American Institute of Mathematical Sciences

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