Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 879649, 14 pages doi:10.1155/2011/879649 Research Article Exponential Stability of Two Coupled Second-Order Evolution Equations Qian Wan and Ti-Jun Xiao Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China Correspondence should be addressed to Ti-Jun Xiao, xiaotj@ustc.edu.cn Received 30 October 2010; Accepted 21 November 2010 Academic Editor: Toka Diagana Copyright q 2011 Q Wan and T.-J Xiao This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited By using the multiplier technique, we prove that the energy of a system of two coupled second order evolution equations one is an integro-differential equation decays exponentially if the convolution kernel k decays exponentially An example is give to illustrate that the result obtained can be applied to concrete partial differential equations Introduction Of concern is the exponential stability of two coupled second-order evolution equations one is an integro-differential equation in Hilbert space H u t Au t αu t − t v t √ β Av t k t − s Au s ds Av t √ β Au t 0, 1.1 1.2 0, with initial data u0 u0 , v v0 , u u1 , v v1 1.3 Here A : D A ⊂ H → H is a positive self-adjoint linear operator, α 0, β > 0, k t is a nonnegative function on 0, ∞ Moreover, the fractional power A1/2 is defined as in the well known operator theory cf, e.g., 1, Advances in Difference Equations An interesting and difficult point for it is to stabilize the whole system via the damping effect given by only one equation 1.1 We remark that there is very few work concerning the situation when the damping mechanism is given by memory terms; see , where a coupled Timoshenko beam system is investigated On the other hand, the stability of the single integro-differential equation has been studied extensively; see, for instance, 4, In this paper, through suitably choosing multipliers for the energy together with other techniques, we obtain the desired exponential decay result for the system 1.1 – 1.3 Nonlinear coupled systems with general decay rates will be discussed in a forthcoming paper In Section 2, we present our exponential decay theorem and its proof An application is given in Section Exponential Decay Result We start with stating our assumptions: A is a self-adjoint linear operator in H, satisfying a u 2, Au, u u∈D A , 2.1 where a > α > 0, β > are constants k t : 0, ∞ → 0, ∞ is locally absolutely continuous, satisfying k > 0, ∞ 1− k t dt − α − β2 > 0, a 2.2 and there exists a positive constant λ, such that k t −λk t , for a.e t 2.3 We define the energy of a mild solution u of 1.1 – 1.3 as E t Eu t : u t 2 t v t k t−s 1− t k s ds √ Au t √ √ Au s − Au t 2.4 ds √ Av t βu t α − β2 The following is our exponential decay theorem ut 2 Advances in Difference Equations Theorem 2.1 Let the assumptions be satisfied Then, √ i for any u0 , v0 ∈ D A and u1 , v1 ∈ H, problem 1.1 – 1.3 admits a unique mild solution on 0, ∞ √ The solution is a classical one, if u0 ∈ D A , v0 ∈ D A and u1 , v1 ∈ D A , ii there exists a constant C > such that the energy E t E e1−Ct , ∀t 2.5 0, for any mild solution of 1.1 – 1.3 Proof We denote u t w t v t A , u0 t w0 t , v0 t √ A α β A , √ β A A u1 t w1 t k tA B t , v1 t 2.6 Then, 1.1 – 1.3 becomes w t t Aw t − B t − s w s ds t ∈ 0, ∞ , 2.7 w w0 , w w1 , in H : H × H From the assumptions, one sees that—A is the generator of a strongly 1,1 continuous cosine function on H, and B · is bounded from D A into Wloc 0, ∞; H Therefore, we justify the assertion i cf., e.g Suppose now that u is a classical solution of 1.1 – 1.3 We observe E t − k t √ Au t 2 t √ √ Au s − Au t k t−s ds 2.8 0, by Assumption and so E t E s , s t T 2.9 Advances in Difference Equations Let ∞ μ: 1− α − β2 , a k t dt − and take < δ < E t 2.10 aμ/2β2 We have u t v t 2 √ Av t 1 − 2δ μ √ Au t 2 √ − δ β2 2a 2 Au t 2.11 , t ∈ 0, ∞ Furthermore, we need the following lemmas Lemma 2.2 For any T such that T S √ Av t βu t D1 E S G1 S T S D2 u t and for any ε1 > 0, there exist positive numbers D1 ε1 , D2 ε1 dt T t S dt √ √ Au s − Au t k t−s G2 ε1 T v t S 2 G3 ε1 dt 2.12 ds dt G4 T S √ Au t dt, for some positive constants Gi i 1, 2, 3, which only depend on α, β, a, and k √ Proof At first, let us take the inner product of both sides of 1.1 with Av t and integrate over S, T Then, noticing 1.2 , we obtain T S √ u t , Av t α β T S T S dt − √ u t , Av t t k t−s T S √ Au t , v t T dt t S √ √ T dt − β S Au t dt k t−s √ √ Au s ds, Au t dt Au s ds, v t β T S √ Av t dt dt 2.13 Advances in Difference Equations For the first item, integrating by parts, we have T S √ u t , Av t √ u t , Av t T dt S T dt − S √ u t , Av t dt 2.14 The second and the fifth items can be treated similarly Therefore, β T S √ Av t α dt S T t S − √ u t , Av t T S √ u t , Av t T − dt √ Au t , v t T dt S √ Au s ds, v t dt T β k t−s t 1− S √ Au t k s ds dt dt 2.15 T t S √ √ Au s − Au t ds, v t k t−s T t S −β √ dt √ √ Au t ds, Au t T S k t−s Au s − √ Au t , v t k t dt dt Then, taking the inner product of both sides of 1.1 with u t and integrating over S, T , we obtain β T S √ Av t , u t − T S u t ,u t T t S − dt T α S ut T dt S dt u t dt √ √ √ k t−s Au s − Au t ds, Au t T S 1− t k s ds √ Au t dt 2.16 dt Advances in Difference Equations Equation 2.15 × β/α T S 2.16 yields that √ Av t − T β α − β α β α S T t S S u t β α dt dt √ Au t , v t T S √ Au s ds, v t dt T S T t S k t−s t 1− dt √ Au t k s ds dt 2.17 √ √ T S β2 − α α k t−s Au s − √ Au t , v t k t α − β2 α − T dt √ u t , Av t T β2 − α α β α dt u t ,u t S − βu t T t S ds, v t Au t dt dt √ √ √ Au s − Au t ds, Au t T √ S k t−s Av t dt − α − β2 T S ut dt dt Next, we will estimate all the terms on the right side of 2.17 From 2.11 , we have the following estimate: T S √ u t , Av t dt √ u t , Av t T T S where M is a positive constant Those terms of the form S ·, · Denote by J the sum of the other terms on the right of 2.17 2ME S , 2.18 dt can be similarly treated Advances in Difference Equations Using Young’s inequality and noting 2.8 , we get, for ε1 > 0, T t S √ √ k t−s T ε1 − t S Au t ds, v t dt √ √ Au s − Au t k t−s ds t S k s ds t √ √ Au s − Au t k t−s T 2ε1 dt T ε1 Au s − S ds dt 2ε1 ε1 k E S T v t S v t T 2ε1 S dt v t dt dt 2.19 The treatment of the other terms of J is similar, giving β α T S α − β2 α k t √ Au t , v t T t S ε1 dt √ Au t T S k β2 2ε1 α2 dt √ √ √ Au s − Au t ds, Au t T v t S dt, k t−s α − β2 2α2 ε1 T S t t k s ds dt √ √ Au s − Au t k t−s T ε1 ds dt S √ Au t dt 2.20 Thus, we obtain J ε1 k β E S α α − β2 2α2 ε1 α k β2 2ε1 α2 where θ t S a T k s ds t k t−s √ Au s − √ Au t ds dt β −α β 2ε1 α T S √ Au t T ε1 S v t 2 2.21 dt T dt u t S T √ Av t dt θ ζ1 β2 √ Au a S dt, max{ α − β2 /α, 0} Make use of the estimate √ Av ζ1 √ Av βu 2 , 2.22 Advances in Difference Equations ζ1 θ < We thus verify our where ζ1 is a positive constant, small enough to satisfy conclusion Lemma 2.3 For any T such that μ T S √ Au t S and for any ε2 > 0, there exist positive numbers D3 ε2 , D4 ε2 , D3 E S dt T t S D4 T β2 2ε2 a √ √ Au s − Au t k t−s S u t dt ε2 T S ds dt 2.23 v t dt Proof We denote w A−1/2 u and take the inner product of both sides of 1.2 with w t , and integrate over S, T It follows that − T S √ Av t , u t T dt dt − v t ,w t S T S v t , w t dt β T S ut 2.24 dt Plugging this equation into 2.16 , we find T 1− S t √ Au t k s ds dt − T u t ,u t S β2 − α T t S T S dt ut β T S T dt S v t ,w t u t dt 2.25 dt √ √ √ Au s − Au t ds, Au t −β T S k t−s dt v t , w t dt Observe T t S √ √ √ Au s − Au t ds, Au t k t−s 2δ3 T S t k s ds t k t−s dt √ √ Au s − Au t ds dt δ3 T S √ Au t dt, 2.26 Advances in Difference Equations where δ3 μ, and for ε2 > T β ε2 v t , w t dt S T S v t β2 2aε2 dt T S u t 2.27 dt The other items on the right of 2.25 can be dealt with as in the proof of Lemma 2.2 Hence, we get the conclusion Lemma 2.4 For any T T u t S S 0, there exist positive numbers D5 , D6 such that T dt D5 E S T t S D6 dt √ √ Au s − Au t k t−s √ Au t T α − β2 a 2 v t S S ds dt √ Av t T dt S 2.28 βu t dt Proof Taking the inner product of both sides of 1.2 with v t and integrating over S, T , we see T S v t T dt T v t ,v t S dt S √ Av t dt β T S √ Au t , v t 2.29 dt Combining this equation and 2.16 gives T S v t T dt T u t ,u t S − S u t T t S dt T dt S v t ,v t T dt 1− S t k s ds √ Au t dt √ √ √ Au s − Au t ds, Au t T S k t−s √ Av t βu t dt α − β2 T S ut dt dt 2.30 This yields the estimate as desired 10 Advances in Difference Equations Lemma 2.5 Let S0 > be fixed For any T D7 ε3 , D8 ε3 such that T u t S D7 E S dt T ε3 S S S0 and for any ε3 > 0, there exist positive numbers T t S √ Au t √ √ Au s − Au t D8 k t−s ε3 dt S Proof Take the inner product of both sides of 1.1 with over S, T This leads to T t S √ Av t T t ds dt 2.31 βu t dt k t−s u s −u t ds and integrate k s ds u t − T t u t, S dt k t − s u s − u t ds dt T t 1− S k s ds √ Au t , T S t k t−s √ Au s − √ Au t ds dt t ut, 2.32 k t − s u s − u t ds dt T t S β k t − s u s − u t ds T −α t u t , S − dt √ √ Au s − Au t ds T S k t−s √ Av t , t dt k t − s u s − u t ds dt Just as in the proofs of the above lemmas, using Young’s inequality and noting that T S we prove the conclusion t k s ds u t S0 dt k s ds T S u t dt, 2.33 Advances in Difference Equations 11 Proof of Theorem 2.1 (continued) From Assumption and 2.8 , we have T t S √ √ Au s − Au t T t S λ − k t−s √ ds dt √ Au t T − λ S k t−s Au s − ds dt 2.34 E t dt E S λ Now, fix S0 > Thanks to Lemmas 2.2 and 2.3, we know that for any T for η > 0, μ √ Au t T S √ Av t T η dt S D4 D3 ηD1 ε2 ηG2 T ε1 S ηD2 v t βu t E S λ S0 and dt η G3 ε1 dt S G4 β2 2ε2 a ηG1 √ Au t T S T S u t 2 dt 2.35 dt Moreover, by the use of Lemmas 2.4 and 2.5, we have √ Au t T μ S dt p0 E S p1 η T S T √ S Av t √ Au t 2 βu t dt dt T p2 S √ 2.36 Av t βu t dt, where p0 D3 ηD1 D4 ηD2 p1 η G3 ε1 G4 p2 ε2 ε1 ηG2 ε2 D5 D6 ε3 ηG2 ε2 ε1 ηG2 α − β2 a β2 2ε2 a D7 1 ε1 D8 ε2 ηG1 β2 2ε2 a β2 2ε2 a ηG2 ε1 ηG1 ηG1 ε3 , λ β2 2ε2 a 2.37 ηG1 , 12 Advances in Difference Equations Let ε−1 , ε1 η ε2 ε2 , ε3 ε5 2.38 Taking ε small enough gives μ , p1 < p2 < η 2.39 Therefore, there is a constant N1 > such that √ Au t T S √ Av t T dt S βu t N1 E S dt 2.40 by 2.36 Using Lemma 2.4 and 2.34 , we deduce that for some N2 > 0, T S u t T dt D5 v t S 2D6 dt E S λ α−β a 2 2.41 √ Au t T S √ T dt S Av t βu t dt N2 E S Next, define u t H t : t v t √ Au t √ √ k t−s Au s − Au t √ Av t 2 βu t 2.42 ds It is easy to see that there exist M1 , M2 > such that M1 E t T S T S E t dt N1 H t dt M1 T S H t dt N2 N1 H t M2 E t Therefore, E S, λ N2 2/λ E S M1 2.43 Advances in Difference Equations 13 S On the other hand, when T S S0 E t dt S S0 , E t dt T S0 N1 S0 − S E S N1 S0 E t dt N2 2/λ E S0 M1 2.44 N2 2/λ E S, M1 that is, T S E t dt NE S , ∀S 2.45 By a standard approximation argument, we see that 2.45 is also true for mild solutions From this integral inequality, we complete the proof cf., e.g., 7, Theorem 8.1 An Example Example 3.1 Consider a coupled system of Petrovsky equations with a memory term ∂2 u t, ξ t Δ2 u t, ξ k t − s Δ2 u t, ξ ds βΔv t, ξ 0, t 0, ξ ∈ Ω, ∂2 v t, ξ t u t, ξ u 0, ξ t αu t, ξ − u0 ξ , Δ2 v t, ξ v t, ξ v 0, ξ βΔu t, ξ Δu t, ξ v0 ξ , Δv t, ξ ∂t u 0, ξ t 0, 0, u1 ξ , 0, ξ ∈ Ω, t 3.1 0, ξ ∈ ∂Ω, ∂t v 0, ξ v1 ξ , ξ ∈ Ω, where Ω is a bounded open domain in ÊN , with sufficiently smooth boundary ∂Ω and α, β, k as in Assumption Let H L2 Ω with the usual inner product and norm Here, we denote by ∂t u the time derivative of u and by Δu the Laplacian of u with respect to space variable ξ Define A : D A ⊂ H → H by A Δ2 , with D A H Ω ∩ H0 Ω 3.2 Then, Assumption is satisfied Therefore, we claim in view of Theorem 2.1 that the energy of the system decays exponentially at infinity 14 Advances in Difference Equations Acknowledgments The authors would like to thank the referees for their comments and suggestions This work was supported partially by the NSF of China 10771202, 11071042 , the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics 08DZ2271900 and the Specialized Research Fund for the Doctoral Program of Higher Education of China 2007035805 References T.-J Xiao and J Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, vol 1701 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1998 K Yosida, Functional Analysis, Classics in Mathematics, Springer, Berlin, Germany, 1995 F Ammar-Khodja, A Benabdallah, J E Munoz Rivera, and R Racke, “Energy decay for Timoshenko ˜ systems of memory type,” Journal of Differential Equations, vol 194, no 1, pp 82–115, 2003 F Alabau-Boussouira and P Cannarsa, “A general method for proving sharp energy decay rates for memory-dissipative evolution equations,” Comptes Rendus de l’Acad´mie des Sciences—Series I Paris, e vol 347, no 15-16, pp 867–872, 2009 F Alabau-Boussouira, P Cannarsa, and D Sforza, “Decay estimates for second order evolution equations with memory,” Journal of Functional Analysis, vol 254, no 5, pp 1342–1372, 2008 C M Dafermos, “An abstract Volterra equation with applications to linear viscoelasticity,” Journal of Differential Equations, vol 7, pp 554–569, 1970 V Komornik, Exact Controllability and Stabilization The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, France, 1994  ... the proofs of the above lemmas, using Young’s inequality and noting that T S we prove the conclusion t k s ds u t S0 dt k s ds T S u t dt, 2.33 Advances in Difference Equations 11 Proof of Theorem... NSF of China 10771202, 11071042 , the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics 08DZ2271900 and the Specialized Research Fund for the Doctoral Program of Higher... 2.3 We define the energy of a mild solution u of 1.1 – 1.3 as E t Eu t : u t 2 t v t k t−s 1− t k s ds √ Au t √ √ Au s − Au t 2.4 ds √ Av t βu t α − β2 The following is our exponential decay theorem