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Potential well and exact boundary controllability for radial semilinear wave equations on Schwarzschild spacetime
1. | Sciences College, Lishui University, Zhejiang 323000, China |
2. | College of Education, Lishui University, Zhejinag 323000, China |
References:
[1] |
G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl., 58 (1979), 249-273. |
[2] |
Y. Choquet-Bruhat, C. Dewitt-Morette and M. Dillard-Bleick, Analysis, Manifolds and Physics, Elsevier Science B.V., Amsterdam, Laussanne, New York, Oxford, Shanon, Tokyo 1996. |
[3] |
T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability inequalityis for parabolic and hyperbolic systems with potentials, Ann. Inst. H. poincare Anal. Non Lineaire, 25 (2008), 1-41.
doi: 10.1016/j.anihpc.2006.07.005. |
[4] |
X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007) ,1578-1614.
doi: 10.1137/040610222. |
[5] |
Y. X. Guo and P. F. Yao, On boundary stability of wave equations with variable coefficients, Acta Math. Sin., Engl. Ser., 18 (2002), 589-598.
doi: 10.1007/s102550200061. |
[6] |
S.Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear klein-Gorden equations, Anal. PDE, 4 (2011), 405-460.
doi: 10.2140/apde.2011.4.405. |
[7] |
C. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212.
doi: 10.1007/s11511-008-0031-6. |
[8] |
Tatsien Li, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS series on applied mathematics, vol. 3, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2010. |
[9] |
J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.
doi: 10.1137/1030001. |
[10] |
Ch. Misner, K. Thorne and J. Wheeler, Gravitation, vol. III, W. H. Freeman and Company, San Francisco, 1973. |
[11] |
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 272-303. |
[12] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[13] |
D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. and Anal., 30 (1968), 148-172. |
[14] |
J. Shatah, Unstable ground state of nonlinear klein-Gorden equations, Trans. Amer. Math. Soc., 290 (1985), 701-710.
doi: 10.2307/2000308. |
[15] |
J. Zhang, Sharp conditions of global existence for nonlinear Schrodinger and Klein-Gorden equations, Nonlinear Anal., 48 (2002), 191-207.
doi: 10.1016/S0362-546X(00)00180-2. |
[16] |
X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, proceedings of the international congress of mathematicians, Hyderabad, India, (2010). |
[17] |
X. Zhang, Remarks on the controllability of some quasilinear equations, Ser. Contemp. Appl. Math. CAM, 15, Higher Ed. Press, Beijing, 2010.
doi: 10.1142/9789814322898_0020. |
[18] |
Y. Zhou and Z. Lei, Local exact boundary controllability for nonlinear wave equations, SIAM J. Control Optim., 46 (2007), 1022-1051.
doi: 10.1137/060650222. |
[19] |
Y. Zhou, W. Xu and Z. Lei, Global exact boundary controllability for cubic semi-linear wave equations and Klein-Gordon equations, Chin. Ann. Math., 31B (2010), 35-58.
doi: 10.1007/s11401-008-0426-x. |
[20] |
Y. Zhou and N. A. Lai, Potential well and exact boundary controllability for semilinear wave equations, Adv. Differential Equations, 16 (2011), 1021-1047. |
[21] |
E. Zuazua, Exact controllability for the semilinear wave equations, J. Math. Pures Appl., 69 (1990), 1-31. |
show all references
References:
[1] |
G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl., 58 (1979), 249-273. |
[2] |
Y. Choquet-Bruhat, C. Dewitt-Morette and M. Dillard-Bleick, Analysis, Manifolds and Physics, Elsevier Science B.V., Amsterdam, Laussanne, New York, Oxford, Shanon, Tokyo 1996. |
[3] |
T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability inequalityis for parabolic and hyperbolic systems with potentials, Ann. Inst. H. poincare Anal. Non Lineaire, 25 (2008), 1-41.
doi: 10.1016/j.anihpc.2006.07.005. |
[4] |
X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007) ,1578-1614.
doi: 10.1137/040610222. |
[5] |
Y. X. Guo and P. F. Yao, On boundary stability of wave equations with variable coefficients, Acta Math. Sin., Engl. Ser., 18 (2002), 589-598.
doi: 10.1007/s102550200061. |
[6] |
S.Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear klein-Gorden equations, Anal. PDE, 4 (2011), 405-460.
doi: 10.2140/apde.2011.4.405. |
[7] |
C. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212.
doi: 10.1007/s11511-008-0031-6. |
[8] |
Tatsien Li, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS series on applied mathematics, vol. 3, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2010. |
[9] |
J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.
doi: 10.1137/1030001. |
[10] |
Ch. Misner, K. Thorne and J. Wheeler, Gravitation, vol. III, W. H. Freeman and Company, San Francisco, 1973. |
[11] |
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 272-303. |
[12] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[13] |
D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. and Anal., 30 (1968), 148-172. |
[14] |
J. Shatah, Unstable ground state of nonlinear klein-Gorden equations, Trans. Amer. Math. Soc., 290 (1985), 701-710.
doi: 10.2307/2000308. |
[15] |
J. Zhang, Sharp conditions of global existence for nonlinear Schrodinger and Klein-Gorden equations, Nonlinear Anal., 48 (2002), 191-207.
doi: 10.1016/S0362-546X(00)00180-2. |
[16] |
X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, proceedings of the international congress of mathematicians, Hyderabad, India, (2010). |
[17] |
X. Zhang, Remarks on the controllability of some quasilinear equations, Ser. Contemp. Appl. Math. CAM, 15, Higher Ed. Press, Beijing, 2010.
doi: 10.1142/9789814322898_0020. |
[18] |
Y. Zhou and Z. Lei, Local exact boundary controllability for nonlinear wave equations, SIAM J. Control Optim., 46 (2007), 1022-1051.
doi: 10.1137/060650222. |
[19] |
Y. Zhou, W. Xu and Z. Lei, Global exact boundary controllability for cubic semi-linear wave equations and Klein-Gordon equations, Chin. Ann. Math., 31B (2010), 35-58.
doi: 10.1007/s11401-008-0426-x. |
[20] |
Y. Zhou and N. A. Lai, Potential well and exact boundary controllability for semilinear wave equations, Adv. Differential Equations, 16 (2011), 1021-1047. |
[21] |
E. Zuazua, Exact controllability for the semilinear wave equations, J. Math. Pures Appl., 69 (1990), 1-31. |
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