非线性波动方程非齐次问题长时间存在性及其应用
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摘要
在本文中,我们将讨论非线性波动方程非齐次问题小初值经典解的长时间存在性.首先我们利用压缩映像原理和广义能量积分证明了系统稳态解的存在唯一性,并借助波动算子的平移不变性、空间旋转不变性和伸缩不变性,通过连续性讨论,证明了三维空间中含非齐次项拟线性波动方程柯西问题经典解的几乎整体存在性和三维空间中具有零形式拟线性波动方程组星形区域非齐次边界外问题经典解的整体存在性.利用类似的思想,我们还得到了受外力各向同性超弹性体经典解的几乎整体存在性及当非线性项满足零条件时系统的经典解当时间趋于无穷大时会收敛于系统的稳态解.其次,我们证明了三次半线性波动方程(Klein-Gordon方程)经典解的指数阶能稳,利用该结论和构造性方法,还得到了系统在一般区域上的全局精确边界能控性,特别地当区域为星形附加区域时,所得到的边界控制函数只需作用在边界的一个相对开子集上.我们的证明方法适用于只要非线性项满足“好符号’增长条件的一般情形.
此外,在附录一,我们借助Morawetz乘子建立了两维空间中扰动线性波动方程的Morawetz能量估计和提升的Morawetz能量估计,利用表现空间方向衰减的带权Sobolev不等式,在只使用波动算子的平移不变性和空间旋转不变性的情况下,得到了两维非线性波动方程星形区域外问题经典解的生命跨度的下界估计,这填补了两维非线性波动外问题的空白.在附录二,我们结合Klainerman-Sideris途径和Alinhac所引入的能表现好导数比一般导数具有更好性质的提高的能量估计,证明了两维拟线性波动方程组多波速几乎对角系统,当非线性项满足第一零条件、不满足第二零条件时经典解几乎整体存在；当非线性项同时满足第一、第二零条件时经典解整体存在.
本文的具体组织如下：
在第一章中,我们介绍非线性波动方程齐次问题的研究历史,并对全文的内容做了个简单的概述.
在第二、第三章中,我们分别得到了三维空间中含非齐次项拟线性波动方程柯西问题经典解的几乎整体存在性和具有零形式拟线性波动方程组星形区域非齐次边界外问题经典解的整体存在性.
在第四章中,我们研究了受外力各向同性超弹性体柯西问题经典解的几乎整体存在性和整体存在性.
在第五章中,我们得到了三次半线性波动方程和三次半线性Klein-Gordon方程在一般区域上的全局精确边界能控性,特别地当区域为星形附加区域时,我们得到了边界控制函数只需作用在边界的一个相对开子集上
在附录一,我们得到了两维空间中的非线性波动方程星形区域外问题经典解的生命跨度的下界估计.在附录二,我们得到了两维空间中的拟线性波动方程组多波速几乎对角系统经典解的几乎整体存在性和整体存在性.
In this Ph.D thesis, we will study the long-time existence of classical solutions to inhomogeneous problem of nonlinear wave equations.
First, with the help of the existence and uniqueness of stationary solutions, by using the classical invariance of wave equations under translations, spatial rotations and changes of scale, we establish the almost global existence of solutions to three dimensional nonlin-ear wave equations with quadratic, divergence-form nonlinearities and time-independent inhomogeneous terms. By a similar idea, we also prove the global existence of classical solu-tions to systems of nonlinear wave equations with multiple speeds exterior to star-shaped regions satisfying time-independent inhomogeneous boundary conditions, provided that nonlinear terms obey the null condition. This idea can be also applied to the system of homogeneous, isotropic hyperelasticity with time-independent external force. Precisely we obtain the almost global existence result of classical solution to this system and when the nonlinear term obeys a type of nonresonance or null condition, we obtain the global existence for this system.
Next, we establish the exponential decay of energy for cubic semi-linear wave equa-tions and cubic Klein-Gordon equations in three space dimensions with some boundary condition by the multiplier method and prove the global exact boundary controllability by the constructive method. Especially when the region is star-complemented, it is ob-tained that the control function only need to be applied on a relatively open subset of the boundary. Our proofs can also be applied to semilinear wave equations and semilinear Klein-Gordon equations with nonlinear terms satisfying "good-sign" growth conditions.
At appendix one, for the perturbed wave equation in two space dimensions, we es-tablish the scaling Morawetz energy estimate by the Morawetz multiplier and introduce a kind of improved scaling Morawetz energy estimate that displays the better properties of good derivative (?)t+(?)r. We also obtain the long-time existence result for small data solutions of two dimensional nonlinear wave equations exterior to star-shaped regions, by only using the invariance of the wave operator under translations and spatial rotations. At appendix two, combining an improved energy inequality and Klainerman-Sideris approach, we obtain the global existence (resp. almost global existence) for the Cauchy problem of almost-diagonal systems of nonlinear wave equations with multiple propagation speeds in two space dimensions satisfying both null conditions (resp. the first null condition).
The arrangement of the thesis is as follows:
First of all, in Chapter 1, a brief introduction is given for the history on the study of the long-time existence of homogeneous nonlinear wave equations and for the results in this thesis. In Chapter 2, we obtain the almost global existence of solutions to Cauchy problem of three dimensional nonlinear wave equations with quadratic, divergence-form nonlinear-ities and time-independent inhomogeneous terms. We also prove the global existence of classical solutions to three dimensional null-form wave equations exterior to star-shaped regions satisfying time-independent inhomogeneous boundary conditions in Chapter 3. In Chapter 4, for the system of homogeneous, isotropic hyperelasticity with time-independent external force, we obtain the almost global existence result and the global existence result when the nonlinear term obeys a type of null condition.
In Chapter 5, we establish the global exact boundary controllability for cubic semi-linear wave equations and cubic semilinear Klein-Gordon equations in the original regions. Especially when the region is star-complemented, we obtain that the control function is only applied on a relatively open subset of the boundary.
In Appendix one, we obtain the long-time existence result for small data solutions of two dimensional nonlinear wave equations exterior to star-shaped regions. In Appendix two, we prove the global existence (resp. almost global existence) for the Cauchy problem of almost-diagonal systems of two dimensional nonlinear wave equations with multiple speeds satisfying both null conditions (resp. the first null condition).
此外,在附录一,我们借助Morawetz乘子建立了两维空间中扰动线性波动方程的Morawetz能量估计和提升的Morawetz能量估计,利用表现空间方向衰减的带权Sobolev不等式,在只使用波动算子的平移不变性和空间旋转不变性的情况下,得到了两维非线性波动方程星形区域外问题经典解的生命跨度的下界估计,这填补了两维非线性波动外问题的空白.在附录二,我们结合Klainerman-Sideris途径和Alinhac所引入的能表现好导数比一般导数具有更好性质的提高的能量估计,证明了两维拟线性波动方程组多波速几乎对角系统,当非线性项满足第一零条件、不满足第二零条件时经典解几乎整体存在；当非线性项同时满足第一、第二零条件时经典解整体存在.
本文的具体组织如下：
在第一章中,我们介绍非线性波动方程齐次问题的研究历史,并对全文的内容做了个简单的概述.
在第二、第三章中,我们分别得到了三维空间中含非齐次项拟线性波动方程柯西问题经典解的几乎整体存在性和具有零形式拟线性波动方程组星形区域非齐次边界外问题经典解的整体存在性.
在第四章中,我们研究了受外力各向同性超弹性体柯西问题经典解的几乎整体存在性和整体存在性.
在第五章中,我们得到了三次半线性波动方程和三次半线性Klein-Gordon方程在一般区域上的全局精确边界能控性,特别地当区域为星形附加区域时,我们得到了边界控制函数只需作用在边界的一个相对开子集上
在附录一,我们得到了两维空间中的非线性波动方程星形区域外问题经典解的生命跨度的下界估计.在附录二,我们得到了两维空间中的拟线性波动方程组多波速几乎对角系统经典解的几乎整体存在性和整体存在性.
In this Ph.D thesis, we will study the long-time existence of classical solutions to inhomogeneous problem of nonlinear wave equations.
First, with the help of the existence and uniqueness of stationary solutions, by using the classical invariance of wave equations under translations, spatial rotations and changes of scale, we establish the almost global existence of solutions to three dimensional nonlin-ear wave equations with quadratic, divergence-form nonlinearities and time-independent inhomogeneous terms. By a similar idea, we also prove the global existence of classical solu-tions to systems of nonlinear wave equations with multiple speeds exterior to star-shaped regions satisfying time-independent inhomogeneous boundary conditions, provided that nonlinear terms obey the null condition. This idea can be also applied to the system of homogeneous, isotropic hyperelasticity with time-independent external force. Precisely we obtain the almost global existence result of classical solution to this system and when the nonlinear term obeys a type of nonresonance or null condition, we obtain the global existence for this system.
Next, we establish the exponential decay of energy for cubic semi-linear wave equa-tions and cubic Klein-Gordon equations in three space dimensions with some boundary condition by the multiplier method and prove the global exact boundary controllability by the constructive method. Especially when the region is star-complemented, it is ob-tained that the control function only need to be applied on a relatively open subset of the boundary. Our proofs can also be applied to semilinear wave equations and semilinear Klein-Gordon equations with nonlinear terms satisfying "good-sign" growth conditions.
At appendix one, for the perturbed wave equation in two space dimensions, we es-tablish the scaling Morawetz energy estimate by the Morawetz multiplier and introduce a kind of improved scaling Morawetz energy estimate that displays the better properties of good derivative (?)t+(?)r. We also obtain the long-time existence result for small data solutions of two dimensional nonlinear wave equations exterior to star-shaped regions, by only using the invariance of the wave operator under translations and spatial rotations. At appendix two, combining an improved energy inequality and Klainerman-Sideris approach, we obtain the global existence (resp. almost global existence) for the Cauchy problem of almost-diagonal systems of nonlinear wave equations with multiple propagation speeds in two space dimensions satisfying both null conditions (resp. the first null condition).
The arrangement of the thesis is as follows:
First of all, in Chapter 1, a brief introduction is given for the history on the study of the long-time existence of homogeneous nonlinear wave equations and for the results in this thesis. In Chapter 2, we obtain the almost global existence of solutions to Cauchy problem of three dimensional nonlinear wave equations with quadratic, divergence-form nonlinear-ities and time-independent inhomogeneous terms. We also prove the global existence of classical solutions to three dimensional null-form wave equations exterior to star-shaped regions satisfying time-independent inhomogeneous boundary conditions in Chapter 3. In Chapter 4, for the system of homogeneous, isotropic hyperelasticity with time-independent external force, we obtain the almost global existence result and the global existence result when the nonlinear term obeys a type of null condition.
In Chapter 5, we establish the global exact boundary controllability for cubic semi-linear wave equations and cubic semilinear Klein-Gordon equations in the original regions. Especially when the region is star-complemented, we obtain that the control function is only applied on a relatively open subset of the boundary.
In Appendix one, we obtain the long-time existence result for small data solutions of two dimensional nonlinear wave equations exterior to star-shaped regions. In Appendix two, we prove the global existence (resp. almost global existence) for the Cauchy problem of almost-diagonal systems of two dimensional nonlinear wave equations with multiple speeds satisfying both null conditions (resp. the first null condition).
引文
[1]方道元,非线性波动方程,杭州：浙江大学出版社,2008.
[2]李大潜和秦铁虎,物理学与偏微分方程(上册),北京：高等教育出版社,1997.
[3]李大潜和秦铁虎,物理学与偏微分方程(下册),北京：高等教育出版社,2000.
[4]Agemi, R., Blow-up of solutions to nonlinear wave equations in two space dimensions, Manuscripta Math.73 (1991), no.2,153-162.
[5]Agemi, R., Global existence of nonlinear elastic waves, Invent. Math.142 (2000),225-250.
[6]Alinhac, S., An example of blowup at infinity for a quasilinear wave equation, Asterisque 284 (2003),1-91.
[7]Alinhac, S., Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. Ⅱ., Acta Math.182 (1999), no.1,1-23.
[8]Alinhac, S., Blowup of small data solutions for a quasilinear wave equation in two space dimensions, Ann. of Math. (2) 149 (1999), no.1,97-127.
[9]Alinhac, S., On the Morawetz-Keel-Smith-Sogge inequality for the wave equation on a curved background, Publ. Res. Inst. Math. Sci.42 (2006), no.3,705-720.
[10]Alinhac, S., The null condition for quasilinear wave equations in two space dimensions Ⅰ, Invent. Math.145 (2001),597-618.
[11]Alinhac, S., The null condition for quasilinear wave equations in two space dimensions Ⅱ, Amer. J. Math.123 (2001), no.6,1071-1101.
[12]Chen, G., Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl.58 (1979),249-273.
[13]Christodoulou, D., Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math.39 (1986), no.2,267-282.
[14]Dehman, B., Lebeau. G. and Zuazua, E., Stabilization and control for subcritical semilinear wave equation, Ann. Sci. ecole Norm. Sup. (4) 36 (2003).525-551.
[15]Du, Y., Metcalfe, J., Sogge, C. D. and Zhou. Y., Concerning the Strauss conjecture and almost global existence for nonlinear Dirichlet-wave equations in 4-dimensions, Comm. Partial Differential Equations 33 (2008), no.7-9,1487-1506.
[16]Du, Y. and Zhou, Y., The lifespan for nonlinear wave equation outside of star-shaped obstacle in thrce space dimensions, Comm. Partial Differential Equations 33 (2008), no.7-9,1455-1486.
[17]Du, Y. and Zhou, Y., The lifespan for nonlinear wave equations with multiple propagation speeds in 3D, Nonlinear Anal.68 (2008), no.7,1793-1801.
[18]Evans, L. C., Partial differential equations, Grad. Stud. Math.19, AMS, Providence, RI, 1998.
[19]Fu, X., Yong, J. and Zhang, X., Exact controllability for multidimensional semilinear hyper-bolic equations, SIAM J. Control Optim.46 (2007),1578-1614.
[20]Hidano, K., An elementary proof of global or almost global existence for quasi-linear wave equations, Tohoku Math. J. (2) 56 (2004), no.2,271-287.
[21]Hidano, K., Scattering problem for the nonlinear wave equation in the finite energy and conformal charge space, J. Funct. Anal.187 (2001), no.2,274-307.
[22]Hidano, K. and Yokoyama, K., A remark on the almost global existence theorems of Keel, Smith and Sogge, Funkcial. Ekvac.48 (2005), no.1,1-34.
[23]Hormander, L., Lectures on nonlinear hyperbolic differential equations, Math. Appl.26, Springer Verlag,1997.
[24]Hoshiga, A., The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in 2-dimensional space, Funkcial. Ekvac.49 (2006),357-384.
[25]Hoshiga, A., The initial value problems for quasi-linear wave equations in two space dimen-sions with small data, Adv. Math. Sci. Appl.5 (1995),67-89.
[26]Hoshiga, A. and Kubo, H., Global solvability for systems of nonlinear wave equations with multiple speeds in two space dimensions, Differential Integral Equations 17 (2004),593-622.
[27]John, F., Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm,. Pure Appl. Math.41 (1988),615-666.
[28]John, F., Blow-up for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math.34 (1981),29-51.
[29]John, F., Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math.28 (1979), no.1-3,235-268.
[30]John, F., Formation of singularities in elastic waves, Lecture Notes in Physics 195 Springer-Verlag, New York (1984),194-210.
[31]John, F., Nonlinear wave equations, Formation of singularities, University Lectures Series 2, A.M.S., Providence, RI,1990.
[32]John, F. and Klainerman, S., Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math.37 (1984), no.4,443-455.
[33]Katayama, S., Global existence for systems of nonlinear wave equations in two space dimen-sions, Publ. RIMS, Kyoto Univ.29 (1993),1021-1041.
[34]Keel, M., Smith, H. F. and Sogge, C. D., Almost global existence for quasilinear wave equa-tions in three space dimensions, J. Amer. Math. Soc.17 (2004), no.1,109-153.
[35]Keel, M., Smith, H. F. and Sogge, C. D., Almost global existence for some semilinear wave equations, J. Anal Math.87 (2002),265-279.
[36]Keel, M., Smith, H. F. and Sogge, C. D., Global existence for a quasilinear wave equation outside of star-shaped domains, J. Funct. Anal.189 (2002), no.1,155-226.
[37]Keel, M., Smith, H. F. and Sogge, C. D., On global existence for nonlinear wave equations outside of convex obstacles, Amer. J. Math.122 (2000), no.4,805-842.
[38]Klainerman, S., Global existence for nonlinear wave equations, Comm. Pure Appl. Math.33 (1980), no.1,43-101.
[39]Klainerman, S., Long-time behavior of solutions to nonlinear evolution equations, Arch. Ra-tional Mech. Anal.78 (1982), no.1,73-98.
[40]Klainerman, S., On "almost global" solutions to quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math.36 (1983), no.3,325-344.
[41]Klainerman, S., Remarks on the global Sobolev inequalities in the Minkowski space Rn+1, Comm. Pure Appl. Math.40 (1987),111-117.
[42]Klainerman, S., Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math.38 (1985),321-332.
[43]Klainerman, S., The null condition and global existence to nonlinear wave equations, Nonlin-ear systems of partial differential equations in applied mathematics, Part 1,293-326, Lectures in Appl. Math.,23, Amer. Math. Soc., Providence, RI,1986.
[44]Klainerman, S. and Ponce, G., Global small amplitude solutions to nonlinear evolution equa-tions, Comm. Pure Appl. Math.36 (1983), no.1,133-141.
[45]Klainerman, S. and Sideris, T. C., On almost global existence for nonrelativistic wave equa-tions in 3D, Comm. Pure Appl. Math.49 (1996), no.3,307-321.
[46]Komornik, V., Exact controllability and stabilization, The multiplier method, Masson, Pairs and John Wiley and Sons, Chicester,1994.
[47]Lasiecka, I. and Triggiani, R., Exact controllability of semilinear abstract systems with ap-plication to waves and plates boundary control problems, Appl. Math. Optim.23 (1991), 109-154.
[48]Lax, P. D., Morawetz, C. S. and Phillips, R. S., Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Comm. Pure Appl. Math.16 (1963),477-486.
[49]Lei, Z., Nonlinear elastic waves and viscoelastic fluids, manuscript.
[50]Li, T. T. and Chen, Y. M., Global classical solutions for nonlinear evolution equations, Long-man Scientific and Technical UK.,1992.
[51]Li, T. T. and Yu, X., Life-span of classical solutions to fully nonlinear wave equations, Comm. Partial Differential Equations 16 (1991), no.6-7,909-940.
[52]Li, T. T. and Zhou, Y., Life-span of classical solutions to fully nonlinear wave equations-Ⅱ, Nonlinear Anal.19 (1992),833-853.
[53]Li, T. T. and Zhou, Y., Life-span of classical solutions to nonlinear wave equations in two space dimensions, J. Math. Pures Appl.73 (1994),223-249.
[54]Li, T. T. and Zhou, Y., Life-span of classical solutions to nonlinear wave equations in two space dimensions Ⅱ,J. Partial Diff. Eqs.6(1993),17-38.
[55]Li, T. T. and Zhou, Y., Nonlinear stability for two-space-dimensional wave equations with higher-order perturbations. Nonlinear World 1 (1994), no.1,35-58.
[56]Lindblad, H., Global solutions of quasilinear wave equations, Amer. J. Math.130 (2008), no. 1,115-157.
[57]Lindblad, H., Global solutions of nonlinear wave equations, Comm. Pure Appl. Math.45 (1992), no.9,1063-1096.
[58]Lindblad, H., On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math.43 (1990), no.4,445-472.
[59]Lions, J. L., Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev.30 (1988),1-68.
[60]Lions, J. L. and Magenes, E., Problemes aux Limites Non homogenes et applications. Vol. 1., Dunod, Paris,1968.
[61]Metcalfe, J., Elastic waves in exterior domains. I. Almost global existence, Int. Math. Res. Not. (2006).
[62]Metcalfe, J., Nakamura, M. and Sogge, C. D., Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains, Forum Math.17 (2005), no.1, 133-168.
[63]Metcalfe, J. and Sogge, C. D., Global existence for Dirichlet-wave equations with quadratic nonlinearities in high dimensions, Math. Ann.336 (2006), no.2,391-420.
[64]Metcalfe, J. and Sogge, C. D., Global existence for high dimensional quasilinear wave equa-tions exterior to star-shaped obstacles, Discrete Contin. Dyn. Sys., to appear.
[65]Metcalfe, J. and Sogge, C.D., Global existence of null-form wave equations in exterior do-mains, Math. Z.256 (2007), no.3,521-549.
[66]Metcalfe, J. and Sogge, C. D., Hyperbolic trapped rays and global existence of quasilinear wave equations, Invent. Math.159 (2005), no.1,75-117.
[67]Metcalfe, J. and Sogge, C. D., Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods, SIAM J. Math. Anal.38 (2006), no.1,188-209.
[68]Metcalfe, J. and Thomases, B., Elastic waves in exterior domains. Ⅱ. Global existence with a null structure, Int. Math. Res. Not. (2007).
[69]Morawetz, C. S., The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math.14 (1961),561-568.
[70]Morawetz, C. S., Ralston, J. V. and Strauss, W. A., Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math.30 (1977), no.4,447-508.
[71]Russell, D. L., Controllability and stabilizability theory for linear partial differential equations, SIAM Rev.20 (1978),639-739.
[72]Russell, D. L., Exact boundary value controllability theorems for wave and heat processes in star-complemented regions, Differential games and control theory,291-319.
[73]Shibata, Y. and Tsutsumi, Y., On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z.191 (1986), no.2,165-199.
[74]Sideris, T. C., Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2) 151 (2000), no.2,849-874.
[75]Sideris, T. C., The null condition and global existence of nonlinear elastic waves, Invent. Math.123 (1996),323-342.
[76]Sideris, T. C. and Tu, S.-Y., Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal.33 (2001), no.2,477-488.
[77]Tahvildar-Zadeh, A. S., Relativistic and nonrelativistic elastodynamics with small shear strains, Ann. Inst. H. Poincare-Phys. Theory 69 (1998),275-307.
[78]Taylor, M. E., Partial differential equations. I. basic theory, Appl. Math. Sci.115, Springer-Verlag, New York,1996.
[79]Zhang, X. and Zuazua, E., Exact controllability of the semi-linear wave equation, in Un-solved Problems in Mathematical Systems and Control Theory, edited by Blondel, V. D. and Megretski, A., Princeton University Press, Princeton, NS, (2004),173-178.
[80]Zhou, Y., Blow up of solutions to the Cauchy problem for nonlinear wave equations, Chinese Ann. Math. Ser. B 22 (2001), no.3,275-280.
[81]Zhou, Y. and Lei, Z., Local exact boundary controllability for nonlinear wave equations, SIAM J. Control Optim.46 (2007),1022-1051.
[82]Zuazua, E., Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincare Anal. Non Lineaire 10 (1993),109-129.
[83]Zuazua, E., Exact Controllability for the semilinear wave equations, J. Math. Pures Appl. (9) 69 (1990),1-31.
[84]Zuazua, E., Exponential decay for semilinear wave equations with localized damping, Comm. Partial Differential Equations 15 (1990),205-235.
[2]李大潜和秦铁虎,物理学与偏微分方程(上册),北京：高等教育出版社,1997.
[3]李大潜和秦铁虎,物理学与偏微分方程(下册),北京：高等教育出版社,2000.
[4]Agemi, R., Blow-up of solutions to nonlinear wave equations in two space dimensions, Manuscripta Math.73 (1991), no.2,153-162.
[5]Agemi, R., Global existence of nonlinear elastic waves, Invent. Math.142 (2000),225-250.
[6]Alinhac, S., An example of blowup at infinity for a quasilinear wave equation, Asterisque 284 (2003),1-91.
[7]Alinhac, S., Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. Ⅱ., Acta Math.182 (1999), no.1,1-23.
[8]Alinhac, S., Blowup of small data solutions for a quasilinear wave equation in two space dimensions, Ann. of Math. (2) 149 (1999), no.1,97-127.
[9]Alinhac, S., On the Morawetz-Keel-Smith-Sogge inequality for the wave equation on a curved background, Publ. Res. Inst. Math. Sci.42 (2006), no.3,705-720.
[10]Alinhac, S., The null condition for quasilinear wave equations in two space dimensions Ⅰ, Invent. Math.145 (2001),597-618.
[11]Alinhac, S., The null condition for quasilinear wave equations in two space dimensions Ⅱ, Amer. J. Math.123 (2001), no.6,1071-1101.
[12]Chen, G., Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl.58 (1979),249-273.
[13]Christodoulou, D., Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math.39 (1986), no.2,267-282.
[14]Dehman, B., Lebeau. G. and Zuazua, E., Stabilization and control for subcritical semilinear wave equation, Ann. Sci. ecole Norm. Sup. (4) 36 (2003).525-551.
[15]Du, Y., Metcalfe, J., Sogge, C. D. and Zhou. Y., Concerning the Strauss conjecture and almost global existence for nonlinear Dirichlet-wave equations in 4-dimensions, Comm. Partial Differential Equations 33 (2008), no.7-9,1487-1506.
[16]Du, Y. and Zhou, Y., The lifespan for nonlinear wave equation outside of star-shaped obstacle in thrce space dimensions, Comm. Partial Differential Equations 33 (2008), no.7-9,1455-1486.
[17]Du, Y. and Zhou, Y., The lifespan for nonlinear wave equations with multiple propagation speeds in 3D, Nonlinear Anal.68 (2008), no.7,1793-1801.
[18]Evans, L. C., Partial differential equations, Grad. Stud. Math.19, AMS, Providence, RI, 1998.
[19]Fu, X., Yong, J. and Zhang, X., Exact controllability for multidimensional semilinear hyper-bolic equations, SIAM J. Control Optim.46 (2007),1578-1614.
[20]Hidano, K., An elementary proof of global or almost global existence for quasi-linear wave equations, Tohoku Math. J. (2) 56 (2004), no.2,271-287.
[21]Hidano, K., Scattering problem for the nonlinear wave equation in the finite energy and conformal charge space, J. Funct. Anal.187 (2001), no.2,274-307.
[22]Hidano, K. and Yokoyama, K., A remark on the almost global existence theorems of Keel, Smith and Sogge, Funkcial. Ekvac.48 (2005), no.1,1-34.
[23]Hormander, L., Lectures on nonlinear hyperbolic differential equations, Math. Appl.26, Springer Verlag,1997.
[24]Hoshiga, A., The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in 2-dimensional space, Funkcial. Ekvac.49 (2006),357-384.
[25]Hoshiga, A., The initial value problems for quasi-linear wave equations in two space dimen-sions with small data, Adv. Math. Sci. Appl.5 (1995),67-89.
[26]Hoshiga, A. and Kubo, H., Global solvability for systems of nonlinear wave equations with multiple speeds in two space dimensions, Differential Integral Equations 17 (2004),593-622.
[27]John, F., Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm,. Pure Appl. Math.41 (1988),615-666.
[28]John, F., Blow-up for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math.34 (1981),29-51.
[29]John, F., Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math.28 (1979), no.1-3,235-268.
[30]John, F., Formation of singularities in elastic waves, Lecture Notes in Physics 195 Springer-Verlag, New York (1984),194-210.
[31]John, F., Nonlinear wave equations, Formation of singularities, University Lectures Series 2, A.M.S., Providence, RI,1990.
[32]John, F. and Klainerman, S., Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math.37 (1984), no.4,443-455.
[33]Katayama, S., Global existence for systems of nonlinear wave equations in two space dimen-sions, Publ. RIMS, Kyoto Univ.29 (1993),1021-1041.
[34]Keel, M., Smith, H. F. and Sogge, C. D., Almost global existence for quasilinear wave equa-tions in three space dimensions, J. Amer. Math. Soc.17 (2004), no.1,109-153.
[35]Keel, M., Smith, H. F. and Sogge, C. D., Almost global existence for some semilinear wave equations, J. Anal Math.87 (2002),265-279.
[36]Keel, M., Smith, H. F. and Sogge, C. D., Global existence for a quasilinear wave equation outside of star-shaped domains, J. Funct. Anal.189 (2002), no.1,155-226.
[37]Keel, M., Smith, H. F. and Sogge, C. D., On global existence for nonlinear wave equations outside of convex obstacles, Amer. J. Math.122 (2000), no.4,805-842.
[38]Klainerman, S., Global existence for nonlinear wave equations, Comm. Pure Appl. Math.33 (1980), no.1,43-101.
[39]Klainerman, S., Long-time behavior of solutions to nonlinear evolution equations, Arch. Ra-tional Mech. Anal.78 (1982), no.1,73-98.
[40]Klainerman, S., On "almost global" solutions to quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math.36 (1983), no.3,325-344.
[41]Klainerman, S., Remarks on the global Sobolev inequalities in the Minkowski space Rn+1, Comm. Pure Appl. Math.40 (1987),111-117.
[42]Klainerman, S., Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math.38 (1985),321-332.
[43]Klainerman, S., The null condition and global existence to nonlinear wave equations, Nonlin-ear systems of partial differential equations in applied mathematics, Part 1,293-326, Lectures in Appl. Math.,23, Amer. Math. Soc., Providence, RI,1986.
[44]Klainerman, S. and Ponce, G., Global small amplitude solutions to nonlinear evolution equa-tions, Comm. Pure Appl. Math.36 (1983), no.1,133-141.
[45]Klainerman, S. and Sideris, T. C., On almost global existence for nonrelativistic wave equa-tions in 3D, Comm. Pure Appl. Math.49 (1996), no.3,307-321.
[46]Komornik, V., Exact controllability and stabilization, The multiplier method, Masson, Pairs and John Wiley and Sons, Chicester,1994.
[47]Lasiecka, I. and Triggiani, R., Exact controllability of semilinear abstract systems with ap-plication to waves and plates boundary control problems, Appl. Math. Optim.23 (1991), 109-154.
[48]Lax, P. D., Morawetz, C. S. and Phillips, R. S., Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Comm. Pure Appl. Math.16 (1963),477-486.
[49]Lei, Z., Nonlinear elastic waves and viscoelastic fluids, manuscript.
[50]Li, T. T. and Chen, Y. M., Global classical solutions for nonlinear evolution equations, Long-man Scientific and Technical UK.,1992.
[51]Li, T. T. and Yu, X., Life-span of classical solutions to fully nonlinear wave equations, Comm. Partial Differential Equations 16 (1991), no.6-7,909-940.
[52]Li, T. T. and Zhou, Y., Life-span of classical solutions to fully nonlinear wave equations-Ⅱ, Nonlinear Anal.19 (1992),833-853.
[53]Li, T. T. and Zhou, Y., Life-span of classical solutions to nonlinear wave equations in two space dimensions, J. Math. Pures Appl.73 (1994),223-249.
[54]Li, T. T. and Zhou, Y., Life-span of classical solutions to nonlinear wave equations in two space dimensions Ⅱ,J. Partial Diff. Eqs.6(1993),17-38.
[55]Li, T. T. and Zhou, Y., Nonlinear stability for two-space-dimensional wave equations with higher-order perturbations. Nonlinear World 1 (1994), no.1,35-58.
[56]Lindblad, H., Global solutions of quasilinear wave equations, Amer. J. Math.130 (2008), no. 1,115-157.
[57]Lindblad, H., Global solutions of nonlinear wave equations, Comm. Pure Appl. Math.45 (1992), no.9,1063-1096.
[58]Lindblad, H., On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math.43 (1990), no.4,445-472.
[59]Lions, J. L., Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev.30 (1988),1-68.
[60]Lions, J. L. and Magenes, E., Problemes aux Limites Non homogenes et applications. Vol. 1., Dunod, Paris,1968.
[61]Metcalfe, J., Elastic waves in exterior domains. I. Almost global existence, Int. Math. Res. Not. (2006).
[62]Metcalfe, J., Nakamura, M. and Sogge, C. D., Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains, Forum Math.17 (2005), no.1, 133-168.
[63]Metcalfe, J. and Sogge, C. D., Global existence for Dirichlet-wave equations with quadratic nonlinearities in high dimensions, Math. Ann.336 (2006), no.2,391-420.
[64]Metcalfe, J. and Sogge, C. D., Global existence for high dimensional quasilinear wave equa-tions exterior to star-shaped obstacles, Discrete Contin. Dyn. Sys., to appear.
[65]Metcalfe, J. and Sogge, C.D., Global existence of null-form wave equations in exterior do-mains, Math. Z.256 (2007), no.3,521-549.
[66]Metcalfe, J. and Sogge, C. D., Hyperbolic trapped rays and global existence of quasilinear wave equations, Invent. Math.159 (2005), no.1,75-117.
[67]Metcalfe, J. and Sogge, C. D., Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods, SIAM J. Math. Anal.38 (2006), no.1,188-209.
[68]Metcalfe, J. and Thomases, B., Elastic waves in exterior domains. Ⅱ. Global existence with a null structure, Int. Math. Res. Not. (2007).
[69]Morawetz, C. S., The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math.14 (1961),561-568.
[70]Morawetz, C. S., Ralston, J. V. and Strauss, W. A., Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math.30 (1977), no.4,447-508.
[71]Russell, D. L., Controllability and stabilizability theory for linear partial differential equations, SIAM Rev.20 (1978),639-739.
[72]Russell, D. L., Exact boundary value controllability theorems for wave and heat processes in star-complemented regions, Differential games and control theory,291-319.
[73]Shibata, Y. and Tsutsumi, Y., On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z.191 (1986), no.2,165-199.
[74]Sideris, T. C., Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2) 151 (2000), no.2,849-874.
[75]Sideris, T. C., The null condition and global existence of nonlinear elastic waves, Invent. Math.123 (1996),323-342.
[76]Sideris, T. C. and Tu, S.-Y., Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal.33 (2001), no.2,477-488.
[77]Tahvildar-Zadeh, A. S., Relativistic and nonrelativistic elastodynamics with small shear strains, Ann. Inst. H. Poincare-Phys. Theory 69 (1998),275-307.
[78]Taylor, M. E., Partial differential equations. I. basic theory, Appl. Math. Sci.115, Springer-Verlag, New York,1996.
[79]Zhang, X. and Zuazua, E., Exact controllability of the semi-linear wave equation, in Un-solved Problems in Mathematical Systems and Control Theory, edited by Blondel, V. D. and Megretski, A., Princeton University Press, Princeton, NS, (2004),173-178.
[80]Zhou, Y., Blow up of solutions to the Cauchy problem for nonlinear wave equations, Chinese Ann. Math. Ser. B 22 (2001), no.3,275-280.
[81]Zhou, Y. and Lei, Z., Local exact boundary controllability for nonlinear wave equations, SIAM J. Control Optim.46 (2007),1022-1051.
[82]Zuazua, E., Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincare Anal. Non Lineaire 10 (1993),109-129.
[83]Zuazua, E., Exact Controllability for the semilinear wave equations, J. Math. Pures Appl. (9) 69 (1990),1-31.
[84]Zuazua, E., Exponential decay for semilinear wave equations with localized damping, Comm. Partial Differential Equations 15 (1990),205-235.