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

Belkacem Said-Houari; Flávio A. Falcão Nascimento

doi:10.3934/cpaa.2013.12.375

Article Contents

2013, Volume 12, Issue 1: 375-403. Doi: 10.3934/cpaa.2013.12.375

This issue Previous Article Spectral method for deriving multivariate Poisson summation formulae Next Article A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth

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

Belkacem Said-Houari^1, and
Flávio A. Falcão Nascimento^2,

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

Received: January 2011

Revised: November 2011

Published: January 2013

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 35L05, 35L15, 35L70, 37B25.

Citation:

Related Papers

Cited by

References

[1]	F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372.doi: 10.1016/j.jfa.2007.09.012.
[2]	S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonl. Anal., 64 (2006), 2314-2331.doi: 10.1016/j.na.2005.08.015.
[3]	L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.doi: 10.1016/j.jde.2010.03.009.
[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), 407-459.doi: 10.1016/j.jde.2007.02.004.
[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), 1-14.
[6]	M. M. Cavalcanti, V. N. Domingos Cavalcanti and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., 68 (2008), 177-193.doi: 10.1016/j.na.2006.10.040.
[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), 661-696.
[8]	M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.doi: 10.1137/S0363012902408010.
[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), 295-308.doi: 10.1006/jdeq.1994.1051.
[10]	T. G. Ha, On viscoelastic wave equation with nonlinear boundary damping and source term, Commun. Pure Appl. Anal., 9 (2010), 1543-1576.doi: 10.3934/cpaa.2010.9.1543.
[11]	M. Kafini and S. A. Messaoudi, A blow-up result for a viscoelastic system in $R^n$, Electron. J. Differential Equations, 113 (2007).
[12]	M. Kafini and S. A. Messaoudi, A blow-up result in a Cauchy viscoelastic problem, Appl. Math. Lett., 21 (2008), 549-553.doi: 10.1016/j.aml.2007.07.004.
[13]	I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533.
[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), 1-23.
[15]	J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites nonlinéaires," Dunod Gautier-Villars, Paris, 1969.
[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), 295-303.doi: 10.1016/j.nonrwa.2010.06.016.
[17]	P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), 419-444 (electronic).doi: 10.1051/cocv:1999116.
[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), 1687-1696.doi: 10.1002/mma.522.
[19]	S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math. Nachrich, 231 (2001), 1-7.doi: 10.1002/1522-2616(200111)231:1<105::AID-MANA105>3.0.CO;2-I.
[20]	S. A. Messaoudi, Blow up and global existence in nonlinear viscoelastic wave equations, Math. Nachrich, 260 (2003), 58-66.doi: 10.1002/mana.200310104.
[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), 3132-3140.doi: 10.1016/j.nonrwa.2008.10.026.
[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), 1759-1800.doi: 10.1080/03605309908821482.
[23]	H. Song and C. Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 3877-3883.doi: 10.1016/j.nonrwa.2010.02.015.
[24]	E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Rational Mech. Anal., 149 (1999), 155-182.doi: 10.1007/s002050050171.
[25]	E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298.doi: 10.1016/S0022-0396(02)00023-2.
[26]	E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J., 44 (2002), 375-395.doi: 10.1017/S0017089502030045.
[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), 1919-1928.doi: 10.1002/mma.1115.