January  2013, 12(1): 375-403. doi: 10.3934/cpaa.2013.12.375

Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction

1. 

Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz

2. 

Department of Mathematics, State University of Ceará- FAFIDAM, 62930-000 Limoeiro do Norte - CE, Brazil

Received  January 2011 Revised  November 2011 Published  September 2012

The goal of this work is to study a model of the viscoelastic wave equation with nonlinear boundary/interior sources and a nonlinear interior damping. First, applying the Faedo-Galerkin approximations combined with the compactness method to obtain existence of regular global solutions to an auxiliary problem with globally Lipschitz source terms and with initial data in the potential well. It is important to emphasize that it is not possible to consider density arguments to pass from regular to weak solutions if one considers regular solutions of our problem where the source terms are locally Lipschitz functions. To overcome this difficulty, we use an approximation method involving truncated sources and adapting the ideas in [13] to show that the existence of weak solutions can still be obtained for our problem. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term, then the solution ceases to exist and blows up in finite time provided that the initial data are large enough.
Citation: Belkacem Said-Houari, Flávio A. Falcão Nascimento. Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction. Communications on Pure & Applied Analysis, 2013, 12 (1) : 375-403. doi: 10.3934/cpaa.2013.12.375
References:
[1]

F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory,, J. Funct. Anal., 254 (2008), 1342. doi: 10.1016/j.jfa.2007.09.012. Google Scholar

[2]

S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source,, Nonl. Anal., 64 (2006), 2314. doi: 10.1016/j.na.2005.08.015. Google Scholar

[3]

L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping,, J. Differential Equations, 249 (2010), 654. doi: 10.1016/j.jde.2010.03.009. Google Scholar

[4]

M. M. Cavalcanti, V. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction,, J. Differential. Equations, 236 (2007), 407. doi: 10.1016/j.jde.2007.02.004. Google Scholar

[5]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping,, E. J. Differential Equations, 44 (2002), 1. Google Scholar

[6]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems,, Nonlinear Anal., 68 (2008), 177. doi: 10.1016/j.na.2006.10.040. Google Scholar

[7]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global solvability and asymptotic stability for the wave equation with nonlinear feedback and source term on the boundary,, Adv. Math. Sci. Appl., 16 (2006), 661. Google Scholar

[8]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation,, SIAM J. Control Optim., 42 (2003), 1310. doi: 10.1137/S0363012902408010. Google Scholar

[9]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source term,, J. Differential. Equations, 109 (1994), 295. doi: 10.1006/jdeq.1994.1051. Google Scholar

[10]

T. G. Ha, On viscoelastic wave equation with nonlinear boundary damping and source term,, Commun. Pure Appl. Anal., 9 (2010), 1543. doi: 10.3934/cpaa.2010.9.1543. Google Scholar

[11]

M. Kafini and S. A. Messaoudi, A blow-up result for a viscoelastic system in $R^n$,, Electron. J. Differential Equations, 113 (2007). Google Scholar

[12]

M. Kafini and S. A. Messaoudi, A blow-up result in a Cauchy viscoelastic problem,, Appl. Math. Lett., 21 (2008), 549. doi: 10.1016/j.aml.2007.07.004. Google Scholar

[13]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential Integral Equations, 6 (1993), 507. Google Scholar

[14]

H. A. Levine and R. A. Smith, A potential well theory for the wave equation with a nonlinear boundary condition,, J. Reine Angew. Math., 374 (1987), 1. Google Scholar

[15]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites nonlinéaires,", Dunod Gautier-Villars, (1969). Google Scholar

[16]

L. Lu, S. Li and S. Chai, On a viscoelastic equation with nonlinear boundary damping and source terms: global existence and decay of the solution,, Nonlinear Anal. Real World Appl., 12 (2011), 295. doi: 10.1016/j.nonrwa.2010.06.016. Google Scholar

[17]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems,, ESAIM Control Optim. Calc. Var., 4 (1999), 419. doi: 10.1051/cocv:1999116. Google Scholar

[18]

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. Google Scholar

[19]

S. A. Messaoudi, Blow up in a nonlinearly damped wave equation,, Math. Nachrich, 231 (2001), 1. doi: 10.1002/1522-2616(200111)231:1<105::AID-MANA105>3.0.CO;2-I. Google Scholar

[20]

S. A. Messaoudi, Blow up and global existence in nonlinear viscoelastic wave equations,, Math. Nachrich, 260 (2003), 58. doi: 10.1002/mana.200310104. Google Scholar

[21]

S. A. Messaoudi and M. I. Mustafa, On the control of solutions of viscoelastic equations with boundary feedback,, Nonlinear Anal. Real World Appl., 10 (2009), 3132. doi: 10.1016/j.nonrwa.2008.10.026. Google Scholar

[22]

M. Milla Miranda and L. P. San Gil Jutuca, Existence and boundary stabilization of solutions for the Kirchhoff equation,, Comm. Partial Differential Equations, 24 (1999), 1759. doi: 10.1080/03605309908821482. Google Scholar

[23]

H. Song and C. Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation,, Nonlinear Anal. Real World Appl., 11 (2010), 3877. doi: 10.1016/j.nonrwa.2010.02.015. Google Scholar

[24]

E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation,, Arch. Rational Mech. Anal., 149 (1999), 155. doi: 10.1007/s002050050171. Google Scholar

[25]

E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms,, J. Differential Equations, 186 (2002), 259. doi: 10.1016/S0022-0396(02)00023-2. Google Scholar

[26]

E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms,, Glasg. Math. J., 44 (2002), 375. doi: 10.1017/S0017089502030045. Google Scholar

[27]

S. Yu, M. Wang and W. Liu, Blow up for a Cauchy viscoelastic problem with a nonlinear dissipation of cubic convolution type,, Math. Methods Appl. Sci., 32 (2009), 1919. doi: 10.1002/mma.1115. Google Scholar

show all references

References:
[1]

F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory,, J. Funct. Anal., 254 (2008), 1342. doi: 10.1016/j.jfa.2007.09.012. Google Scholar

[2]

S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source,, Nonl. Anal., 64 (2006), 2314. doi: 10.1016/j.na.2005.08.015. Google Scholar

[3]

L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping,, J. Differential Equations, 249 (2010), 654. doi: 10.1016/j.jde.2010.03.009. Google Scholar

[4]

M. M. Cavalcanti, V. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction,, J. Differential. Equations, 236 (2007), 407. doi: 10.1016/j.jde.2007.02.004. Google Scholar

[5]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping,, E. J. Differential Equations, 44 (2002), 1. Google Scholar

[6]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems,, Nonlinear Anal., 68 (2008), 177. doi: 10.1016/j.na.2006.10.040. Google Scholar

[7]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global solvability and asymptotic stability for the wave equation with nonlinear feedback and source term on the boundary,, Adv. Math. Sci. Appl., 16 (2006), 661. Google Scholar

[8]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation,, SIAM J. Control Optim., 42 (2003), 1310. doi: 10.1137/S0363012902408010. Google Scholar

[9]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source term,, J. Differential. Equations, 109 (1994), 295. doi: 10.1006/jdeq.1994.1051. Google Scholar

[10]

T. G. Ha, On viscoelastic wave equation with nonlinear boundary damping and source term,, Commun. Pure Appl. Anal., 9 (2010), 1543. doi: 10.3934/cpaa.2010.9.1543. Google Scholar

[11]

M. Kafini and S. A. Messaoudi, A blow-up result for a viscoelastic system in $R^n$,, Electron. J. Differential Equations, 113 (2007). Google Scholar

[12]

M. Kafini and S. A. Messaoudi, A blow-up result in a Cauchy viscoelastic problem,, Appl. Math. Lett., 21 (2008), 549. doi: 10.1016/j.aml.2007.07.004. Google Scholar

[13]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential Integral Equations, 6 (1993), 507. Google Scholar

[14]

H. A. Levine and R. A. Smith, A potential well theory for the wave equation with a nonlinear boundary condition,, J. Reine Angew. Math., 374 (1987), 1. Google Scholar

[15]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites nonlinéaires,", Dunod Gautier-Villars, (1969). Google Scholar

[16]

L. Lu, S. Li and S. Chai, On a viscoelastic equation with nonlinear boundary damping and source terms: global existence and decay of the solution,, Nonlinear Anal. Real World Appl., 12 (2011), 295. doi: 10.1016/j.nonrwa.2010.06.016. Google Scholar

[17]

P. Martinez, A new method to obtain decay rate estimates for dissipative systems,, ESAIM Control Optim. Calc. Var., 4 (1999), 419. doi: 10.1051/cocv:1999116. Google Scholar

[18]

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. Google Scholar

[19]

S. A. Messaoudi, Blow up in a nonlinearly damped wave equation,, Math. Nachrich, 231 (2001), 1. doi: 10.1002/1522-2616(200111)231:1<105::AID-MANA105>3.0.CO;2-I. Google Scholar

[20]

S. A. Messaoudi, Blow up and global existence in nonlinear viscoelastic wave equations,, Math. Nachrich, 260 (2003), 58. doi: 10.1002/mana.200310104. Google Scholar

[21]

S. A. Messaoudi and M. I. Mustafa, On the control of solutions of viscoelastic equations with boundary feedback,, Nonlinear Anal. Real World Appl., 10 (2009), 3132. doi: 10.1016/j.nonrwa.2008.10.026. Google Scholar

[22]

M. Milla Miranda and L. P. San Gil Jutuca, Existence and boundary stabilization of solutions for the Kirchhoff equation,, Comm. Partial Differential Equations, 24 (1999), 1759. doi: 10.1080/03605309908821482. Google Scholar

[23]

H. Song and C. Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation,, Nonlinear Anal. Real World Appl., 11 (2010), 3877. doi: 10.1016/j.nonrwa.2010.02.015. Google Scholar

[24]

E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation,, Arch. Rational Mech. Anal., 149 (1999), 155. doi: 10.1007/s002050050171. Google Scholar

[25]

E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms,, J. Differential Equations, 186 (2002), 259. doi: 10.1016/S0022-0396(02)00023-2. Google Scholar

[26]

E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms,, Glasg. Math. J., 44 (2002), 375. doi: 10.1017/S0017089502030045. Google Scholar

[27]

S. Yu, M. Wang and W. Liu, Blow up for a Cauchy viscoelastic problem with a nonlinear dissipation of cubic convolution type,, Math. Methods Appl. Sci., 32 (2009), 1919. doi: 10.1002/mma.1115. Google Scholar

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