3.2 it is easy to see that u ν = u ν on Γ t, (3.9 an for u H 2 ( t H 4 ( t (see Komornik [8] page 26 we have u =, 2 u u ν2 = u ν ivν = u on Γ t. (3.1 Lemma 3.3 For any function g C 1 (R an u C 1 (, : H 2 ( t H 4 ( t we have that t t g(t s u(s u t sx = 1 2 g(t t u(t 2 x 1 2 g u 1 2 t [ ( t ] g u g(ss u 2. t Proof. Differentiating the term g u an applying the lemma 3.1 we obtain Using (3.1 we have t g u = t g(t s u(t u(s 2 sx t t t γ g(t s u(t u(s 2 (x νsγ t. γ Γ t t t g u = g (t s u(t u(s 2 sx t t ( t 2 g(t s u t (t u(ssx t from where it follows that t 2 g(t s u t (t u(ssx = t t {g u t t t g(t ss t g (t s u(t u(s 2 sx g(t u(t 2 x. t t u(t 2 x g(t ss u(t 2 x} t The proof is now complete. Let us introuce the functional of energy E(t = u t 2 L 2 ( t u 2 L 2 ( t (1 v t 2 L 2 ( t v 2 L 2 ( t (1 t t g 1 (ss u 2 L 2 ( g t 1 u M( u 2 L 2 ( t v 2 L 2 ( t 2 t H(u vx. g 2 (ss v 2 L 2 ( g t 2 v EJQTDE, 27 No. 9, p. 18
Lemma 3.4 Let us take (u,v (H 2 ( H 4 ( 2, (u 1,v 1 (H 2 ( 2 an let us suppose that assumptions (1.15, (1.16, (1.17, (1.19, (1.2, (2.1 an (3.5 hol. Then any strong solution of system (1.1-(1.4 satisfies t E(t 2α( u t 2 L 2 ( v t t 2 L 2 ( t Γt γ Γt γ Γt γ γ (ν x( u t 2 u 2 Γ t γ (ν x( v t 2 v 2 Γ t 2 γ (ν xh(u vγ t = g 1 (t u 2 x g 1 u g 2 (t v 2 x g 2 v. t t Proof. Multiplying the equation (1.1 by u t, performing an integration by parts over t an using the lemmas 3.1, 3.2 an 3.3 we obtain 1 2 t u t 2 L 2 ( t M( u 2 L 2 ( v 2 L 2 ( t t u 2 L 2 ( t 1 2 t u 2 L 2 ( Γt γ 2γ (ν x( u t 2 u 2 Γ t α u t 2 L 2 ( 1 t 2 g 1(t u 2 x 1 t 2 g 1 u 1 [ ( t ] g 1 u g 1 (ss u 2 x 2 t t h(u vu t x =. t Similarly, using equation (1.2 instea (1.1 we get 1 2 t v t 2 L 2 ( t M( u 2 L 2 ( v 2 L 2 ( 1 t 2 t v 2 L 2 ( t 1 2 t v 2 L 2 ( Γt γ 2γ (ν x( v t 2 v 2 Γ t α v t 2 L 2 ( 1 t 2 g 2(t v 2 x 1 t 2 g 2 v 1 [ ( t ] g 2 v g 2 (ss v 2 x 2 t t h(u vv t x =. t Summing these two last equalities, taking into account the lemma 3.1 an hypothesis on h we obtain the conclusion of lemma. Let us consier the follow functional ψ(t = 2 u t ux α u 2 L 2 ( 2 t v t vx α v 2 L 2 (. t t t EJQTDE, 27 No. 9, p. 19
Lemma 3.5 Let us take (u,v (H 2 ( H 4 ( 2, (u 1,v 1 (H 2 ( 2 an let us suppose that assumptions (1.15, (1.16, (1.17, (1.19, (1.2, (2.1 an (3.5 hol. Then any strong solution of system (1.1-(1.4 satisfies 1 2 t ψ(t u t 2 L 2 ( v t t 2 L 2 ( t u 2 L 2 ( t v 2 L 2 ( t ( t 1 u 2 L 2 ( u 2 t L 2 ( t g 1 (ss (g1 u 1 2 v 2 L 2 ( t v L 2 ( t ( t 1 2 g 2 (ss (g2 v 1 2. Proof. Multiplying the equations (1.1 by u an (1.2 by v, integrating by parts over t an summing up the results we obtain Noting that t t t t 1 2 t ψ(t = u t 2 L 2 ( v t t 2 L 2 ( t u 2 L 2 ( t v 2 L 2 ( t M( u 2 L 2 ( v 2 L 2 ( t u 2 L 2 ( t t g 1 (t s u(s u(tsx t g 1 (t s u(s u(tsx = g 2 (t s v(s v(tsx = M( u 2 L 2 ( v 2 L 2 ( t v 2 L 2 ( t t g 2 (t s v(s v(tsx t h(u v(u vx. t t t t t t t g 1 (t s( u(s u(t usx ( t g 1 (ss u 2 x, g 2 (t s( v(s v(t vsx ( t g 2 (ss v 2 x, an taking into account that t ( t 1 g 1 (t s( u(s u(t ux u 2 L 2 ( t g 1 (ss (g1 u 1 2, t t ( t 1 g 2 (t s( v(s v(t vx v 2 L 2 ( t g 2 (ss (g2 v 1 2, t M( u 2 L 2 ( v 2 L 2 ( t ( u 2 L 2 ( v 2 L 2 ( t M( u 2 L 2 ( v 2 L 2 (, t h(u v(u vx H(u vx t t EJQTDE, 27 No. 9, p. 2
follows the conclusion of lemma. Let us introuce the functional L(t = NE(t ψ(t, (3.11 with N >. It is not ifficult to see that L(t verifies k E(t L(t k 1 E(t, (3.12 for k an k 1 positive constants. Now we are in a position to show the main result of this paper. Theorem 3.1 Let us take (u,v (H 2 ( H 4 ( 2, (u 1,v 1 (H 2 ( 2 an let us suppose that assumptions (1.15, (1.16, (1.17, (1.19, (1.2, (2.1, (3.1, (3.2, (3.4 an (3.5 hol. Then any strong solution of system (1.1-(1.4 satisfies where C an ζ are positive constants. E(t Ce ζt E(, t Proof. Using the lemmas (3.4 an (3.5 we get t L(t 2Nα u t 2 L 2 ( t C 1Ng 1 u u t 2 L 2 ( t u 2 L 2 ( M( u 2 t L 2 ( v 2 L 2 ( t ( t g 1 (ss u 2 L 2 ( t u 2 L 2 ( t ( t 1 2 g 1 (ss (g1 u 1 2 2Nα v t 2 L 2 ( C t 1Ng 2 v v t 2 L 2 ( t ( t v 2 L 2 ( t g 2 (ss v 2 L 2 ( t ( t 1 v 2 2 L 2 ( t g 2 (ss (g2 v 1 2 H(u vx. t EJQTDE, 27 No. 9, p. 21
Using Young inequality we obtain for ɛ > t L(t 2Nα u t 2 L 2 ( t C 1Ng 1 u u t 2 L 2 ( t Choosing N large enough an ɛ small we obtain u 2 L 2 ( M( u 2 t L 2 ( v 2 L 2 ( t ( t g 1 (ss u 2 L 2 ( t ɛ 2 u 2 L 2 ( g 1 L 1 (, t g 1 u 2ɛ 2Nα v t 2 L 2 ( C t 1Ng 2 v v t 2 L 2 ( t ( t v 2 L 2 ( t g 2 (ss v 2 L 2 ( t ɛ 2 v 2 L 2 ( g 2 L 1 (, t g 2 v 2ɛ H(u vx. t t L(t λ E(t (3.13 where λ is a positive constant inepenent of t. From (3.12 an (3.13 follows that L(t L(e λ k 1 t, t. From equivalence relation (3.12 our conclusion follows. The proof now is complete. Acknowlegement: The authors thank the referee of this paper for his valuable suggestions which improve this paper. References [1] D. Burgreen, Free vibrations of a pinene column with constant istance between pinenes. J. Appl. Mech., 1951; 18:135-139. [2] M. M. Cavalcanti, Existence an uniform ecay for the Euler-Bernoulli viscoelastic equation with nonlocal bounary issipative. Discrete an Contin. Dynam. System, 22; 8(3:675-695. [3] C. M. Dafermos & J. A. Nohel, Energy methos for nonlinear hyperbolic Volterra integroifferential equations. Comm. Partial Differential Equations, 1979; 4:218-278. [4] C. M. Dafermos & J. A. Nohel, A nonlinear for nonlinear hyperbolic Volterra Equation in viscoelasticity. Amer. J. Math. Supplement, 1981; 87-116. EJQTDE, 27 No. 9, p. 22
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[18] L. A. Meeiros, J. Lìmaco & S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, Part two. J. of Comp. Analysis an Appl., 22, 4(3:211-263. [19] J. E. Muñoz Rivera, E.C. Lapa & R. Barreto, Decay Rates for Viscoelastic Plates with Memory. Journal of Elasticity, 1996; 44:61-87. [2] J.E. Muñoz Rivera, Energy ecay rates in linear thermoelasticity. Funkc. Ekvacioj, 1992; 35(1:19-3. [21] R. D. Passo & M. Ughi, Problèmes e Dirichlet pour une classe équations paraboliques non linéaires ans es ouverts non cylinriques. C. R. Aca. Sc. Paris, 1989; 38:555-558. [22] V. Komornik, On the nonlinear bounary stabilization of Kirchhoff plates. Nonlinear Differential Equation an Appl., 1994; 1:323-337. [23] M. L. Santos & F. M. O. Junior, A bounary conition with memory for Kirchhoff plates equations. Applie Mathematics an Computation, 24; 148, 475-496. [24] S. K. Woinowsky, The effect of axial force on the vibration of hinge bars. Appl. Mech., 195; 17:35-36. (Receive October 4, 26 EJQTDE, 27 No. 9, p. 24