Abstract
Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coefficients. In particular, some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.
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Barbu, V., Lasiecka, I. and Rammaha, A. M., Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms, Indiana Univ. Math. J., 56(3), 2007, 995–1021.
Cavalcanti, M. M., Khemmoudj, A. and Medjden, M., Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions, J. Math. Anal. Appl., 328(2), 2007, 900–930.
Chai, S., Guo, Y. and Yao, P. F., Boundary feedback stabilization of shallow shells, SIAM J. Control Optim., 42(1), 2003, 239–259.
Chai, S. and Liu, K., Observability inequalities for the transmission of shallow shells, Sys. Control Lett., 55(9), 2006, 726–735.
Chai, S. and Liu, K., Boundary feedback stabilization of the transmission problem of Naghdi’s model, J. Math. Anal. Appl., 319(1), 2006, 199–214.
Chai, S. and Yao, P. F., Observability inequalities for thin shells, Sci. China Ser. A, 46(3), 2003, 300–311.
Feng, S. J. and Feng, D. X., Nonlinear internal damping of wave equations with variable coefficients, Acta Math. Sin., Engl. Ser., 20(6), 2004, 1057–1072.
Gallot, S., Hulin, D. and Lafontaine, J., Riemannian Geometry, Second Edition, Springer-Verlag, Heidelberg, 1990.
Green, R. E. and Wu, H., C ∞ convex functions and manifolds of positive curvature, Acta Math., 137(1), 1976, 209–245.
Gulliver, R., Lasiecka, I., Littman, W. et al, The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber, Geometric Methods in Inverse Problems and PDE Control, 73–181, IMA Vol. Math. Appl., 137, Springer-Verlag, New York, 2004.
Haraux, A., Stabilization of trajectories for some weakly damped hyperbolic equations, J. Diff. Eqs., 59(2), 1985, 145–154.
Ho, L. F., Observabilité frontière de l’équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 302(12), 1986, 443–446.
Lagnese, J., Control of wave processes with distributed controls supported on a subregion, SIAM J. Control Optim., 21(1), 1983, 68–85.
Lasiecka, I. and Ong, J., Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation, Commun. Part. Diff. Eqs., 24(11–12), 1999, 2069–2107.
Lasiecka, I., Triggiani, R., and Yao, P. F., Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235(1), 1999, 13–57.
Liu, K. S., Locally distributed control and damping for the conservative system, SIAM J. Control Optim., 35(5), 1997, 1574–1590.
Nakao, M., Energy decay for the linear and semilinear wave equation in exterior domains with some localized dissipations, Math. Z., 238(4), 2001, 781–797.
Nakao, M., Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305(3), 1996, 403–417.
Nakao, M., Global attractors for nonlinear wave equations with nonlinear dissipative terms, J. Diff. Eqs., 227(1), 2006, 204–229.
Nakao, M., Existence of global solutions for the Kirchhoff-type quasilinear wave equation in exterior domains with a half-linear dissipation, Kyushu J. Math., 58(2), 2004, 373–391.
Nakao, M. and Ono, K., Global existence to the Cauchy problem for the semilinear dissipative wave equation, Math. Z., 214(1), 1993, 325–342.
Nicaise, S. and Pignotti, C., Internal and boundary observability estimates for the heterogeneous Maxwell’s system, Appl. Math. Optim., 54(1), 2006, 47–70.
Nirenberg, L., On elliptic partial differential equations, Ann. Sc. Norm. Sup. Pisa, 13(2), 1959, 115–162.
Morawetz, C., Time decay for nonlinear Klein-Gordon equations, Proc. Roy. Soc. London, 306A, 1968, 503–518.
Rammaha, M. A. and Strei, T. A., Global existence and nonexistence for nonlinear wave equations with damping and source terms, Trans. Amer. Math. Soc., 354(9), 2002, 3621–3637.
Slemrod, M., Weak asymptotic decay via a “related invariance principle” for a wave equation with nonlinear, nonmonotone damping, Proc. Roy. Soc. Edinburgh Sect. A, 113(1–2), 1989, 87–97.
Tcheugoué Tébou, L. R., Stabilization of the wave equation with localized nonlinear damping, J. Diff. Eqs., 145(2), 1998, 502–524.
Todorova, G., Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, Nonlinear Anal. Ser. A, 41(7–8), 2000, 891–905.
Todorova, G. and Yordanov, B., The energy decay problem for wave equations with nonlinear dissipative terms in ℝn, Indiana Univ. Math. J., 56(1), 2007, 389–416.
Todorova, G. and Yordanov, B., Critical exponent for a nonlinear wave equation with damping, J. Diff. Eqs., 174(2), 2001, 464–489.
Wu, H., Shen, C. L. and Yu, Y. L., An Introduction to Riemannian Geometry (in Chinese), Beijing University Press, Beijing, 1989.
Yao, P. F., On the observability inequalities for the exact controllability of the wave equation with variable coefficients, SIAM J. Control Optim., 37(6), 1999, 1568–1599.
Yao, P. F., Observatility inequality for shallow shells, SIAM J. Control Optim., 38(6), 2000, 1729–1756.
Yao, P. F., Observability Inequalities for the Euler-Bernoulli Plate with Variable Coefficients, Contemporary Mathematics, Vol. 268, A. M. S., Providence, RI, 2000, 383–406.
Yao, P. F., Boundary controllability for the quasilinear wave equation, Appl. Math. Optim., to appear. arXiv:math.AP/ 0603280
Yao, P. F., Global smooth solutions for the quasilinear wave equation with boundary dissipation, J. Diff. Eqs., 241(1), 2007, 62–93.
Zhang, Z. F. and Yao, P. F., Global smooth solutions of the quasilinear wave equation with internal velocity feedbacks, SIAM J. Control Optim., 47(4), 2008, 2044–2077.
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Projects supported by the National Natural Science Foundation of China (Nos. 60225003, 60821091, 10831007, 60774025) and KJCX3-SYW-S01.
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Yao, P. Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation. Chin. Ann. Math. Ser. B 31, 59–70 (2010). https://doi.org/10.1007/s11401-008-0421-2
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DOI: https://doi.org/10.1007/s11401-008-0421-2