弹性波在不同介质中的传播及其稳定性分析
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摘要
本文主要研究弹性波在不同介质中的传播及其稳定性分析.运用Galerkin逼近方法证明了边界值问题的解的存在性与唯一性,而且运用Nakao引理和乘子技巧证明了能量的稳定性,即能量的一般衰减性,包括能量的指数衰减和多项式衰减.
第二章主要研究带有非线性阻尼边界条件的弹性波的边界值问题及其能量的衰减性更确切地说,我们研究上面方程的强解和弱解的存在性与唯一性及其能量的衰减性.在本章中,我们克服的主要困难有三点：
第一,由于我们考虑的是非线性阻尼边界条件,通常的Galerkin逼近方法在这里失效,因此,需要通过变量代换转化为初值为零的等价问题；
第二,由于边界条件上的非线性阻尼项g(u1)和非线性源项f(u)的出现,在通过极限的过程中,带来一些困难.为了克服这些困难,我们用紧性和单调性讨论：
最后,因为局部耗散项b(x)u1,经典的能量方法在这里失效,因此运用扰动的能量方法和乘子技巧克服这个困难,从而得到能量的指数衰减.
第三章研究了一类广泛的弹性波在弹性介质中的边界值问题及其稳定性分析.运用非线性半群方法得到强解或者弱解的存在性与唯一性,然后运用扰动的能量方法和乘子技巧得到能量的指数衰减,推广了第二章的主要结果.更确切地说,我们研究带有梯度项和非线性边界阻尼条件的Klein-Gordon型方程证明的主要困难除了跟上述第二章的情形一样的困难外,还有以下两点：
第一,由于上述方程中出现了梯度项和非线性源项,逼近解的构造更加复杂,因此,运用非线性半群方法得到强解或者弱解的存在性与唯一性。更确切地说,我们把上述方程转化为抽象的柯西问题接下来,只要证明算子A在Hilbert空间Η=V×H中产生G0类压缩半群.也就是说,证明算子A在Hilbert空间Η=V×H中满足这里R(I+A)为I+A的值域,I为恒等算子.
第二,由于非线性阻尼边界条件和耗散项的出现,能量估计更加困难更加具有技巧性,运用扰动的能量方法和乘子技巧得到能量的指数衰减.
第四章研究了弹性波在粘弹性介质中的边界值问题及其稳定性分析.更确切地说,我们研究了带有非线性局部阻尼项和依赖于速度的粘弹性材料的密度的非线性粘弹性波方程
由于方程中出现局部非线性阻尼项α(x)u1|u1|k,非线性源项bu|u|r和非线性项M(τ),松弛函数g(t),使得能量估计更加困难,分别运用Galerkin逼近方法和扰动的能量方法研究了解的存在性与唯一性及其能量的一般衰减性.而且,对于某些初值和对松弛函数的适当的假设,我们证明了能量的衰减率取决于松弛函数g(t)的衰减率.更确切地说,若松弛函数g(t)是指数性衰减为零,则能量也是指数性衰减为零,若松弛函数g(t)是多项式衰减为零,则能量也是多项式衰减为零,这个结果推广了早期文献的结果。详见第四章.
第五章研究了弹性波在各向同性的不可压介质中的边界值问题及其稳定性分析.更确切地说,我们研究了下面的非线性弹性波方程证明的主要困难有两个：
第一,由于各向同性和不可压的介质的性质,需要先验估计时,我们运用Sobolev嵌入理论和乘子技巧克服一些困难,从而得到解的存在性与唯一性：
第二,由于局部的非线性耗散效应ρ(x,u1),进行能量估计时遇到一些困难.为了克服这些困难,运用Nakao引理和乘子技巧得到能量的一般衰减性.证明的关键在于如何得到：能量关于时间t是一般的衰减的结论(包括能量的指数衰减和能量的多项式衰减).这里我们采用了第二、三、四章的基本思想(乘子技巧),但是我们的能量估计方法是新颖的,而且由于各向同性的不可压介质中的弹性波以及非线性局部耗散效应,我们不能直接应用经典的能量估计方法,这里我们巧妙地构造了一系列的乘子和运用Nakao引理来克服这个困难.
在最后一部分(附录),若忽略压力项,考虑到弹性波在各向异性介质中的传播,我们研究了弹性波在各向异性介质中的边界值问题及其稳定性分析.更确切地说,我们研究了如下的非线性弹性波方程运用非线性半群方法证明解的存在性与唯一性及其能量的指数衰减性.也就是说,我们分为两个步骤证明：
第一步,我们证明解的存在性与唯一性.为了进一步分析,我们把上面的方程转化为下面的抽象的柯西问题基于非线性半群理论,我们将证明算子A产生Hilbert空间Η上的G0类压缩半群.
第二步,我们证明能量的指数衰减.也就是说,我们证明下面的估计这里c,ω为正常数,s(t)=eA1为Hilbert空间Η上的C0类压缩半群.
In this dissertation, we investigate the propagation and stability analysis of elastic waves in different medium. By using Galerkin's approximation method, we prove the existence and uniqueness of solutions of boundary value problem (BVP). Moreover, by means of Nakao Lemma and the multiplier technique, we prove the energy stability, that is, general energy decay, including the exponential energy decay and polynomial energy decay.
In chapter two, we investigate the BVP and stability analysis of the elastic waves with nonlinear boundary damping given by More precisely, we investigate existence, uniqueness and energy decay of strong solutions and weak solutions of the above equation. Main difficulties involved in studying our problem are as follows: First, due to the nonlinear boundary damping, the usual Galerkin's approxi-mation method does not work here. Therefore, this approximation requites a change of variables to transformation our problem into an equivelent problem with the initial value equalizing zero.
Second, we overcome some difficulties, such as the presence of nonlinear boundary damping g(ut) and nonlinear source term f(u) that bring up serious difficulties when passing the limit, which overcome combining arguments of compactly and monotonicity.
Third, due to the locally dissipative term b(x)ut, the classic energy method does not work here. We apply the perturbed energy method and the multiplier technique to overcome difficulties and obtain the exponential energy decay.
In chapter three, we investigate the BVP and stability analysis of the extended elastic waves in elasticity. We obtain the existence and uniqueness of strong solutions or weak solutions by means of nonlinear semigroup method. Then, we obtain the exponential energy decay by using of the perturbed energy method and multiplier technique and we extend our main results in chapter two. More precisely, we investigate the Klein-Gordon type with grade term and nonlinear boundary damping given by Except the same difficulties as the case in chapter two, main difficulties lie in two sides:
First, in the presence of grade term and nonlinear source term, the construction of approximating solutions become complex. So, we obtain the existence and uniqueness of strong solutions or weak solutions by means of nonlinear semiroup method. More precisely, we formulate the above equation as an abstract Cauchy problem
Next, we shall prove that operator A generate a Co semigroup of contractions on Hilbert spaceH=V×H. That is, it is sufficient to prove that Where R(I+A) is the rage of the operator I+A, I is the identity operator.
Second, in the presence of noninear boundary damping and dissipative term, the energy estimate become difficult and skillful. We obtain the exponential energy decay by using of the perturbed energy method and the multiplier technique.
In chapter four, we investigate the BVP and stabiltiy analysis of the elastic waves in viscoelasticity medium. More precisely, we investigate the nonlinear viscoelastic equation with nonlinear localized damping and velocity-depende-nt material density given by
In the presence of nonlinear localized damping a(x)ut|ut|k,nonlinear source term bu|u|r, nonlinear term M(τ) and the relaxation functiong(t), the energy estimate become difficult. We obtain the existence, uniqueness and general energy decay of the solutions by means of the Galerkin's approximation method and the perturbed energy method respectively. Furthermore, for certain initial value and suitable conditions on the relaxation function, we show that the energy decays exponentially or polynomially depending the rate of the decay of the relaxation function. More precisely, the energy decays exponentially to zero provided the relaxation function g(t) decays exponentia-lly to zero. When the relaxation function g(t) decays polynomially, we show the energy decays polynomially to zero with the same rate of decay. This result improves the earlier ones in the literatures. More details are present in chapter four.
In chapter five, we study the BVP and stability of the elastic waves in isotropic incompressible medium. More precisely, we investigate the nonlinear elastic wave equation as follows: There are two difficulties in our proof.
First, dut to the properties of isotropic and incompressible, we apply the Sobolev embedding theory and the multiplier technique to overcome some difficulties when we need a prior estimate. Then, we obtain the existence and uniqueness of the solutions.
Second, due to nonlinear localized dissipative effectsρ(x,ut), we overcome some difficulties when we show the energy estimates. Applying Nakao lemma and the multiplier technique, we obtain the general enegy decay. The key is how to get:the energy is general decay with respect to the time t, including the exponential energy decay and the polynomial energy decay. Here, we apply the basic ideas in chapter 2,3,4(i.e the multiplier technique), but our energy estimate method is new. Moreover, due to the elastic waves in isotropic incompressible medium and nonlinear localized dissipative effects, we cannot use the classic energy method estimate directly. Here, we skillfully construct a series of multipliers and use Nakao lemma to overcome some difficulties.
At the last part(i.e Appendix), in absence of pressure term and in the presence ofρ(x, ut)= a(x)ut, we consider the propagation of the elastic waves in anisotropic medium. That is, we investigate the BVP and stability analysis of the elastic waves in anisotropic medium. More precisely, we investigate nonlinear elastic wave equation as follows: We prove the exitence, uniqueness and the exponential energy decay of the solutions by using nonlinear semigroup method. That is, we divide our proof into two steps.
In step 1, we prove the exitence, uniqueness of the solutions. To facilitate our analysis, we formulate the above equation as an abstract Cauchy problem given by Based on nonlinear semigroup theory, we shall prove that the opertor A generates a C0 semigroup of contractions on Hilbert space H.
In step 2, we prove the exponential energy decay of the solutions. That is, we prove the following estimate where C andωare positive constants, S(t)= eAt is a C0 semigroup of contractions on Hilbert spac H.
第二章主要研究带有非线性阻尼边界条件的弹性波的边界值问题及其能量的衰减性更确切地说,我们研究上面方程的强解和弱解的存在性与唯一性及其能量的衰减性.在本章中,我们克服的主要困难有三点：
第一,由于我们考虑的是非线性阻尼边界条件,通常的Galerkin逼近方法在这里失效,因此,需要通过变量代换转化为初值为零的等价问题；
第二,由于边界条件上的非线性阻尼项g(u1)和非线性源项f(u)的出现,在通过极限的过程中,带来一些困难.为了克服这些困难,我们用紧性和单调性讨论：
最后,因为局部耗散项b(x)u1,经典的能量方法在这里失效,因此运用扰动的能量方法和乘子技巧克服这个困难,从而得到能量的指数衰减.
第三章研究了一类广泛的弹性波在弹性介质中的边界值问题及其稳定性分析.运用非线性半群方法得到强解或者弱解的存在性与唯一性,然后运用扰动的能量方法和乘子技巧得到能量的指数衰减,推广了第二章的主要结果.更确切地说,我们研究带有梯度项和非线性边界阻尼条件的Klein-Gordon型方程证明的主要困难除了跟上述第二章的情形一样的困难外,还有以下两点：
第一,由于上述方程中出现了梯度项和非线性源项,逼近解的构造更加复杂,因此,运用非线性半群方法得到强解或者弱解的存在性与唯一性。更确切地说,我们把上述方程转化为抽象的柯西问题接下来,只要证明算子A在Hilbert空间Η=V×H中产生G0类压缩半群.也就是说,证明算子A在Hilbert空间Η=V×H中满足这里R(I+A)为I+A的值域,I为恒等算子.
第二,由于非线性阻尼边界条件和耗散项的出现,能量估计更加困难更加具有技巧性,运用扰动的能量方法和乘子技巧得到能量的指数衰减.
第四章研究了弹性波在粘弹性介质中的边界值问题及其稳定性分析.更确切地说,我们研究了带有非线性局部阻尼项和依赖于速度的粘弹性材料的密度的非线性粘弹性波方程
由于方程中出现局部非线性阻尼项α(x)u1|u1|k,非线性源项bu|u|r和非线性项M(τ),松弛函数g(t),使得能量估计更加困难,分别运用Galerkin逼近方法和扰动的能量方法研究了解的存在性与唯一性及其能量的一般衰减性.而且,对于某些初值和对松弛函数的适当的假设,我们证明了能量的衰减率取决于松弛函数g(t)的衰减率.更确切地说,若松弛函数g(t)是指数性衰减为零,则能量也是指数性衰减为零,若松弛函数g(t)是多项式衰减为零,则能量也是多项式衰减为零,这个结果推广了早期文献的结果。详见第四章.
第五章研究了弹性波在各向同性的不可压介质中的边界值问题及其稳定性分析.更确切地说,我们研究了下面的非线性弹性波方程证明的主要困难有两个：
第一,由于各向同性和不可压的介质的性质,需要先验估计时,我们运用Sobolev嵌入理论和乘子技巧克服一些困难,从而得到解的存在性与唯一性：
第二,由于局部的非线性耗散效应ρ(x,u1),进行能量估计时遇到一些困难.为了克服这些困难,运用Nakao引理和乘子技巧得到能量的一般衰减性.证明的关键在于如何得到：能量关于时间t是一般的衰减的结论(包括能量的指数衰减和能量的多项式衰减).这里我们采用了第二、三、四章的基本思想(乘子技巧),但是我们的能量估计方法是新颖的,而且由于各向同性的不可压介质中的弹性波以及非线性局部耗散效应,我们不能直接应用经典的能量估计方法,这里我们巧妙地构造了一系列的乘子和运用Nakao引理来克服这个困难.
在最后一部分(附录),若忽略压力项,考虑到弹性波在各向异性介质中的传播,我们研究了弹性波在各向异性介质中的边界值问题及其稳定性分析.更确切地说,我们研究了如下的非线性弹性波方程运用非线性半群方法证明解的存在性与唯一性及其能量的指数衰减性.也就是说,我们分为两个步骤证明：
第一步,我们证明解的存在性与唯一性.为了进一步分析,我们把上面的方程转化为下面的抽象的柯西问题基于非线性半群理论,我们将证明算子A产生Hilbert空间Η上的G0类压缩半群.
第二步,我们证明能量的指数衰减.也就是说,我们证明下面的估计这里c,ω为正常数,s(t)=eA1为Hilbert空间Η上的C0类压缩半群.
In this dissertation, we investigate the propagation and stability analysis of elastic waves in different medium. By using Galerkin's approximation method, we prove the existence and uniqueness of solutions of boundary value problem (BVP). Moreover, by means of Nakao Lemma and the multiplier technique, we prove the energy stability, that is, general energy decay, including the exponential energy decay and polynomial energy decay.
In chapter two, we investigate the BVP and stability analysis of the elastic waves with nonlinear boundary damping given by More precisely, we investigate existence, uniqueness and energy decay of strong solutions and weak solutions of the above equation. Main difficulties involved in studying our problem are as follows: First, due to the nonlinear boundary damping, the usual Galerkin's approxi-mation method does not work here. Therefore, this approximation requites a change of variables to transformation our problem into an equivelent problem with the initial value equalizing zero.
Second, we overcome some difficulties, such as the presence of nonlinear boundary damping g(ut) and nonlinear source term f(u) that bring up serious difficulties when passing the limit, which overcome combining arguments of compactly and monotonicity.
Third, due to the locally dissipative term b(x)ut, the classic energy method does not work here. We apply the perturbed energy method and the multiplier technique to overcome difficulties and obtain the exponential energy decay.
In chapter three, we investigate the BVP and stability analysis of the extended elastic waves in elasticity. We obtain the existence and uniqueness of strong solutions or weak solutions by means of nonlinear semigroup method. Then, we obtain the exponential energy decay by using of the perturbed energy method and multiplier technique and we extend our main results in chapter two. More precisely, we investigate the Klein-Gordon type with grade term and nonlinear boundary damping given by Except the same difficulties as the case in chapter two, main difficulties lie in two sides:
First, in the presence of grade term and nonlinear source term, the construction of approximating solutions become complex. So, we obtain the existence and uniqueness of strong solutions or weak solutions by means of nonlinear semiroup method. More precisely, we formulate the above equation as an abstract Cauchy problem
Next, we shall prove that operator A generate a Co semigroup of contractions on Hilbert spaceH=V×H. That is, it is sufficient to prove that Where R(I+A) is the rage of the operator I+A, I is the identity operator.
Second, in the presence of noninear boundary damping and dissipative term, the energy estimate become difficult and skillful. We obtain the exponential energy decay by using of the perturbed energy method and the multiplier technique.
In chapter four, we investigate the BVP and stabiltiy analysis of the elastic waves in viscoelasticity medium. More precisely, we investigate the nonlinear viscoelastic equation with nonlinear localized damping and velocity-depende-nt material density given by
In the presence of nonlinear localized damping a(x)ut|ut|k,nonlinear source term bu|u|r, nonlinear term M(τ) and the relaxation functiong(t), the energy estimate become difficult. We obtain the existence, uniqueness and general energy decay of the solutions by means of the Galerkin's approximation method and the perturbed energy method respectively. Furthermore, for certain initial value and suitable conditions on the relaxation function, we show that the energy decays exponentially or polynomially depending the rate of the decay of the relaxation function. More precisely, the energy decays exponentially to zero provided the relaxation function g(t) decays exponentia-lly to zero. When the relaxation function g(t) decays polynomially, we show the energy decays polynomially to zero with the same rate of decay. This result improves the earlier ones in the literatures. More details are present in chapter four.
In chapter five, we study the BVP and stability of the elastic waves in isotropic incompressible medium. More precisely, we investigate the nonlinear elastic wave equation as follows: There are two difficulties in our proof.
First, dut to the properties of isotropic and incompressible, we apply the Sobolev embedding theory and the multiplier technique to overcome some difficulties when we need a prior estimate. Then, we obtain the existence and uniqueness of the solutions.
Second, due to nonlinear localized dissipative effectsρ(x,ut), we overcome some difficulties when we show the energy estimates. Applying Nakao lemma and the multiplier technique, we obtain the general enegy decay. The key is how to get:the energy is general decay with respect to the time t, including the exponential energy decay and the polynomial energy decay. Here, we apply the basic ideas in chapter 2,3,4(i.e the multiplier technique), but our energy estimate method is new. Moreover, due to the elastic waves in isotropic incompressible medium and nonlinear localized dissipative effects, we cannot use the classic energy method estimate directly. Here, we skillfully construct a series of multipliers and use Nakao lemma to overcome some difficulties.
At the last part(i.e Appendix), in absence of pressure term and in the presence ofρ(x, ut)= a(x)ut, we consider the propagation of the elastic waves in anisotropic medium. That is, we investigate the BVP and stability analysis of the elastic waves in anisotropic medium. More precisely, we investigate nonlinear elastic wave equation as follows: We prove the exitence, uniqueness and the exponential energy decay of the solutions by using nonlinear semigroup method. That is, we divide our proof into two steps.
In step 1, we prove the exitence, uniqueness of the solutions. To facilitate our analysis, we formulate the above equation as an abstract Cauchy problem given by Based on nonlinear semigroup theory, we shall prove that the opertor A generates a C0 semigroup of contractions on Hilbert space H.
In step 2, we prove the exponential energy decay of the solutions. That is, we prove the following estimate where C andωare positive constants, S(t)= eAt is a C0 semigroup of contractions on Hilbert spac H.
引文
[1]R. Teman. Infinite dimensional dynamics systems in mechanics and physics. Springer-Verlag, NewYork,1988.
[2]Z. Y. Zhang. New exact traveling wave solutions for the nonlinear Klein-Gordon equation. Turk. J Phys,2008,32:235-240.
[3]C. S. Morawetz. The decay of solutions of the exterior initial-boundary-value problem for the wave equation. Comm. Pure Appl. Math.,1961,14:561-568.
[4]G. Dassios. Local energy decay for scattering of elastic waves. J. Differential Equations,1983,49:124-141.
[5]M. Nakao. Decay of solutions of the wave equations with a local nonlinear dissipation. Math. Ann.,1996,305:403-417.
[6]M. Nakao. Decay of solutions of the wave equation with some localized dissipations. Nonlinear Anal.,1997,30:3775-3784.
[7]M. Nakao. Stabilization of local energy in an exterior domain for the wave equation with a localized dissipation. J. Differential Equations,1998,148:388-406.
[8]M. Nakao. Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations. Math. Z.,2001,238:781-797.
[9]R. Ikehata. Energy decay of solutions for the semilinear dissipative wave equations in an exterior domain. Funkcial. Ekvac.,2001,44:487-499.
[10]R. Ikehata. Small data global existence of solutions for dissipative wave equations in an exterior domain. Funkcial. Ekvac.,2002,45:259-269.
[11]R. Ikehata. Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain. J. Differential Equations,2003,188:390-405.
[12]R. Ikehata. Global existence of solutions for 2-D semilinear wave equations with dissipation localized near infinity in an exterior domain. Math. Methods Appl. Sci.. 2006,29:479-496.
[13]Y. H. Kang, M. J. Lee, I. H. Jung, Sharp energy decay estimates for the wave equation with a local degenerate dissipation. Comput. Math. Appl.,2009.57:21-57.
[14]E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures Appl.,1991,70:513-529.
[15]A.B. Aliev, A.Kh. Khanmamedov. Energy estimates for solutions of the mixed problem for linear seconder order hyperbolic equation. Math. Note,1996.59(4):345-349.
[16]M. M. Cavalcanti, V. N. Domings, Cavalcanti, J. A. Soriano. On solvability and stability of solutions to hyperbolic-parabolic problems with boundary damping. In Atas diXIX Congresso B.rasileiro de Mathematica Applicadae Computational,1996.
[17]M. M. Cavalcanti, J. A. Soriano. On solvability and stability of the wave equation with lower terms and boundary damping. Rev. Math. Appl.,1997,18:61-78.
[18]M. M. Cavalcanti, V. N. Domings, Cavalcanti, J. A. Soriano. Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients. EJDE, 1998,1998(8):1-21.
[19]M.M.Cavalcanti, V.N.Domings, Cavalcanti, J. S. Prates Filho, J. A. Soriano. On existence and asymptotic stability of solutions of the degenerate wave equatin with nonlinear boundary conditions. J. Math. Anal. Appl.,2003,281:108-124.
[20]M.M.Cavalcanti, V.N. Domings, Cavalcanti, R. Fukuoka, D. Toundykov. Stabilization of the damped wave equation with Cauchy Ventcel boundary conditions. J. Evol. Equ.,2009,9:143-169.
[21]M. Aassila. A note on the boundary stabilization of a compactly coupled system of wave equations. Appl. Math. Lett.,1999,12:19-24.
[22]G. Kirchhoff. Vorlesuagen Uber Mechanik. Tehbner, Leipzig,1883.
[23]L.A. Medeiros, J. Limaco, S. B. Menezes. Vibrations of elastic strings:Mathematical Aspects, Part One. Journal of Computational Analysis and Applications,2002,4:91-127.
[24]L.A. Medeiros, J. Limaco, S. B. Menezes. Vibrations of elastic strings:Mathematical Aspects, Part Two. Journal of Computational Analysis and Applications,2002,4:211-263.
[25]Z. Y. Zhang, Z. H. Liu, X. J. Miao. Estimate on the dimension of global attractor for nonlinear dissipative Kirchhoff equation. Acta Appl. Math.,2010,110:271-282
[26]Z. Y. Zhang, Z. H. Liu, X. J. Miao, D. M. Yu. On solvability and stabilization of a class of hyperbolic hemivariational inequalities in elasticity. Funkcial. Ekvac,2011, Accepted.
[27]Z. Y. Zhang, Z. H. Liu, X. J. Miao,Y. Z. Chen, Existence and asymptotic behavior of solutions to generalized Kirchhoff equation, Nonlinear Studies,2010,Accepted.
[28]Z. Y. Zhang. Central manifold for the elastic string with dissipative effect. Pacific Journal of Applied Mathematics,2010,2(4):329-343.
[29]K. Ono. On global existence, asymptotic stability and blowing-up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation. Math. Methods Appl. Sci.,1997,20:151-177.
[30]K. Ono. Global existence and decay properties of solutions for some mildly degenerate nonlinear dissipative Kirchhoff strings. Funkcial. Ekvac,1997,40:255-270.
[31]K. Ono. Global existence, decay and blowing-up of solutions for some mildly degenerate nonlinear Kirchhoff strings. J. Differential Equations,1997,137:273-301.
[32]K. Ono. On global existence, asymptotic behavior, and global non-existence of solutions for damped nonlinear wave eqautions of Kirchhoff type in the whole space. Math. Methods Appl. Sci.,2000,23:535-560.
[33]K. Ono. On global existence and asymptotic stability of solutions of mildly degenerat-e nonlinear wave eqautions of Kirchhoff type. Asymptotic Anal.1998,16:299-314.
[34]K. Nishihara. Degenerate qusilinear hyperbolic equation with strong damping. Funkcial. Ekvac,1984,27:125-145.
[35]K. Nishihara. Decay properties of solutions of some qusilinear hyperbolic equation with strong damping. Nonlinear Anal.TMA,21993,21:17-21.
[36]K. Nishihara, K. Ono. Asymptotic behaviors of solutions of some nonlinear oscillatio-n equations with strong damping. Adv. Math. Sci. Appl.,1994,4:285-295.
[37]K.Nishihara,Y.Yamamda. On global solutions of some degenerate qusilinear hyperbolic eqautions with dissipative terms. Funkcial. Ekvac,1990,33:151-159.
[38]N. I. Karachalios, N. M. Stavrakakis. Existence of global attractors for semilinear dissipative wave equations on RN. J. Differential Equations,1999,157:183-205.
[39]N. I. Karachalios, N. M. Stavrakakis. Global existence and blow-up results for some nonlinear wave equations on RN. Adv. Diff. Equations,2001,6:155-174.
[40]N. I. Karachalios, N. M. Stavrakakis. Asymptotic behavior of solutions of some nonl-inearly damped wave equations on RN. Topological Methods in Nonlinear Analysis, 2001,18:73-87.
[41]N. I.Karachalios, N. M. Stavrakakis. Estimates on the dimension of a semilinear dissipative wave equationon RN. Discret and Continuous Dynamical Systems,2002,8: 935-951.
[42]S. Berrimi, S. A. Messaoudi. Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal.,2006,64:2314-2331.
[43]S. A. Messaoudi. General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal.,2008,69:2589-2598.
[44]Z. Y. Zhang. Z. H. Liu, X. J. Miao, Y. Z. Chen. A note on decay properties for the solutions of a class of partial differential equation with memory..J.Appl.Math. Comput. In press (2010)(DOI 10.1007/sl2190-010-0422-7).
[45]M. L. Santos, J. Rerreira, C. A. Raposo. Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary, Abstract Application Analysis.2005,2005(8):901-919.
[46]J. Y. Park, J. R. Kang. Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Acta Appl Math,2010,110:1393-1406.
[47]J. L. Lions. Exact controllability stabilization and perturbations for distributed systems. SIAM J. Control Optim.,1988,30(1):1-68.
[48]J.L. Lions. On some hyperbolic equations with a pressure term, Proceedings of the conference dedicated to Louis Nirenberg held in Trento,Italy, September 3-8,1990. Harlow: Longman Scientic and Technical,Pitman Res. Notes Math. Ser 269,1992,196-208.
[49]J. C. Oliveira, R. C. Chara o. Stabilization of a locally damped incompressible wave equation. J. Math. Anal. Appl.,2005,303:699-725.
[50]M. A. Astraburuaga, R. C. Chara o. Stabilization of the total energy for a system of elasticity with localized dissipation. Differential and Integral Equations,2002,15:1357-1376.
[51]L. R. T. Tebou. Well-posedness and energy decay estimates for the damped wave equation with Ln localizing coefficient. Comm. Partial Differential Equations,1998,23: 1839-1855.
[52]E. Bisognin, V. Bisognin, R. C. Chara o. Uniform stabilization for elastic waves system with highly nonlinear localized dissipation. Portugal. Math.,2003,60:99-124.
[53]M. Sofonea and EL-H. Essoufi, A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal.,2004,9:229-242.
[54]P. Rosenau, J. M. Hyman. Compactons:Solitons with finite wave length. Phys. Rev. Lett.,1993,70:564-567.
[55]J. Quinn, D. L. Russell.Asymptotic stability and energy decay for solutions of hyperbolic equations with boundary damping. Proc. Roy. Soc. Edinburgh,1987,77:97-127.
[56]G. Chen. Control and stabilization for the wave equation in a bounded domain. SIAM J. Control Optim.,1979,17:114-122.
[57]J. E. Lagnese. Decay of solution of wave equations in a bounded region with boundary dissipation. J. Differential Equations,1983,50:163-182.
[58]J. E. Lagnese. Note on boundary stabilization of wave equations. SIAM J. Control Optim.,1988,26:1250-1257.
[59]V. Kmornik, E. Zuazua. A direct method for boundary stabilization of the wave equation. J. Math. Pure Appl.,1990,69:33-54.
[60]M. M. Cavalcanti, J. A. Soriano. On solvability and stability of the wave equation with lower terms and boundary damping. Rev. Math. Appl.,1997,18:61-78.
[61]G. Chen. A note on the stabilization of the wave equation in a bounded domain. SIAM J. Control Optim.,1981,19:197-213.
[62]I. Lasiecka. Stabilization of hyperbolic and parabolic systems with nonlinearly perturbed boundary conditions. J. Differential Equations,1988,75:53-87.
[63]D. L. Russell. Controllability and stability theory for linear partial differential equations:Recent progress and open questions. SIAM Rev.,1978,20:639-739.
[64]R. Triggiani. Wave equation on a bounded domain with boundary dissipation:An operator approach. J. Math. Anal. Appl.,1989,137:438-461.
[65]E. Zuazua. Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim.,1990,28:66-477.
[66]G. Chen, H.Wong. Asymptotic behavior of solutions of the one dimensional wave equation with a nonlinear boundary stabilizer. SIAM J. Control Optim.,1989,27:758-775.
[67]J. E. Lagnese, G. Leugering. Uniform stabilization of nonlinear beam by nonlinear boundary feedback. J. Differential Equations,1991,91:355-388.
[68]Y. You. Energy decay and exact controllability for the Petrovsky equation in a bounded domain. Adv. Appl. Math.,1990,11:372-388.
[69]A. Favini, M. A. Horor, I. Lasiecka, D. Tataru. Global existence and regularity of solutions to a Vonkarman system with nonlinear boundary dissipation. Differential Integral Equations,1996,9:267-269.
[70]I. Lasiecka, D.Tataru. Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential Integral Equations,1993,6:507-533.
[71]V. Georgiev, G. Todorova. Existence of solution of the wave equation with nonlinear damping and source term. J. Differential Equations,1994,109:295-308.
[72]G. C. Gorain. Exponential energy decay estimate for the solutions of internally damped wave equation in a bounded domain. J. Math. Anal. Appl.,1997,216:510-520.
[73]R. A. Adams. Sobolev Space. Academic Press, NewYork,1975.
[74]J. L. Lions, E. Magenes. Problemes Aux Limites Non Homogenes Applications. Dunod, Paris,1968.
[75]P. Martinez. A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim. Calc. Var.,1999,4:419-449.
[76]Z.Y Zhang, X. J. Miao. Global existence and uniform decay for wave equation with dissipative term and boundary damping. Computers and Mathematics with Applications, 2010,59(2):1003-1018.
[77]M. M. Cavalcanti, V. N. D. Cavalcanti, P. Martinez. Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term. J. Differential Equations.2004,203:119-158.
[78]M. M. Cavalcanti, V. N. D. Cavalcanti, I. Lasiecka. Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J. Differential Equations,2007,236:407-459.
[79]A.Guesmia. A new approach of stabilization of nondissipative distributed systems. C. R. Acad. Sci. Paris, Serie I,2001,332:633-636.
[80]A.Guesmia. A new approach of stabilization of nondissipative distributed systems. SIAM J. Control and Optimization,2003,42(1):24-52.
[81]M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, J. A. Soriano. Asymptotic stability of the wave equation on compact manifolds and locally distributed damping:a sharp result. Arch. Ration. Mech. Anal.,2010,197:925-964.
[82]M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, J. A. Soriano. Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-A sharp result.Trans. Amer. Math. Soc.,2009,361:4561-4580.
[83]M. M. Cavalcanti, V. N. D. Cavalcanti, J. S. Prates Filho, J. A. Soriano. Existence and uniform decay rates forviscoelastic problems with nonlinear boundary damping. Differential and Integral Equations,2001,14(1):85-116.
[84]V. Georgiev, G. Todorova. Existence of solutions of the wave equation with nonlinear damping and source terms. J. Differential Equations,1994,109:295-308.
[85]Z. Y. Zhang, Z. H. Liu, X. J. Miao, Y. Z. Chen. Stability analysis of heat flow with boundary time-varying delay effect. Nonlinear Anal. TMA,2010,73:1878-1889
[86]H.Brezis. Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Pub. Co., Amsterdam,1973.
[87]V. Barbu. Nonlinear semigroups and differential equations in Banach spaces, Ed. Academici,1976.
[88]R. E. Showalter. Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. AMS, Providence,1997.
[89]Z. Denkowski, S. Migorski, N.S. Papageorgiou. An Introduction to Nonlinear Analysis:Applications. Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York,2003.
[90]I. Chueshov, M. Eller, I. Lasiecka. Attractors and their structure for semilinear wave equations with nonlinear boundary dissipation. Bol. Soc. Paran. Mat.,2004,22(1):38-57. I.Chueshov, M. Eller, I. Lasiecka. On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Comm. Partial Differential Equations, 2002,27(9-10):1901-1951.
[91]L. Bociu, I. Lasiecka. Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping. J. Differential Equations,2010,249 (3):654-683.
[92]M. Fabrizio, A. Morro.Mathematic problems in linear viscoelasticity, SIAM Studies in Applied Mathematics. Vol.12, SIAM, Pennsylvania,1992.
[93]M. Renaray, W. J. Hrusa, J. A. Nohel. Mathematic problems in viscoelasticity. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol.35, Longman Scientific and Technical, Harlow, John Wiley and Sons, New York.1987.
[94]J. M. Ball. Initial-boundary value problem for an extensible beam. J. Math. Anal. Appl.,1973,42:61-90.
[95]J. M. Ball. Stabilty theory for an extensible beam. J. Diff. Eqs.,1973,14:399-418.
[96]P. Biller. Remark on the decay for generalized damped string and beam equations. Nonlinear Anal.,1986,10:839-842.
[97]E. H. Brito. Decay estimates for generalized damped extensible string and beam equation. Nonlinear Anal.,1984,8:1489-1496.
[98]S. Gerbi, B. Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete and Continuous Dynamical Systems,2011, repri nt.
[99]S. A. Messaoudi. Decay of the solution energy for a nonlinearly damped wave equation, Arab. J. Sci, Eng.,2001,26:63-68.
[100]A. Benaissa, S. A. Messaoudi. Exponential decay of the solutions of a nonlinearly damped wave equation. NoDEA,2005,12:391-399.
[101]A. Haraux, E. Zuazua. Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal.,1999,150:191-206.
[102]R. Ikehata. Imroved decay rates for solutions to one-dimensional linear and semilinear dissipative wave equations in all space. J. Math. Anal. Appl.,2003,272:557-570.
[103]R. Ikehata. New decay rates for linear damped wave equations and applications to nonlinear problem. Math. Methods Appl. Sci.,2004,27(4):865-889.
[104]R. Ikehata. Local energy decay for linear wave equations with non-compactly supported initial data. Math. Methods Appl. Sci.,2004,27(16):1881-1892.
[105]W. J. Liu. General decay of solutions of a nonlinear system of viscoelastic equations. Acta Appl. Anal.,2010,110:153-165.
[106]M. M. Cavalcanti, V. N. D. Cavalcanti, J. A. Soriano.Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. EJD E,2002, 44(1):1-14.
[107]S. Berrimi, S. A. Messaoudi. Exponential decay of solutions to a viscoelastic equation withnonlinear localized damping, EJDE,2004,88:1-10.
[108]M. M. Cavalcanti, H. P. Oquendo. Frictional verus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim.,2003,42(4):1310-1324.
[109]S. A. Messaoudi. General decay of solutions of a viscoelastic equations. J. Math. Anal. Appl.,2008,34:1457-1467.
[110]W. J. Liu. General decay of solutions to a viscoelastic equation with nonlinear localized damping. Annales Acaem. Sci. Fennic. Math.,2009,34:291-302.
[111]W. J. Liu. Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping. J.Appl.Math. Comput.,2010,32:59-68.
[112]N.-E. Tatar. Exponential decay for a viscoelastic problem with a singular kenerl. ZAMP,2009,60:640-650.
[113]S. A. Messaoudi, N.-E. Tatar. Uniform stabilization of solutions of a nonlinear system of viscoelastic equations. Appl. Anal.,2008,87(3):247-263.
[114]M. M. Cavalcanti, V. N. D. Cavalcanti, J. Ferreira. Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci.,2001,24: 1043-1053.
[115]S. A. Messaoudi, N.-E. Tatar. Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math. Methods.Appl. Sci.,2007,30:665-680.
[116]S. A. Messaoudi, N.-E. Tatar. Exponential or polynomial decay for a quasilinear viscoelastic equation. Nonlinear Anal.,2008,68:785-793.
[117]X. S. Han, M. X. Wang. General decay of energy for a viscoelastic equation with nonlinear damping. Math. Methods Appl. Sci.,2009,32(3):346-358.
[118]W. J. Liu. General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. J. Math. Phys.,2009,50:113506.
[119]X. S. Han, M. X. Wang.General decay of energy for a viscoelastic equation with nonlinear damping. Journal of the Franklin Institute,2010,347(5):806-817.
[120]X. S. Han, M. X. Wang. Global existence and asymptotic behavior for a couple hyperbolic system with localized damping. Nonlinear Anal. TMA,2010,72(2):965-986.
[121]X. S. Han, M. X. Wang. Energy decay rate for a couple hyperbolic system with nonlinear damping. Nonlinear Anal. TMA,2009,70:3264-3272.
[122]S. A. Messaoudi. Blow-up and global existence in a nonlinear viscoelastic wave equation.Math. Nachr.,2003,260:58-66.
[123]S. A. Messaoudi. Blow-up in a nonlinearly damped wave equation. Math. Nachr. 2001,231:1-7.
[124]Y. J. Wang. A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy. Appl. Math. Lett.,2009,22(9):1394-1400.
[125]H. T. Song, C. K. Zhong.Blow-up of solutions of a nonlinear viscoelastic wave equation. Nonlinear Anal. RWA,2010, 11(5):3877-3883.
[126]S. A. Messaoudi. Blow-up of positive-initial energy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl.,2006,320:902-905.
[127]A. Aassila, M. M. Cavalcanti, J. A. Soriano. Asymptotic stability and energy decay rates for solutions of the wave equations with memory in star-shaped domains. SIAM J. Control andOptim.,2000,38(5):1581-1602.
[128]S. T. Wu, Blow-up of solutions for an integro-differential equation with a nonlinear source. EJDE,2006,2006(45):1-9.
[129]M. M. Cavalcanti, V. N. Domings, Cavalcanti, J. S. Prates Filho, J. A. Soriano. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differential Integral Equations,2001,14:85-116.
[130]X. S. Han, M. X. Wang. Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Nonlinear Anal. TMA,2009,70(9):3090-3098.
[131]W. J. Liu. Uniform decay of solutions for a quasilinear system of viscoelastic equations, Nonlinear Anal. TMA,2009,71:2257-2267.
[132]W. J. Liu. General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal. TMA,2010,73:1890-1904.
[133]S. A. Messaoudi, M. I. Mustafa. On the control of solutions of viscoelastic equations with boundary feedback. Nonlinear Anal. RWA.,2009,10:3132-3140.
[134]S. A. Messaoudi, A. Soufyane.Boundary stabilization of solutions of a nonlinear system of Timoshenko type. Nonlinear Anal.,2007,67:2107-2121.
[135]S. A. Messaoudi, A. Soufyane. General decay of solutions of a wave equation with a boundary control of memory type. Nonlinear Anal. RWA,2010,11:2896-2904
[136]S. A. Messaoudi, M. I. Mustafa. On convexity for energy decay rates of a viscoelastic equation with boundary feedback. Nonlinear Anal.,2010,72:3602-3611.
[137]L. Q. Lu, S. J. Li, S. G. Chai. On a viscoelastic equation with nonlinear boundary damping and source terms:Global existence and decay of the solution. Nonlinear Anal. RWA,2011,12(l):295-303.
[138]T. G. Ha. On viscoealstic wave equation with nonlinear boundary damping and source term. Communications on Pure and Applied Anal.,2010,9(6):1543-1576.
[139]R.C.Coimbra, G.P. Menzala. On Huygen's Principle and perturbed elastic waves Differential and Integral Equation,1992,5:631-646.
[140]A. Haraux. E. Zuazua. Decay estimates for some semilinear damped hyperbolic problems. Arch. Ration. Mech. Anal.,1988,100:191-206.
[141]E. Zuazua. Stability and decay for a class of nonlinear hyperbolic problems. Asym ptotic Anal.,1998,1:161-185.
[142]E. Zuazua. Exponential decay for the semilinear wave equation with locally distributing damping. Comm. Partial Differential Equations,1995,15:513-529.
[143]Y. Yamada. On the decay of solutions for some nonlinear evolution equation of seconder. Nagoya Math. J.,1979,73:67-98.
[144]V. Komornik. Decay estimates for the wave equation with internal damping. Int. Ser. Numer. Math.,1994,118:253-266.
[145]M. Nakao. A difference inequality and its applications to nonlinear evolutions. J. Math. Soc. Japan.,1978,30:747-762.
[146]M. Nakao. Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equation. Math. Z.,1991,206:265-276.
[147]M. Nakao. Global and periodic solutions for the nonlinear wave equations with some localized nonlinear dissipation. J. Differential Equations,2003,190:81-107.
[148]M. Nakao. Global attractors for the nonlinear wave equations with nonlinear dissipative terms. J. Differential Equations,2006,227:204-229.
[149]M. Nakao. Energy decay to the Cauchy problem of nonlinear Klein-Gordon equations with asublinear dissipative term. Adv. Math. Sci. Appl.,2009,19:479-501.
[150]M. Nakao, I. H. Jung. Energy decay for the wave equation in exterior domains with some half-linear dissipation. Differential and Integral Equations,2003,16:927-948.
[151]M. Nakao, K. Ono. Global existence to the Cauchy problem of the semilinear wave equation with a nonlinear dissipation. Funkcial. Ekvac,1995,38:417-431.
[152]M. Nakao. Remarks on the energy decay for nonlinear wave equations with localized dissipative terms. Nonlinear. Anal. TMA,2010,73(7):2158-2169.
[153]V. Girault, P.A. Raviart. Finite element methods for Navier-Stokes equations. Springer-Verlag, Berlin,1986.
[154]P. Constantin, C. Foias. Navier-Stokes equations. University of Chicago Press, Chicago,1988.
[155]E. Coddington, N. Levinson. Theory of ordinary differential equations. McGraw-Hill, NewYork,1955.
[156]J. L. Lions. Quelques methodes de resolution des problemes aux limites non lineaires. Gauthier-Villars, Paris,1969.
[157]R. Temam. Navier-Stokes equations. Stud. Math. Appl., Vol.2, North-Holland, Amsterdam,1984.
[158]J. L. Lions. Problemes aux limites dans les equations aux derivees partielles. Press. Univ. Montreal,1962.
[159]J. L. Lions, E. Magenes. Problemes Aux Limites Non Homogenes Applications. Dunod, Paris,1968.
[160]M.V. Ferreira, G.P. Menzala. Uniform stabilization of an electromagnetic-elasticity problem in exterior domains. Discrete Contin. Dyn. Syst.,2007,18(4):719-746
[161]C.R. Luz,G.P. Menzala. Large time behavior of anisotropic electro-magnetic/elasticity equations in exterior domains. J. Math. Anal. Appl.,2009,359:464-481
[162]W.J.Liu, M. Krsti'c. Strong stabilization of the system of linear elasticity by a Dirchlet boundary feedback. IMA J. Appl. Math.,2000,65:109-121
[163]W.J.Liu.The exponential stabilization of the higher-dimensional linear thermo-viscoelasticity. J.Math. Pures Appl.,1998,77:355-386
[164]W.J.Liu, G. H. Williams.The exponential stability of the problem of transmission of the wave equation. Bull. Australian Math. Soc.,1998,57:305-327
[165]W.J.Liu, E. Zuazua. Uniform stabilization of the higher dimensional system of thermoelasticity with an on linear boundary feedback. Quart. Appl. Math.,2001,59(2):269-314
[166]W.J.Liu.Partial exact contrllabiltiy and exponential stability of the higher-dimension al linear thermoelasticity. ESAIM Control Optim. Calc.Var.,1998,3:23-48
[167]Z.Y.Zhang,Z.H.Liu,X.J.Miao,Y.Z.Chen.Global existence and uniform stabilization of a generalized dissipative Klein-Gordon equation type with boundary damping. J. Math. Phys.,2011,52(2):023502
[168]P.F.Yao. Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation.Chin.Ann.Math.Ser.B,2010,31(1):59-70
[169]P.F.Yao. On the observability inequalities for the exact controllability of the wave equation with variable coeffcients.SIAMJ.ControlOptim.,1999,37 (6):1568-1599
[170]S.J.Feng,D.X.Feng.Boundary stabilization of wave equations with variable coefficients. Sci. China Ser.A,2001,44(3):345-350
[171]I. Hamchi.Uniform decay rates fo rsecond-order hyperbolic equations with variable coefficients. Asymptot. Anal.,2008,57(1-2):71-82
[172]I. Lasiecka, R. Triggiani, P.F.Yao. Inverse/observability estimates for second-order hyperbolic equations with variable systems. J. Math.Anal.Appl.,1999,235:13-57
[173]P.F.Yao. Observability inequalities for the shallow shell. SIAM J. Control Optim. 2000,38(6):1729-1756
[174]P.F.Yao.Global smooth solutions for the quasilinear wave equation with boundary dissipation. J. Diff. Eqs.,2007,241:62-93
[175]P.F.Yao.Boundary controllability for the quasilinear wave equation. Appl. Math. Optim.,2010,61:191-233
[176]Z.F.Zhang,P.F.Yao. Global smooth solutions of the quasilinear wave equation with internal velocity feedbacks. SIAMJ. Control Optim.,2008,47(4):2044-2077
[177]S.Chai,Y.Guo,P.F.Yao. Boundary feedback stabilization of shallow shells. SIAM J. Control Optim.,2003,42(1):239-259
[178]S.Chai,P.F.Yao.Observability inequalities for thin shells. Sci. China Ser. A,2003,46 (3):300-311
[179]S.J.Feng,D.X.Feng.Nonlinear internal damping of wave equations with variable coecients. Acta Math. Sin., Engl.Ser.,2004,20(6):1057-1072
[180]P.F.Yao.Observability Inequalities for the Euler-Bernoulli Plate with Variable Coefficients. Contemporary Mathematies,2000,268:383-406
[181]Y.X.Guo,P.F.Yao.On boundary stability of wave equations with variable coefficients. Acta Math.Sin.,Engl.Ser.,2002,18(4):589-598
[182]F.L.Huang.Strong asymptotic stability of linear dynamical systems in Banach spaces. J. Diff. Eqs.,1985,104:307-324
[183]J. Pruss.On the spectrum of Co-semigroups. Trans. Amer. Math. Soc.,1984,284:847-857
[184]A. Pazy. Semi groups of linear operators and applications to partial differential equations. Springer,New York,1983
[185]L.C. Evans.Partial differential equations. Rhode Island, American Methematical Society, Providence,1998
[186]L. Gearhart.Spectral theory for contration semigroups on Hilbert space.Trans. Amer. Math. Soc.,1978,236:385-394
[187]Z. Liu, S. Zheng. Semigroups associated with dissipative systems. Chapman and Hall/CRC,1999
[2]Z. Y. Zhang. New exact traveling wave solutions for the nonlinear Klein-Gordon equation. Turk. J Phys,2008,32:235-240.
[3]C. S. Morawetz. The decay of solutions of the exterior initial-boundary-value problem for the wave equation. Comm. Pure Appl. Math.,1961,14:561-568.
[4]G. Dassios. Local energy decay for scattering of elastic waves. J. Differential Equations,1983,49:124-141.
[5]M. Nakao. Decay of solutions of the wave equations with a local nonlinear dissipation. Math. Ann.,1996,305:403-417.
[6]M. Nakao. Decay of solutions of the wave equation with some localized dissipations. Nonlinear Anal.,1997,30:3775-3784.
[7]M. Nakao. Stabilization of local energy in an exterior domain for the wave equation with a localized dissipation. J. Differential Equations,1998,148:388-406.
[8]M. Nakao. Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations. Math. Z.,2001,238:781-797.
[9]R. Ikehata. Energy decay of solutions for the semilinear dissipative wave equations in an exterior domain. Funkcial. Ekvac.,2001,44:487-499.
[10]R. Ikehata. Small data global existence of solutions for dissipative wave equations in an exterior domain. Funkcial. Ekvac.,2002,45:259-269.
[11]R. Ikehata. Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain. J. Differential Equations,2003,188:390-405.
[12]R. Ikehata. Global existence of solutions for 2-D semilinear wave equations with dissipation localized near infinity in an exterior domain. Math. Methods Appl. Sci.. 2006,29:479-496.
[13]Y. H. Kang, M. J. Lee, I. H. Jung, Sharp energy decay estimates for the wave equation with a local degenerate dissipation. Comput. Math. Appl.,2009.57:21-57.
[14]E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures Appl.,1991,70:513-529.
[15]A.B. Aliev, A.Kh. Khanmamedov. Energy estimates for solutions of the mixed problem for linear seconder order hyperbolic equation. Math. Note,1996.59(4):345-349.
[16]M. M. Cavalcanti, V. N. Domings, Cavalcanti, J. A. Soriano. On solvability and stability of solutions to hyperbolic-parabolic problems with boundary damping. In Atas diXIX Congresso B.rasileiro de Mathematica Applicadae Computational,1996.
[17]M. M. Cavalcanti, J. A. Soriano. On solvability and stability of the wave equation with lower terms and boundary damping. Rev. Math. Appl.,1997,18:61-78.
[18]M. M. Cavalcanti, V. N. Domings, Cavalcanti, J. A. Soriano. Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients. EJDE, 1998,1998(8):1-21.
[19]M.M.Cavalcanti, V.N.Domings, Cavalcanti, J. S. Prates Filho, J. A. Soriano. On existence and asymptotic stability of solutions of the degenerate wave equatin with nonlinear boundary conditions. J. Math. Anal. Appl.,2003,281:108-124.
[20]M.M.Cavalcanti, V.N. Domings, Cavalcanti, R. Fukuoka, D. Toundykov. Stabilization of the damped wave equation with Cauchy Ventcel boundary conditions. J. Evol. Equ.,2009,9:143-169.
[21]M. Aassila. A note on the boundary stabilization of a compactly coupled system of wave equations. Appl. Math. Lett.,1999,12:19-24.
[22]G. Kirchhoff. Vorlesuagen Uber Mechanik. Tehbner, Leipzig,1883.
[23]L.A. Medeiros, J. Limaco, S. B. Menezes. Vibrations of elastic strings:Mathematical Aspects, Part One. Journal of Computational Analysis and Applications,2002,4:91-127.
[24]L.A. Medeiros, J. Limaco, S. B. Menezes. Vibrations of elastic strings:Mathematical Aspects, Part Two. Journal of Computational Analysis and Applications,2002,4:211-263.
[25]Z. Y. Zhang, Z. H. Liu, X. J. Miao. Estimate on the dimension of global attractor for nonlinear dissipative Kirchhoff equation. Acta Appl. Math.,2010,110:271-282
[26]Z. Y. Zhang, Z. H. Liu, X. J. Miao, D. M. Yu. On solvability and stabilization of a class of hyperbolic hemivariational inequalities in elasticity. Funkcial. Ekvac,2011, Accepted.
[27]Z. Y. Zhang, Z. H. Liu, X. J. Miao,Y. Z. Chen, Existence and asymptotic behavior of solutions to generalized Kirchhoff equation, Nonlinear Studies,2010,Accepted.
[28]Z. Y. Zhang. Central manifold for the elastic string with dissipative effect. Pacific Journal of Applied Mathematics,2010,2(4):329-343.
[29]K. Ono. On global existence, asymptotic stability and blowing-up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation. Math. Methods Appl. Sci.,1997,20:151-177.
[30]K. Ono. Global existence and decay properties of solutions for some mildly degenerate nonlinear dissipative Kirchhoff strings. Funkcial. Ekvac,1997,40:255-270.
[31]K. Ono. Global existence, decay and blowing-up of solutions for some mildly degenerate nonlinear Kirchhoff strings. J. Differential Equations,1997,137:273-301.
[32]K. Ono. On global existence, asymptotic behavior, and global non-existence of solutions for damped nonlinear wave eqautions of Kirchhoff type in the whole space. Math. Methods Appl. Sci.,2000,23:535-560.
[33]K. Ono. On global existence and asymptotic stability of solutions of mildly degenerat-e nonlinear wave eqautions of Kirchhoff type. Asymptotic Anal.1998,16:299-314.
[34]K. Nishihara. Degenerate qusilinear hyperbolic equation with strong damping. Funkcial. Ekvac,1984,27:125-145.
[35]K. Nishihara. Decay properties of solutions of some qusilinear hyperbolic equation with strong damping. Nonlinear Anal.TMA,21993,21:17-21.
[36]K. Nishihara, K. Ono. Asymptotic behaviors of solutions of some nonlinear oscillatio-n equations with strong damping. Adv. Math. Sci. Appl.,1994,4:285-295.
[37]K.Nishihara,Y.Yamamda. On global solutions of some degenerate qusilinear hyperbolic eqautions with dissipative terms. Funkcial. Ekvac,1990,33:151-159.
[38]N. I. Karachalios, N. M. Stavrakakis. Existence of global attractors for semilinear dissipative wave equations on RN. J. Differential Equations,1999,157:183-205.
[39]N. I. Karachalios, N. M. Stavrakakis. Global existence and blow-up results for some nonlinear wave equations on RN. Adv. Diff. Equations,2001,6:155-174.
[40]N. I. Karachalios, N. M. Stavrakakis. Asymptotic behavior of solutions of some nonl-inearly damped wave equations on RN. Topological Methods in Nonlinear Analysis, 2001,18:73-87.
[41]N. I.Karachalios, N. M. Stavrakakis. Estimates on the dimension of a semilinear dissipative wave equationon RN. Discret and Continuous Dynamical Systems,2002,8: 935-951.
[42]S. Berrimi, S. A. Messaoudi. Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal.,2006,64:2314-2331.
[43]S. A. Messaoudi. General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal.,2008,69:2589-2598.
[44]Z. Y. Zhang. Z. H. Liu, X. J. Miao, Y. Z. Chen. A note on decay properties for the solutions of a class of partial differential equation with memory..J.Appl.Math. Comput. In press (2010)(DOI 10.1007/sl2190-010-0422-7).
[45]M. L. Santos, J. Rerreira, C. A. Raposo. Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary, Abstract Application Analysis.2005,2005(8):901-919.
[46]J. Y. Park, J. R. Kang. Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Acta Appl Math,2010,110:1393-1406.
[47]J. L. Lions. Exact controllability stabilization and perturbations for distributed systems. SIAM J. Control Optim.,1988,30(1):1-68.
[48]J.L. Lions. On some hyperbolic equations with a pressure term, Proceedings of the conference dedicated to Louis Nirenberg held in Trento,Italy, September 3-8,1990. Harlow: Longman Scientic and Technical,Pitman Res. Notes Math. Ser 269,1992,196-208.
[49]J. C. Oliveira, R. C. Chara o. Stabilization of a locally damped incompressible wave equation. J. Math. Anal. Appl.,2005,303:699-725.
[50]M. A. Astraburuaga, R. C. Chara o. Stabilization of the total energy for a system of elasticity with localized dissipation. Differential and Integral Equations,2002,15:1357-1376.
[51]L. R. T. Tebou. Well-posedness and energy decay estimates for the damped wave equation with Ln localizing coefficient. Comm. Partial Differential Equations,1998,23: 1839-1855.
[52]E. Bisognin, V. Bisognin, R. C. Chara o. Uniform stabilization for elastic waves system with highly nonlinear localized dissipation. Portugal. Math.,2003,60:99-124.
[53]M. Sofonea and EL-H. Essoufi, A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal.,2004,9:229-242.
[54]P. Rosenau, J. M. Hyman. Compactons:Solitons with finite wave length. Phys. Rev. Lett.,1993,70:564-567.
[55]J. Quinn, D. L. Russell.Asymptotic stability and energy decay for solutions of hyperbolic equations with boundary damping. Proc. Roy. Soc. Edinburgh,1987,77:97-127.
[56]G. Chen. Control and stabilization for the wave equation in a bounded domain. SIAM J. Control Optim.,1979,17:114-122.
[57]J. E. Lagnese. Decay of solution of wave equations in a bounded region with boundary dissipation. J. Differential Equations,1983,50:163-182.
[58]J. E. Lagnese. Note on boundary stabilization of wave equations. SIAM J. Control Optim.,1988,26:1250-1257.
[59]V. Kmornik, E. Zuazua. A direct method for boundary stabilization of the wave equation. J. Math. Pure Appl.,1990,69:33-54.
[60]M. M. Cavalcanti, J. A. Soriano. On solvability and stability of the wave equation with lower terms and boundary damping. Rev. Math. Appl.,1997,18:61-78.
[61]G. Chen. A note on the stabilization of the wave equation in a bounded domain. SIAM J. Control Optim.,1981,19:197-213.
[62]I. Lasiecka. Stabilization of hyperbolic and parabolic systems with nonlinearly perturbed boundary conditions. J. Differential Equations,1988,75:53-87.
[63]D. L. Russell. Controllability and stability theory for linear partial differential equations:Recent progress and open questions. SIAM Rev.,1978,20:639-739.
[64]R. Triggiani. Wave equation on a bounded domain with boundary dissipation:An operator approach. J. Math. Anal. Appl.,1989,137:438-461.
[65]E. Zuazua. Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim.,1990,28:66-477.
[66]G. Chen, H.Wong. Asymptotic behavior of solutions of the one dimensional wave equation with a nonlinear boundary stabilizer. SIAM J. Control Optim.,1989,27:758-775.
[67]J. E. Lagnese, G. Leugering. Uniform stabilization of nonlinear beam by nonlinear boundary feedback. J. Differential Equations,1991,91:355-388.
[68]Y. You. Energy decay and exact controllability for the Petrovsky equation in a bounded domain. Adv. Appl. Math.,1990,11:372-388.
[69]A. Favini, M. A. Horor, I. Lasiecka, D. Tataru. Global existence and regularity of solutions to a Vonkarman system with nonlinear boundary dissipation. Differential Integral Equations,1996,9:267-269.
[70]I. Lasiecka, D.Tataru. Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential Integral Equations,1993,6:507-533.
[71]V. Georgiev, G. Todorova. Existence of solution of the wave equation with nonlinear damping and source term. J. Differential Equations,1994,109:295-308.
[72]G. C. Gorain. Exponential energy decay estimate for the solutions of internally damped wave equation in a bounded domain. J. Math. Anal. Appl.,1997,216:510-520.
[73]R. A. Adams. Sobolev Space. Academic Press, NewYork,1975.
[74]J. L. Lions, E. Magenes. Problemes Aux Limites Non Homogenes Applications. Dunod, Paris,1968.
[75]P. Martinez. A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim. Calc. Var.,1999,4:419-449.
[76]Z.Y Zhang, X. J. Miao. Global existence and uniform decay for wave equation with dissipative term and boundary damping. Computers and Mathematics with Applications, 2010,59(2):1003-1018.
[77]M. M. Cavalcanti, V. N. D. Cavalcanti, P. Martinez. Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term. J. Differential Equations.2004,203:119-158.
[78]M. M. Cavalcanti, V. N. D. Cavalcanti, I. Lasiecka. Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J. Differential Equations,2007,236:407-459.
[79]A.Guesmia. A new approach of stabilization of nondissipative distributed systems. C. R. Acad. Sci. Paris, Serie I,2001,332:633-636.
[80]A.Guesmia. A new approach of stabilization of nondissipative distributed systems. SIAM J. Control and Optimization,2003,42(1):24-52.
[81]M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, J. A. Soriano. Asymptotic stability of the wave equation on compact manifolds and locally distributed damping:a sharp result. Arch. Ration. Mech. Anal.,2010,197:925-964.
[82]M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, J. A. Soriano. Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-A sharp result.Trans. Amer. Math. Soc.,2009,361:4561-4580.
[83]M. M. Cavalcanti, V. N. D. Cavalcanti, J. S. Prates Filho, J. A. Soriano. Existence and uniform decay rates forviscoelastic problems with nonlinear boundary damping. Differential and Integral Equations,2001,14(1):85-116.
[84]V. Georgiev, G. Todorova. Existence of solutions of the wave equation with nonlinear damping and source terms. J. Differential Equations,1994,109:295-308.
[85]Z. Y. Zhang, Z. H. Liu, X. J. Miao, Y. Z. Chen. Stability analysis of heat flow with boundary time-varying delay effect. Nonlinear Anal. TMA,2010,73:1878-1889
[86]H.Brezis. Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Pub. Co., Amsterdam,1973.
[87]V. Barbu. Nonlinear semigroups and differential equations in Banach spaces, Ed. Academici,1976.
[88]R. E. Showalter. Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. AMS, Providence,1997.
[89]Z. Denkowski, S. Migorski, N.S. Papageorgiou. An Introduction to Nonlinear Analysis:Applications. Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York,2003.
[90]I. Chueshov, M. Eller, I. Lasiecka. Attractors and their structure for semilinear wave equations with nonlinear boundary dissipation. Bol. Soc. Paran. Mat.,2004,22(1):38-57. I.Chueshov, M. Eller, I. Lasiecka. On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Comm. Partial Differential Equations, 2002,27(9-10):1901-1951.
[91]L. Bociu, I. Lasiecka. Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping. J. Differential Equations,2010,249 (3):654-683.
[92]M. Fabrizio, A. Morro.Mathematic problems in linear viscoelasticity, SIAM Studies in Applied Mathematics. Vol.12, SIAM, Pennsylvania,1992.
[93]M. Renaray, W. J. Hrusa, J. A. Nohel. Mathematic problems in viscoelasticity. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol.35, Longman Scientific and Technical, Harlow, John Wiley and Sons, New York.1987.
[94]J. M. Ball. Initial-boundary value problem for an extensible beam. J. Math. Anal. Appl.,1973,42:61-90.
[95]J. M. Ball. Stabilty theory for an extensible beam. J. Diff. Eqs.,1973,14:399-418.
[96]P. Biller. Remark on the decay for generalized damped string and beam equations. Nonlinear Anal.,1986,10:839-842.
[97]E. H. Brito. Decay estimates for generalized damped extensible string and beam equation. Nonlinear Anal.,1984,8:1489-1496.
[98]S. Gerbi, B. Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete and Continuous Dynamical Systems,2011, repri nt.
[99]S. A. Messaoudi. Decay of the solution energy for a nonlinearly damped wave equation, Arab. J. Sci, Eng.,2001,26:63-68.
[100]A. Benaissa, S. A. Messaoudi. Exponential decay of the solutions of a nonlinearly damped wave equation. NoDEA,2005,12:391-399.
[101]A. Haraux, E. Zuazua. Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal.,1999,150:191-206.
[102]R. Ikehata. Imroved decay rates for solutions to one-dimensional linear and semilinear dissipative wave equations in all space. J. Math. Anal. Appl.,2003,272:557-570.
[103]R. Ikehata. New decay rates for linear damped wave equations and applications to nonlinear problem. Math. Methods Appl. Sci.,2004,27(4):865-889.
[104]R. Ikehata. Local energy decay for linear wave equations with non-compactly supported initial data. Math. Methods Appl. Sci.,2004,27(16):1881-1892.
[105]W. J. Liu. General decay of solutions of a nonlinear system of viscoelastic equations. Acta Appl. Anal.,2010,110:153-165.
[106]M. M. Cavalcanti, V. N. D. Cavalcanti, J. A. Soriano.Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. EJD E,2002, 44(1):1-14.
[107]S. Berrimi, S. A. Messaoudi. Exponential decay of solutions to a viscoelastic equation withnonlinear localized damping, EJDE,2004,88:1-10.
[108]M. M. Cavalcanti, H. P. Oquendo. Frictional verus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim.,2003,42(4):1310-1324.
[109]S. A. Messaoudi. General decay of solutions of a viscoelastic equations. J. Math. Anal. Appl.,2008,34:1457-1467.
[110]W. J. Liu. General decay of solutions to a viscoelastic equation with nonlinear localized damping. Annales Acaem. Sci. Fennic. Math.,2009,34:291-302.
[111]W. J. Liu. Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping. J.Appl.Math. Comput.,2010,32:59-68.
[112]N.-E. Tatar. Exponential decay for a viscoelastic problem with a singular kenerl. ZAMP,2009,60:640-650.
[113]S. A. Messaoudi, N.-E. Tatar. Uniform stabilization of solutions of a nonlinear system of viscoelastic equations. Appl. Anal.,2008,87(3):247-263.
[114]M. M. Cavalcanti, V. N. D. Cavalcanti, J. Ferreira. Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci.,2001,24: 1043-1053.
[115]S. A. Messaoudi, N.-E. Tatar. Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math. Methods.Appl. Sci.,2007,30:665-680.
[116]S. A. Messaoudi, N.-E. Tatar. Exponential or polynomial decay for a quasilinear viscoelastic equation. Nonlinear Anal.,2008,68:785-793.
[117]X. S. Han, M. X. Wang. General decay of energy for a viscoelastic equation with nonlinear damping. Math. Methods Appl. Sci.,2009,32(3):346-358.
[118]W. J. Liu. General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. J. Math. Phys.,2009,50:113506.
[119]X. S. Han, M. X. Wang.General decay of energy for a viscoelastic equation with nonlinear damping. Journal of the Franklin Institute,2010,347(5):806-817.
[120]X. S. Han, M. X. Wang. Global existence and asymptotic behavior for a couple hyperbolic system with localized damping. Nonlinear Anal. TMA,2010,72(2):965-986.
[121]X. S. Han, M. X. Wang. Energy decay rate for a couple hyperbolic system with nonlinear damping. Nonlinear Anal. TMA,2009,70:3264-3272.
[122]S. A. Messaoudi. Blow-up and global existence in a nonlinear viscoelastic wave equation.Math. Nachr.,2003,260:58-66.
[123]S. A. Messaoudi. Blow-up in a nonlinearly damped wave equation. Math. Nachr. 2001,231:1-7.
[124]Y. J. Wang. A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy. Appl. Math. Lett.,2009,22(9):1394-1400.
[125]H. T. Song, C. K. Zhong.Blow-up of solutions of a nonlinear viscoelastic wave equation. Nonlinear Anal. RWA,2010, 11(5):3877-3883.
[126]S. A. Messaoudi. Blow-up of positive-initial energy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl.,2006,320:902-905.
[127]A. Aassila, M. M. Cavalcanti, J. A. Soriano. Asymptotic stability and energy decay rates for solutions of the wave equations with memory in star-shaped domains. SIAM J. Control andOptim.,2000,38(5):1581-1602.
[128]S. T. Wu, Blow-up of solutions for an integro-differential equation with a nonlinear source. EJDE,2006,2006(45):1-9.
[129]M. M. Cavalcanti, V. N. Domings, Cavalcanti, J. S. Prates Filho, J. A. Soriano. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differential Integral Equations,2001,14:85-116.
[130]X. S. Han, M. X. Wang. Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Nonlinear Anal. TMA,2009,70(9):3090-3098.
[131]W. J. Liu. Uniform decay of solutions for a quasilinear system of viscoelastic equations, Nonlinear Anal. TMA,2009,71:2257-2267.
[132]W. J. Liu. General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal. TMA,2010,73:1890-1904.
[133]S. A. Messaoudi, M. I. Mustafa. On the control of solutions of viscoelastic equations with boundary feedback. Nonlinear Anal. RWA.,2009,10:3132-3140.
[134]S. A. Messaoudi, A. Soufyane.Boundary stabilization of solutions of a nonlinear system of Timoshenko type. Nonlinear Anal.,2007,67:2107-2121.
[135]S. A. Messaoudi, A. Soufyane. General decay of solutions of a wave equation with a boundary control of memory type. Nonlinear Anal. RWA,2010,11:2896-2904
[136]S. A. Messaoudi, M. I. Mustafa. On convexity for energy decay rates of a viscoelastic equation with boundary feedback. Nonlinear Anal.,2010,72:3602-3611.
[137]L. Q. Lu, S. J. Li, S. G. Chai. On a viscoelastic equation with nonlinear boundary damping and source terms:Global existence and decay of the solution. Nonlinear Anal. RWA,2011,12(l):295-303.
[138]T. G. Ha. On viscoealstic wave equation with nonlinear boundary damping and source term. Communications on Pure and Applied Anal.,2010,9(6):1543-1576.
[139]R.C.Coimbra, G.P. Menzala. On Huygen's Principle and perturbed elastic waves Differential and Integral Equation,1992,5:631-646.
[140]A. Haraux. E. Zuazua. Decay estimates for some semilinear damped hyperbolic problems. Arch. Ration. Mech. Anal.,1988,100:191-206.
[141]E. Zuazua. Stability and decay for a class of nonlinear hyperbolic problems. Asym ptotic Anal.,1998,1:161-185.
[142]E. Zuazua. Exponential decay for the semilinear wave equation with locally distributing damping. Comm. Partial Differential Equations,1995,15:513-529.
[143]Y. Yamada. On the decay of solutions for some nonlinear evolution equation of seconder. Nagoya Math. J.,1979,73:67-98.
[144]V. Komornik. Decay estimates for the wave equation with internal damping. Int. Ser. Numer. Math.,1994,118:253-266.
[145]M. Nakao. A difference inequality and its applications to nonlinear evolutions. J. Math. Soc. Japan.,1978,30:747-762.
[146]M. Nakao. Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equation. Math. Z.,1991,206:265-276.
[147]M. Nakao. Global and periodic solutions for the nonlinear wave equations with some localized nonlinear dissipation. J. Differential Equations,2003,190:81-107.
[148]M. Nakao. Global attractors for the nonlinear wave equations with nonlinear dissipative terms. J. Differential Equations,2006,227:204-229.
[149]M. Nakao. Energy decay to the Cauchy problem of nonlinear Klein-Gordon equations with asublinear dissipative term. Adv. Math. Sci. Appl.,2009,19:479-501.
[150]M. Nakao, I. H. Jung. Energy decay for the wave equation in exterior domains with some half-linear dissipation. Differential and Integral Equations,2003,16:927-948.
[151]M. Nakao, K. Ono. Global existence to the Cauchy problem of the semilinear wave equation with a nonlinear dissipation. Funkcial. Ekvac,1995,38:417-431.
[152]M. Nakao. Remarks on the energy decay for nonlinear wave equations with localized dissipative terms. Nonlinear. Anal. TMA,2010,73(7):2158-2169.
[153]V. Girault, P.A. Raviart. Finite element methods for Navier-Stokes equations. Springer-Verlag, Berlin,1986.
[154]P. Constantin, C. Foias. Navier-Stokes equations. University of Chicago Press, Chicago,1988.
[155]E. Coddington, N. Levinson. Theory of ordinary differential equations. McGraw-Hill, NewYork,1955.
[156]J. L. Lions. Quelques methodes de resolution des problemes aux limites non lineaires. Gauthier-Villars, Paris,1969.
[157]R. Temam. Navier-Stokes equations. Stud. Math. Appl., Vol.2, North-Holland, Amsterdam,1984.
[158]J. L. Lions. Problemes aux limites dans les equations aux derivees partielles. Press. Univ. Montreal,1962.
[159]J. L. Lions, E. Magenes. Problemes Aux Limites Non Homogenes Applications. Dunod, Paris,1968.
[160]M.V. Ferreira, G.P. Menzala. Uniform stabilization of an electromagnetic-elasticity problem in exterior domains. Discrete Contin. Dyn. Syst.,2007,18(4):719-746
[161]C.R. Luz,G.P. Menzala. Large time behavior of anisotropic electro-magnetic/elasticity equations in exterior domains. J. Math. Anal. Appl.,2009,359:464-481
[162]W.J.Liu, M. Krsti'c. Strong stabilization of the system of linear elasticity by a Dirchlet boundary feedback. IMA J. Appl. Math.,2000,65:109-121
[163]W.J.Liu.The exponential stabilization of the higher-dimensional linear thermo-viscoelasticity. J.Math. Pures Appl.,1998,77:355-386
[164]W.J.Liu, G. H. Williams.The exponential stability of the problem of transmission of the wave equation. Bull. Australian Math. Soc.,1998,57:305-327
[165]W.J.Liu, E. Zuazua. Uniform stabilization of the higher dimensional system of thermoelasticity with an on linear boundary feedback. Quart. Appl. Math.,2001,59(2):269-314
[166]W.J.Liu.Partial exact contrllabiltiy and exponential stability of the higher-dimension al linear thermoelasticity. ESAIM Control Optim. Calc.Var.,1998,3:23-48
[167]Z.Y.Zhang,Z.H.Liu,X.J.Miao,Y.Z.Chen.Global existence and uniform stabilization of a generalized dissipative Klein-Gordon equation type with boundary damping. J. Math. Phys.,2011,52(2):023502
[168]P.F.Yao. Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation.Chin.Ann.Math.Ser.B,2010,31(1):59-70
[169]P.F.Yao. On the observability inequalities for the exact controllability of the wave equation with variable coeffcients.SIAMJ.ControlOptim.,1999,37 (6):1568-1599
[170]S.J.Feng,D.X.Feng.Boundary stabilization of wave equations with variable coefficients. Sci. China Ser.A,2001,44(3):345-350
[171]I. Hamchi.Uniform decay rates fo rsecond-order hyperbolic equations with variable coefficients. Asymptot. Anal.,2008,57(1-2):71-82
[172]I. Lasiecka, R. Triggiani, P.F.Yao. Inverse/observability estimates for second-order hyperbolic equations with variable systems. J. Math.Anal.Appl.,1999,235:13-57
[173]P.F.Yao. Observability inequalities for the shallow shell. SIAM J. Control Optim. 2000,38(6):1729-1756
[174]P.F.Yao.Global smooth solutions for the quasilinear wave equation with boundary dissipation. J. Diff. Eqs.,2007,241:62-93
[175]P.F.Yao.Boundary controllability for the quasilinear wave equation. Appl. Math. Optim.,2010,61:191-233
[176]Z.F.Zhang,P.F.Yao. Global smooth solutions of the quasilinear wave equation with internal velocity feedbacks. SIAMJ. Control Optim.,2008,47(4):2044-2077
[177]S.Chai,Y.Guo,P.F.Yao. Boundary feedback stabilization of shallow shells. SIAM J. Control Optim.,2003,42(1):239-259
[178]S.Chai,P.F.Yao.Observability inequalities for thin shells. Sci. China Ser. A,2003,46 (3):300-311
[179]S.J.Feng,D.X.Feng.Nonlinear internal damping of wave equations with variable coecients. Acta Math. Sin., Engl.Ser.,2004,20(6):1057-1072
[180]P.F.Yao.Observability Inequalities for the Euler-Bernoulli Plate with Variable Coefficients. Contemporary Mathematies,2000,268:383-406
[181]Y.X.Guo,P.F.Yao.On boundary stability of wave equations with variable coefficients. Acta Math.Sin.,Engl.Ser.,2002,18(4):589-598
[182]F.L.Huang.Strong asymptotic stability of linear dynamical systems in Banach spaces. J. Diff. Eqs.,1985,104:307-324
[183]J. Pruss.On the spectrum of Co-semigroups. Trans. Amer. Math. Soc.,1984,284:847-857
[184]A. Pazy. Semi groups of linear operators and applications to partial differential equations. Springer,New York,1983
[185]L.C. Evans.Partial differential equations. Rhode Island, American Methematical Society, Providence,1998
[186]L. Gearhart.Spectral theory for contration semigroups on Hilbert space.Trans. Amer. Math. Soc.,1978,236:385-394
[187]Z. Liu, S. Zheng. Semigroups associated with dissipative systems. Chapman and Hall/CRC,1999