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Exponential decay for solutions to semilinear damped wave equation
1. | Laboratoire de Mathématiques, Université de Savoie, 73376 Le Bourget du Lac, France |
2. | Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia |
References:
[1] |
J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Quart. J. Math. Oxford. Ser. (2), 28 (1977), 473-486. |
[2] |
A. Benaissa and S. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 391-399.
doi: 10.1007/s00030-005-0008-5. |
[3] |
J. Esquivel-Avila, Qualitative analysis of a nonlinear wave equation, Discrete. Contin. Dyn. Syst., 10 (2004), 787-804.
doi: 10.3934/dcds.2004.10.787. |
[4] |
F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. I. H. Poincaré, 23 (2006), 185-207. |
[5] |
V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.
doi: 10.1006/jdeq.1994.1051. |
[6] |
A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal., 100 (1988), 191-206.
doi: 10.1007/BF00282203. |
[7] |
R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear. Anal., 27 (1996), 1165-1175.
doi: 10.1016/0362-546X(95)00119-G. |
[8] |
R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475-491. |
[9] |
J. Esquivel-Avila, The dynamics of nonlinear wave equation, J. Math. Anal. Appl., 279 (2003), 135-150.
doi: 10.1016/S0022-247X(02)00701-1. |
[10] |
V. K. Kalantarov and O. A. Ladyzhenskaya, The occurence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Soviet. Math., 10 (1978), 53-70.
doi: 10.1007/BF01109723. |
[11] |
M. Kopáčkova, Remarks on bounded solutions of a semilinear dissipative hyperbolic equation, Comment. Math. Univ. Carolin., 30 (1989), 713-719. |
[12] |
H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{\mathcalt\mathcal t}=-Au+\mathcal F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.
doi: 10.2307/1996814. |
[13] |
H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.
doi: 10.1137/0505015. |
[14] |
S. Messaoudi and B. Said Houari, Global non-existence of solutions of a class of wave equations with non-linear damping and source terms, Math. Methods Appl. Sci., 27b (2004), 1687-1696.
doi: 10.1002/mma.522. |
[15] |
K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20 (1997), 151-177.
doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.3.CO;2-S. |
[16] |
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.
doi: 10.1007/BF02761595. |
[17] |
G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, C. R. Acad Sci. Paris Ser., 326 (1998), 191-196. |
[18] |
G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave with nonlinear damping and source terms, J. Math. Anal. Appl., 239 (1999), 213-226.
doi: 10.1006/jmaa.1999.6528. |
[19] |
E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-182.
doi: 10.1007/s002050050171. |
[20] |
Z. Yang, Existence and asymptotic behavior of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms, Math. Meth. Appl. Sci., 25 (2002), 795-814.
doi: 10.1002/mma.306. |
[21] |
E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial. Diff. Eq., 15 (1990), 205-235. |
show all references
References:
[1] |
J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Quart. J. Math. Oxford. Ser. (2), 28 (1977), 473-486. |
[2] |
A. Benaissa and S. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 391-399.
doi: 10.1007/s00030-005-0008-5. |
[3] |
J. Esquivel-Avila, Qualitative analysis of a nonlinear wave equation, Discrete. Contin. Dyn. Syst., 10 (2004), 787-804.
doi: 10.3934/dcds.2004.10.787. |
[4] |
F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. I. H. Poincaré, 23 (2006), 185-207. |
[5] |
V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.
doi: 10.1006/jdeq.1994.1051. |
[6] |
A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal., 100 (1988), 191-206.
doi: 10.1007/BF00282203. |
[7] |
R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear. Anal., 27 (1996), 1165-1175.
doi: 10.1016/0362-546X(95)00119-G. |
[8] |
R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475-491. |
[9] |
J. Esquivel-Avila, The dynamics of nonlinear wave equation, J. Math. Anal. Appl., 279 (2003), 135-150.
doi: 10.1016/S0022-247X(02)00701-1. |
[10] |
V. K. Kalantarov and O. A. Ladyzhenskaya, The occurence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Soviet. Math., 10 (1978), 53-70.
doi: 10.1007/BF01109723. |
[11] |
M. Kopáčkova, Remarks on bounded solutions of a semilinear dissipative hyperbolic equation, Comment. Math. Univ. Carolin., 30 (1989), 713-719. |
[12] |
H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{\mathcalt\mathcal t}=-Au+\mathcal F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.
doi: 10.2307/1996814. |
[13] |
H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.
doi: 10.1137/0505015. |
[14] |
S. Messaoudi and B. Said Houari, Global non-existence of solutions of a class of wave equations with non-linear damping and source terms, Math. Methods Appl. Sci., 27b (2004), 1687-1696.
doi: 10.1002/mma.522. |
[15] |
K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20 (1997), 151-177.
doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.3.CO;2-S. |
[16] |
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.
doi: 10.1007/BF02761595. |
[17] |
G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, C. R. Acad Sci. Paris Ser., 326 (1998), 191-196. |
[18] |
G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave with nonlinear damping and source terms, J. Math. Anal. Appl., 239 (1999), 213-226.
doi: 10.1006/jmaa.1999.6528. |
[19] |
E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-182.
doi: 10.1007/s002050050171. |
[20] |
Z. Yang, Existence and asymptotic behavior of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms, Math. Meth. Appl. Sci., 25 (2002), 795-814.
doi: 10.1002/mma.306. |
[21] |
E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial. Diff. Eq., 15 (1990), 205-235. |
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