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

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.
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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show all references

References:
[1]

J. Math. Pures Appl., 58 (1979), 249-273.  Google Scholar

[2]

Elsevier Science B.V., Amsterdam, Laussanne, New York, Oxford, Shanon, Tokyo 1996. Google Scholar

[3]

Ann. Inst. H. poincare Anal. Non Lineaire, 25 (2008), 1-41. doi: 10.1016/j.anihpc.2006.07.005.  Google Scholar

[4]

SIAM J. Control Optim., 46 (2007) ,1578-1614. doi: 10.1137/040610222.  Google Scholar

[5]

Acta Math. Sin., Engl. Ser., 18 (2002), 589-598. doi: 10.1007/s102550200061.  Google Scholar

[6]

Anal. PDE, 4 (2011), 405-460. doi: 10.2140/apde.2011.4.405.  Google Scholar

[7]

Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.  Google Scholar

[8]

AIMS series on applied mathematics, vol. 3, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2010.  Google Scholar

[9]

SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar

[10]

vol. III, W. H. Freeman and Company, San Francisco, 1973.  Google Scholar

[11]

Israel J. Math., 22 (1975), 272-303.  Google Scholar

[12]

SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.  Google Scholar

[13]

Arch. Rational Mech. and Anal., 30 (1968), 148-172.  Google Scholar

[14]

Trans. Amer. Math. Soc., 290 (1985), 701-710. doi: 10.2307/2000308.  Google Scholar

[15]

Nonlinear Anal., 48 (2002), 191-207. doi: 10.1016/S0362-546X(00)00180-2.  Google Scholar

[16]

proceedings of the international congress of mathematicians, Hyderabad, India, (2010).  Google Scholar

[17]

Ser. Contemp. Appl. Math. CAM, 15, Higher Ed. Press, Beijing, 2010. doi: 10.1142/9789814322898_0020.  Google Scholar

[18]

SIAM J. Control Optim., 46 (2007), 1022-1051. doi: 10.1137/060650222.  Google Scholar

[19]

Chin. Ann. Math., 31B (2010), 35-58. doi: 10.1007/s11401-008-0426-x.  Google Scholar

[20]

Adv. Differential Equations, 16 (2011), 1021-1047.  Google Scholar

[21]

J. Math. Pures Appl., 69 (1990), 1-31.  Google Scholar

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