 Applied Mathematics, 2010, 1, 529-533 doi:10.4236/am.2010.16070 Published Online December 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM On the Exponential Decay of Solutions for Some Kirchhoff-Type Modelling Equations with Strong Dissipation Yaojun Ye Department of Math ematics and Informa tion Science, Zhejiang University of Science and Technology, Hangzhou, China E-mail: yeyaojun2002@yahoo.com.cn Received August 24, 2010; revised October 19, 2010; accepted October 23, 2010 Abstract This paper deals with the initial boundary value problem for a class of nonlinear Kirchhoff-type equations with strong dissipative and source terms 222() ||,,0tt tuuuaubuuxt  in a bounded domain, where ,0aband 2 are constants. We obtain the global existence of solutions by construct-ing a stable set in10()H and show the energy exponential decay estimate by applying a lemma of V. Ko-mornik. Keywords: Kirchhoff-type Equation; Initial Boundary Value Problem; Stable Set; Exponential Decay Estimate 1. Introduction Let be a bounded domain in nR with smooth boun-dary. In this paper, we investigate the existence and the energy exponential decay estimate of global solutions for the initial boundary value problem of the following Kirchhoff-type equation with strong dissipative and source terms in a boun ded do m a i n 222,,0,tt tuuuaubuuxt  (1.1) 01(,0)(),(,0)(),,tux uxux uxx (1.2) (,) 0,,0,uxt xt (1.3) where ,0ab and 2are constants, ()s is a 1C-class function on [0, ) satisfying  00,,0,ssmssd s  (1.4) with 01m constant. When 1n, the equation (1.1) describes a small am-plitude vibration of an elastic string (). The original equation is 2202202LuuEhu uhPdsftLxtx   where 0xLand 0,t(,)uxt is the lateral dis-placement at the space coordinate x and the time t, is the mass density, h is the cross-section area, L is the length, 0P is the initial axial tension, is the resistance modulus, E is the Young modulus and f is the external force. Many authors have studied the existence and unique-ness of solutions of (1.1)-(1.3) by using various methods. When ,0ab, and (), 1,rssr K. Nishihara and Y. Yamada  have proved the existence and the polynomi-al decay of global solution under the assumptions that the initial data 0u and 1uare sufficiently small and 00u. However, the method in  can not be applied directly to the case that the equations have the blow-up term 2||uu. M. Aassila and A. Benaissa  extend the global existence part of  to the case where () 0s with 200u and the nonlinear dissipative term 2||ttuu. K. Ono and K. Nishihara  have proved the global existence and decay structure of solutions of (1.1)-(1.3) without small condition of data using Galerkin method. K. Ono  has obtained the global existence of solutions for the problem (1.1)-(1.3) with dissipative term tu instead oftu. In the case 0a, for large  and 0sr, P. Y. J. YE Copyright © 2010 SciRes. AM 530 D. Ancona and S. Spagnolo  proved that if 01 0,nuuC R are small, then problem (1.1)-(1.3) has a global solution. When0,sM. Ghisi and M. Gob-bino  proved the existence and uniqueness of a global solution ,uxt of (1.1)-(1.3) for small initial data  12101 00,uu HHH with 200mu and the asymptotic behavior , ,,0,0tttutu tutu in   121201 00,uu HHHL as t, where either 0u or 20u. The case () 0sr has been considered by M. Hosoya and Y. Yamada  under the following condi-tion: 20,5;0,4.4nnn They proved that, if the initial data are small enough, the problem (1.1)-(1.3) has a global solution which de-cays exponentially ast. In this paper, we prove the global existence for the problem (1.1) -(1.3) by applying the potential well theory introduced by D. H. Sattinger  and L. Payne and D. H. Sattinger . Meanwhile, we obtain the exponential decay estimate of global solutions by using the different method from paper . We adopt the usual notation and convention. Let mH denote the Sobolev space with the norm 22() ()|| ,mHLmuDu 0()mHdenotes the closure in mHof 0()C. For sim-plicity of notations, hereafter we denote by p the Lebesgue space ()pL norm,  denotes 2()L norm and we write equivalent norm  instead of 10()H norm 10().HMoreover, M denotes various positive constants depending on the known constants and it may be different at each appeara nc e. 2. Preliminary In order to state and prove our main results, we first de-fine the following functionals 20,Kumu bu 202mbJuuu for 10uH. Then we define the stable set S by  10,0, 0,SuHKuJud  where  100inf sup,0dJuuH. We denote the total energy functional associated with (1.1)-(1.3) by  22201122utbEtusds u  (2.1) for 10,0uH t, and  202102011022ubEu sdsu  is the total energy of the initial data. Lemma 2.1 Let qbe a number with 2,q 2n and 22,2.2nqnnThen the re exists a cons - tant C depending on  and q such that 1010(),()qHuCu uH . Lemma 2.2  Let :yt RRbe a nonincreas-ing function and assume that there is a constant0A, such that () (),0,sytdt Ayss then 1() (0),0.tyty et We state a local existence result, which is known as a standard one. Theorem 2.1 Suppose that satisfies 22,2;2 ,2.2nnnn   (2.2) If 1201 0(,)()()uu HL , then there exists 0T such that the problem (1.1)-(1.3) has a unique local solu-tion ()ut in the class 1200, ;,0, ;.tuC THuC TL  (2.3) Lemma 2.3 Let (, )utx be a solutions of problem (1.1)-(1.3). Then ()Et is a nonincreasing function for 0t and  .tdEtau tdt   (2.4) Proof By multiplying equation (1.1) by tu and inte-grating over , we get  0.tdEtau tdt Therefore, Et is a nonincreasing function on t. Lemma 2.4 Let 10uH, if (2.2) holds, then 0.d Proof Since 220,2mbJuuu Y. J. YE Copyright © 2010 SciRes. AM 531so, we get 210.dJumu bud  Let 0dJud, which implies that 1122120.ubmu    As 1, an elementary calculation shows that 220dJud. Hence, we have from Lem ma 2.1 that  212221001222202sup 220.2ubJuJ uumbCm    we get from t he definit i on of d that 0.d In order to prove the existence of global solutions for the problem (1.1)-(1. 3), we need the following Lemma. Lemma 2.5 Supposed that (2.2) hold, If 201,uSuL and 0Ed, then uS, for each 0, .tT Proof Assume that there exists a number *0,.tT such that ut Son 0, *t and *utS. Then, in virtue of the continuity of ut, we see *utS. From the definition of Sand the continuity of Jut and Kut in t, we have either *Jut d or *0Kut. It follows from (1.4) and (2.1) that   20***2*0.mbJutut utEt Ed  (2.5) So, the case *Jut d is impossible. Assume that *0Kut holds, then we get that 220*1 .dJut mud We obtain from *0dJutdthat 1. Since 2102*2*0.dJut mutd Consequently, we get from (2.5) that 10sup ***JutJ utJutd which contradicts the definition of d. Therefore, the case (()) 0Kut is impossible as well. Thus, we con-clude that ()ut S on [0, ).T 3. Main Results and Proof Theorem 3.1 Suppose that (2.2) holds, and ut is a local solution of problem (1.1)-(1.3) on 0,T. If 201,uSuL and 0Ed, then ,uxt is a global solu tion of the problem (1.1)-(1.3 ). Proof It suffices to show that  22tutu t is bounded independently of t. Under the hypotheses in Theorem 3.1, we get from Lemma 2.5 that ut Son 0, .T So the following formula holds on 0, .T 2022020(())() ()2() ()2(2) () ,2mbJutut utmbut utmut   (3.1) Therefore, we have from (3.1) that 2202(2)1() ()221()( ())()(0).2ttmut ututJutEt Ed (3.2) Hence, we get 2202()()max2,.(2)tut utdm  The above inequality and the continuation principle lead to the existence of global solution, that is, T . Therefore, the solution ut is a global solution of the problem (1.1)-(1.3). The following Theorem shows the exponential decay estimate of global solutions for problem (1.1)-(1.3). Theorem 3.2 If the hypo theses in Theor em 3.1 ar e va-lid, then the global solutions of problem (1.1)-(1.3) has the following expon ential decay property  10,tMEt Ee where 0M is a constant. Proof Multiplying by u on both sides of the Equa-tion (1.1) and integrating over [0, )T , we obtain that Y. J. YE Copyright © 2010 SciRes. AM 532 2220||,Ttt tSuuuua ubuudxdt (3.3) where 0.ST Since 2.TTTttt StSSuudxdtuu dxudxdt  (3.4) So, substituting the Formula (3.4) into the right-hand side of (3.3), we get that 2222220221.TtSTttSTTtS Sbuuuudtuauu dxdtuu dxbudt (3.5) It follows from (3.2) that    200 022 20222utEt Edmm m     (3.6) By exploiting Lemma 2.1 and (3.6), we easily arrive at   2222202,2bubC utbCututbCdu tm  (3.7) We obtain from (3.6) and (3.7) that 2220220022002221()(2)222()(2)(2)22 ().(2)bubC d utmbCdE tmmbC dEtmm      (3.8) We derive from (1.4) that 2220,usdsuu  (3.9) It follows fro m (3.5), (3.8) and (3.9) that  22002221 22| |.TSTTtt tSSbC dEtdtmmuauudxdtuu dx  (3.10) We have from Lemma 2.1 and (3.2) that   222220020112221222max ,1,2TTtStSTtSTSuu dxuumCuumCEt MESm (3.11) Substituting the estimate (3.11) into (3.10), we conclude that  220022221 22.TSTttSbC dEtdtmmuauudxdtMES (3.12) We get from Lemma 2.1 and Lemma 2.3 that  22 222222222.TTTtt tSSSudxdtudt CudtCCET ESESaa  (3.13) From Young inequality, Lemma 2.1, Lemma 2.3 and (3.6), We receive that   2200222.2TTttSSTSTSau udxdtauMudtaEtdtMESETmaEtdtMESm     (3.14) Choosing small enough  such that  2200 0221,22abCdmm m  then, substituting (3.13) and (3.14) into (3.12), .TSEtdt MES (3.15) Let T, then we have from (3.15) that .SEtdt MES  (3.16) Thus, we receive from (3.16) and Lemma 3.1 that  10,0,.tMEt E et (3.17) The proof of Theorem 3.2 is finished. 4. Acknowledgments This Research was supported by Natural Science Foun- Y. J. YE Copyright © 2010 SciRes. AM 533dation of Zhejiang Province (No.Y6100016), The Sci- ence and Research Project of Zhejiang Province Educa-tion Commission (No. Y200803804 and Y200907298), The Research Fundation of Zhejiang University of Sci- ence and Technology (No. 200803) and the Middleaged and Young Leader in Zhejiang University of Science an d Technology (2008-2012). 5. References  K. Narasimha, “Nonlinear Vibration of an Elastic String,” Journal of Sound and Vibration, Vol. 8, No. 1, 1968, pp. 134-146.  K. Nishihara and Y. Yamada, “On Global Solutions of Some Degenerate Quasilinear Hyperbolic Equations with Dissipative Terms,” Funkcialaj Ekvacioj, Vol. 33, No. 1, 1990, pp. 151-159.  M. Aassila and A. 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