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May  2014, 13(3): 1317-1325. doi: 10.3934/cpaa.2014.13.1317

Potential well and exact boundary controllability for radial semilinear wave equations on Schwarzschild spacetime

Ning-An Lai 1, and  Jinglei Zhao 2,

1. 

Sciences College, Lishui University, Zhejiang 323000, China
2. 

College of Education, Lishui University, Zhejinag 323000, China

Received  August 2013 Revised  October 2013 Published  December 2013

In this paper, we study the exact boundary controllability for the cubic focusing semilinear wave equation on Schwarzschild black hole background in radially symmetrical case. When the initial data and the final data are in the so called potential well, we find that the sufficient condition for the global existence is also sufficient to ensure the exact boundary controllability of the problem. Moreover, under the assumption of radial symmetry, our problem is changed to one space dimension case, and then the control time can be that of the linear wave equation.
Keywords: potential well., exact boundary controllability, Schwarzschild spacetime, Semilinear wave equation.
Mathematics Subject Classification: 35L05, 35L71, 93B05, 93C2.
Citation: Ning-An Lai, Jinglei Zhao. Potential well and exact boundary controllability for radial semilinear wave equations on Schwarzschild spacetime. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1317-1325. doi: 10.3934/cpaa.2014.13.1317
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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.  Google Scholar

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J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar

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Ch. Misner, K. Thorne and J. Wheeler, Gravitation, vol. III, W. H. Freeman and Company, San Francisco, 1973.  Google Scholar

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L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 272-303.  Google Scholar

[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.  Google Scholar

[13]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. and Anal., 30 (1968), 148-172.  Google Scholar

[14]

J. Shatah, Unstable ground state of nonlinear klein-Gorden equations, Trans. Amer. Math. Soc., 290 (1985), 701-710. doi: 10.2307/2000308.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Y. Zhou and N. A. Lai, Potential well and exact boundary controllability for semilinear wave equations, Adv. Differential Equations, 16 (2011), 1021-1047.  Google Scholar

[21]

E. Zuazua, Exact controllability for the semilinear wave equations, J. Math. Pures Appl., 69 (1990), 1-31.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar

[10]

Ch. Misner, K. Thorne and J. Wheeler, Gravitation, vol. III, W. H. Freeman and Company, San Francisco, 1973.  Google Scholar

[11]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 272-303.  Google Scholar

[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.  Google Scholar

[13]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. and Anal., 30 (1968), 148-172.  Google Scholar

[14]

J. Shatah, Unstable ground state of nonlinear klein-Gorden equations, Trans. Amer. Math. Soc., 290 (1985), 701-710. doi: 10.2307/2000308.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Y. Zhou and N. A. Lai, Potential well and exact boundary controllability for semilinear wave equations, Adv. Differential Equations, 16 (2011), 1021-1047.  Google Scholar

[21]

E. Zuazua, Exact controllability for the semilinear wave equations, J. Math. Pures Appl., 69 (1990), 1-31.  Google Scholar

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