Exponential decay for solutions to semilinear damped wave equation

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June 2012, 5(3): 559-566. doi: 10.3934/dcdss.2012.5.559

Exponential decay for solutions to semilinear damped wave equation

Stéphane Gerbi ^1, and Belkacem Said-Houari ^2,

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

Received March 2010 Revised May 2010 Published October 2011

Abstract
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This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4].

Keywords: stable set, decay rate, Strong damping, positive initial energy., global existence.

Mathematics Subject Classification: Primary: 35L70, 35B4.

Citation: Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559

References:

[1]	J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations,, Quart. J. Math. Oxford. Ser. (2), 28 (1977), 473.
[2]	A. Benaissa and S. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation,, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 391. 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. 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.
[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. 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. doi: 10.1007/BF00282203.
[7]	R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms,, Nonlinear. Anal., 27 (1996), 1165. 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.
[9]	J. Esquivel-Avila, The dynamics of nonlinear wave equation,, J. Math. Anal. Appl., 279 (2003), 135. 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. doi: 10.1007/BF01109723.
[11]	M. Kopáčkova, Remarks on bounded solutions of a semilinear dissipative hyperbolic equation,, Comment. Math. Univ. Carolin., 30 (1989), 713.
[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. 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. 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., 27 (2004), 1687. 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. 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. 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.
[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. 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. 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. 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.

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.
[2]	A. Benaissa and S. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation,, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 391. 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. 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.
[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. 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. doi: 10.1007/BF00282203.
[7]	R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms,, Nonlinear. Anal., 27 (1996), 1165. 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.
[9]	J. Esquivel-Avila, The dynamics of nonlinear wave equation,, J. Math. Anal. Appl., 279 (2003), 135. 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. doi: 10.1007/BF01109723.
[11]	M. Kopáčkova, Remarks on bounded solutions of a semilinear dissipative hyperbolic equation,, Comment. Math. Univ. Carolin., 30 (1989), 713.
[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. 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. 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., 27 (2004), 1687. 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. 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. 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.
[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. 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. 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. 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.

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