On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback |
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| Institution: | 1. Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, PR China;2. School of Information Technology and Management, University of International Business and Economics, Beijing 100029, PR China;3. School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa;1. Dipartimento di Economia Politica e Statistica, Università degli Studi di Siena, Siena, Italy;2. Dipartimento di Economia e Finanza, LUISS University, Rome, Italy;1. Department of Applied Mathematics, Donghua University, Shanghai 201620, PR China;2. College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, PR China;3. College of Information Science and Technology, Donghua University, Shanghai 201620, PR China;1. Dept. Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014, University of Jyväskylä, Finland;2. Dept. Mathematics and System Analysis, P.O. Box 11100, FI-00076, Aalto University, Finland |
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| Abstract: | The uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered. The existence of both weak and strong solutions to the system is proven by the Galerkin method. The exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the Riemannian manifold. This equivalent energy function shows particularly that the system is essentially a dissipative system. This result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003) 24–52] where the system is claimed to be nondissipative. |
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