Eletroni Journal of Differential Equations, Vol. 2(2), No. 73, pp.. ISSN: 72-669. URL: http://eje.math.swt.eu or http://eje.math.unt.eu ftp eje.math.swt.eu (login: ftp) Asymptoti behavior of solutions to wave equations with a memory onition at the bounary Mauro e Lima Santos Abstrat In this paper, we stuy the stability of solutions for wave equations whose bounary onition inlues a integral that represents the memory effet. We show that the issipation is strong enough to proue exponential eay of the solution, provie the relaxation funtion also eays exponentially. When the relaxation funtion eays polynomially, we show that the solution eays polynomially an with the same rate. Introution The main purpose of this work is stuy the asymptoti behavior of solution of the wave equation with a bounary onition of memory type. For this, we onsier the following initial bounary-value problem u tt µ(t)u xx = in (, ) (, ), (.) u(, t) =, u(, t) + g(t s)µ(s)u x(, s)s =, t > (.2) u(x, ) = u (x), u t (x, ) = u (x) in (, ). (.3) The integral in (.2) is a bounary onition whih inlues the memory effet. Here, by u we enote the isplaement an by g the relaxation funtion. By µ = µ(t) we represent a funtion of W, lo (, : R), suh that µ(t) µ > an µ (t) for all t. We refer to [4] for the physial motivation of this moel. Fritional issipative bounary onition for the wave equation was stuie by several authors, see for example [4, 5, 8, 9,,, 2, 5, 6] among others. In these works existene of solutions an exponential stabilization were prove for linear an for nonlinear equations. In ontrast with the large literature for fritional issipative, for bounary onition with memory, we have only a few works as for example [2, 3, 7, 3, 4]. Let us explain briefly eah of the Mathematis Subjet Classifiations: 39A. Key wors: Wave equations, asymptoti behavior. 2 Southwest Texas State University. Submitte September 2, 2. Publishe November 26, 2. Supporte by grant 56/-4 from the CNPq/LNCC Brazil
2 Asymptoti behavior of solutions EJDE 2/73 above works. In [2] Ciarletta establishe theorems of existene, uniqueness an asymptoti stability for a linear moel of heat onution. In this ase the memory onition esribes a bounary that an absorb heat an ue to the hereitary term, an retain part of it. In [3] Fabrizio & Morro onsiere a linear eletromagneti moel with bounary onition of memory type an prove the existene, uniqueness an asymptoti stability of solutions. While in [3] showe the existene of global smooth solution for the one imensional nonlinear wave equation, provie the initial ata (u, u ) is small in the H 3 H 2 -norm, moreover he showe that the solution tens to zero as time goes to infinity. In all the above works was left open the rate of eay. Rivera & Anrae [7] onsier a nonlinear one imensional wave equation with a visoelasti bounary onition. They prove the existene, uniqueness of global smooth solution, provie the initial ata (u, u ) is small in the H 2 H -norm an also that the solution eays uniformly in time (exponentially an algebraially). Finally, in [4] Qin prove a blow up result for the nonlinear one imensional wave equation with memory bounary onition. Our main result is to show that the solution of system (.)-(.3) eays uniformly in time, with rates epening on the rate of eay of the relaxation funtion. More preisely, enoting by k the resolvent kernel of g (the erivative of the relaxation funtion) we show that the solution eays exponentially to zero provie k eays exponentially to zero. When k eays polynomially, we show that the orresponing solution also eays polynomially to zero with the same rate of eay. The metho use here is base on the onstrution of a suitable Lyapunov funtional L satisfying t L(t) L(t) + 2 e γt or t L(t) L(t) + α + 2 ( + t) α+ for some positive onstants, 2, α an γ. To stuy the existene of solution of (.)-(.3), we introue the spae V := {v H (, ); v() = }. The notation use in this paper is stanar an an be foun in Lions s book [6]. In the sequel by (sometime, 2,...) we enote various positive onstants inepenent of t an on the initial ata. The organization of this paper is as follows. In setion 2 we establish a existene an regularity result. In setion 3 prove the uniform rate of exponential eay. Finally in setion 4 we prove the uniform rate of polynomial eay. 2 Existene an Regularity In this setion we shall stuy the existene an regularity of solutions for the (.)-(.3). We assume that the kernel g is positive an k satisfies: < k(t) b e γt, b k(t) k (t) b 2 k(t), (2.) b 3 k (t) k (t) b 4 k (t)
EJDE 2/73 Mauro L. Santos 3 for some positive onstants b i, i =,,..., 4, an γ. To failitate our analysis, we introue the following binary operators: (f ϕ)(t) = (f ϕ)(t) = f(t s) ϕ(t) ϕ(s) 2 s, f(t s)ϕ(s)s, where is the onvolution prout. Volterra integral equation Differentiating (.2) we arrive to the µ(t)u x (, t) + g() g µ(t)u x (, t) = g() u t(, t). Using the Volterra inverse operator, we obtain With τ = g() µ(t)u x (, t) = g() {u t(, t) + k u t (, t)}. an using the above ientity, we write µ(t)u x (, t) = τ{u t (, t) + k()u(, t) k(t)u () + k u(, t)}. (2.2) The following lemma state an important property of the onvolution operator. Lemma 2. For f, ϕ C ([, [: R) we have f(t s)ϕ(s)sϕ t = 2 f(t) ϕ(t) 2 + 2 f ϕ 2 t [ f ϕ ( f(s)s) ϕ 2 ]. The proof of this lemma follows by ifferentiating the term f ϕ. We summarize the well-poseness of (.)-(.3) in the following theorem. Theorem 2.2 Let (u, u ) V L 2 (, ), then there exists only one solution to the (.)-(.3) satisfying u C([, T ] : V ) C ([, T ] : L 2 (, )). Moreover, if (u, u ) H 2 (, ) V V satisfies the ompatibility onition then u C([, T ] : H 2 (, ) V ) C ([, T ] : V ). µ()u,x () = τu (), (2.3) This theorem an be showe using the stanar Galerkin metho, for this reason we omit it here.
4 Asymptoti behavior of solutions EJDE 2/73 3 Exponential Deay In this setion we show that the solution of (.)-(.3) eays exponentially. Our point of eparture will be to establish some inequalities for the solution of system (.)-(.3). For this en, we introue the funtional F (t) = 2 u t 2 + µ(t) u x 2 x + τ 2 (k(t) u(, t) 2 k (t) u(, t)). Lemma 3. For a strong solution of the system (.)-(.3), t F (t) τ 2 u t(, t) 2 + τ 2 k2 (t) u () 2 + τ 2 k (t) u(, t) 2 τ 2 k (t) u(, t) + 2 µ u x 2 x. Proof. Multiplying (.) by u t an integrating over [, ] we obtain 2 t u t 2 + µ(t) u x 2 x = µ(t)u x (, t)u t (, t) + 2 µ u x 2 x. (3.) Using (2.2) an Lemma 2. the onlusion follows. Q.E.D. The following Lemma plays an important role in the onstrution of the Lyapunov funtional. Let us efine the funtional ψ(t) = xu x u t x. Lemma 3.2 The strong solution of (.)-(.3) satisfies t ψ(t) 2 ( u t 2 + µ(t) u x 2 )x + u t (, t) 2 + k 2 (t) u () 2 +k()k(t) u(, t) 2 + k() k u(, t). Proof. From (.) it follows that t ψ(t) = Note that = 2 = 2 xu xt u t x + x x u t 2 x + 2 xu x u tt x xµ(t) x u x 2 x (3.2) ( u t 2 + µ(t) u x 2 )x + 2 u t(, t) 2 + 2 µ(t) u x(, t) 2. k()u(, t) k u(, t)
EJDE 2/73 Mauro L. Santos 5 = k (t s)[u(, s) u(, t)]s k(t)u(, t) ( /2[ k k (s) s) u(, t)] /2 + k(t) u(, t) (3.3) k(t) k() /2 [ k u(, t)] /2 + k(t) u(, t). Using (2.2) an (3.3), it follows that µ(t) u x (, t) 2 (3.4) { u t (, t) 2 + k 2 (t) u () 2 + k() k u(, t) + k()k(t) u(, t) 2 }. Substituting (3.4) into (3.2), the onlusion of the lemma follows. Let us introue the funtional Q.E.D. L(t) = NF (t) + ψ(t), (3.5) with N >. It is not iffiult to see that L(t) satisfies q F (t) L(t) q F (t), (3.6) for some positive onstants q an q. Finally, we shall show the main result of this Setion. Theorem 3.3 Assume that the initial ata (u, u ) V L 2 (, ) an that the resolvent k satisfies (2.). Then there exist positive onstants α an γ 2 suh that F (t) α e γ2t F (), t. Proof. We will suppose that (u, u ) H 2 (, ) V V an satisfies (2.3); our onlusion will follow by stanar ensity arguments. Using Lemmas 3. an 3.2 we get t L(t) τ 2 N u t(, t) 2 + τ 2 Nk2 (t) u () 2 + τ 2 Nk (t) u(, t) 2 τ 2 Nk u(, t) 2 ( u t 2 + µ(t) u x 2 )x + u t (, t) 2 +k 2 (t) u () 2 + k()k(t) u(, t) 2 + k() k u(, t). (3.7) Then, hoosing N large enough we obtain t L(t) q 2F (t) + k 2 (t)f (), where q 2 > is a small onstant. Here we use (2.) to onlue the following estimates for the orresponing two terms appearing in Lemma 3.. τ 2 k u(, t) k u(, t), τ 2 k u(, t) 2 k u(, t) 2.
6 Asymptoti behavior of solutions EJDE 2/73 Finally, in view of (3.6) we onlue that t L(t) γ L(t) + k 2 (t)f (). Using the exponential eay of the resolvent kernel k we onlue L(t) {L() + }e γ2t for all t, where γ 2 = min(γ, γ ). From (3.6) the onlusion follows. Q.E.D. 4 Polynomial rate of eay The proof of the existene of global solutions for (.)-(.3) with resolvent kernel k eaying polynomially is essentially the same as in Setion 2. Here our attention will be fouse on the uniform rate of eay when the resolvent k eays polynomially suh as ( + t) p. In this ase, we will show that the solution also eays polynomially with the same rate. We shal use the following hypotheses: < k(t) b ( + t) p, b k p k (t) b 2 k p, (4.) b 3 ( k (t)) p+2 k (t) b 4 ( k (t)) p+2 where p > an b i, i =,..., 4, are positive onstants. Also we assume that k (t) r t < if r > p +. (4.2) The following lemmas will play an important role in the sequel. Lemma 4. Let m an h be integrable funtions, r < an q >. Then, for t, m(t s)h(s) s ( m(t s) r + q ) q ( q+ h(s) s ) m(t s) r q+ h(s) s. Proof. Let v(s) := m(t s) r q r q+ h(s) q+, w(s) := m(t s) q+ h(s) Then using Höler s inequality with δ = arrive to the onlusion. Q.E.D. q+. q q+ for v an δ = q + for w, we
EJDE 2/73 Mauro L. Santos 7 Lemma 4.2 Let p >, r < an t. Then for r >, ( k u(, t)) + ( 2 an for r =, ( k u(, t)) p+2 ( 2 k (s) r s u 2 L ((,T ),H (,)) u(., s) 2 H (,) st u(., s) 2 H (,) ) ( k + u(, t)), ) ( k + u(, t)) Proof. The above inequality is a immeiate onsequene of Lemma 4. with m(s) := k (s), h(s) := u(x, t) u(x, s) 2, q := ( r)(p + ), an t fixe. Q.E.D. Lemma 4.3 Let α >, β α +, an f be ifferentiable funtion satisfying f (t) f(t) + 2 f() α + α ( + t) β f() for t an some positive onstants, 2. Then there exists a onstant 3 > suh that for t, 3 f(t) ( + t) α f(). Proof. Let t an Then F (t) = f(t) + 2 2 α ( + t) α f(). where we use β α +. Hene F f() α Integration yiels F = f 2 2 ( + t) (α+) f() f + f() α 2 ( + t) (α+) f(), α (f + α + ( + t) (α+) f() + α ) F + α. F () α F (t) F () ( + t) α ( + t) α f() hene f(t) 3 ( + t) α f() for some 3, whih proves Lemma 4.3. Q.E.D.
8 Asymptoti behavior of solutions EJDE 2/73 Theorem 4.4 Assume that (u, u ) V L 2 (, ) an that the resolvent k satisfies (4.). Then there exists a positive onstant for whih F (t) F (). ( + t) Proof. We will suppose that (u, u ) H 2 (, ) V V an satisfies (2.3); our onlusion will follow by stanar ensity arguments. We efine the funtional L as (3.5) an we have the equivalene to the energy term F as given in (3.6) again. The negative term k(t) u(, t) 2 an be obtaine from Lemma 3.2 an estimate k(t) u(, t) 2 µ(t) u x 2 x. From Lemmas 3. an 3.2, using the properties of k from the assumption (4.) for the term τ 2 k u(, t), we obtain ( t L(t) Applying Lemma 4.2 with r > we obtain ( k ) + u(, t) u t 2 + µ(t) u x 2 x + k(t) u(, t) 2 +N( k ) + u(, t) ) + 2 k 2 (t)f (). (4.3) ( k r s) F () (with = (r)). On the other han, we have (k(t) u(, t) 2 + F () u t 2 + µ(t) u x 2 x) + ( k(t) u(, t) 2 + (( k ) u(, t)) + ) u t 2 + µ(t) u x 2 x. From (4.2) an (4.3) an using the last two inequalities, we onlue that t L(t) F () [(k(t) u(, t) 2 + +(( k ) u(, t)) + ] + k 2 (t)f (). u t 2 + µ(t) u x 2 x) +
EJDE 2/73 Mauro L. Santos 9 Using (3.6) an the above inequality an be written as t L(t) L() L(t) + + k 2 (t)f (). (4.4) Applying the Lemma 4.3 with f = L, α = ( r)(p + ) an β = 2p we have Choosing r suh that L(t) L(). (4.5) ( + t) α p + < r < p p +, an taking into aount the inequality (4.5), we obtain F (s)s L(s)s L(), (4.6) t u(., t) 2 H (,) tl(t) L(), (4.7) u(., s) 2 H (,) L(t)t L(). (4.8) Estimates (4.6)-(4.8) together with Lemma 4.2 (ase r = ) imply that ( k ) + u(, t) F () (( k ) u(, t)) +. Using the same arguments as in the erivation of (4.4), we have t L(t) Applying Lemma 4.3, we obtain Then from (3.6), we onlue L() L(t) F (t) whih ompletes the present proof. L(t) + + k 2 (t)f (). L(). ( + t) F (), ( + t) Q.E.D. Aknowlegements: The author expresses the appreiation to the referee for his value suggestions whih improve this paper. Referenes [] C. M. Dafermos & J. A. Nohel, Energy methos for nonlinear hyperboli Volterra integro-ifferential equation. Comm. Partial Differential Equation 4, 29-278 (979).
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EJDE 2/73 Mauro L. Santos Mauro L. Santos Departamento e Matemátia, Universiae Feeral o Pará Campus Universitario o Guamá Rua Augusto Corrêa, Cep 6675-, Pará, Brazil e-mail: ls@ufpa.br