Global existence and energy decay of a nondissipative Cauchy viscoelastic problem
Authors:
Mohammad Kafini and Muhammad I. Mustafa
Journal:
Quart. Appl. Math. 73 (2015), 739-757
MSC (2010):
Primary 35B05, 35L05, 35L15, 35L70
DOI:
https://doi.org/10.1090/qam/1420
Published electronically:
September 11, 2015
MathSciNet review:
3432281
Full-text PDF Free Access
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Additional Information
Abstract: A viscoelastic Cauchy problem subjected to a nonlinear source term is investigated. The memory term in the system involves a kernel which is regular, as is usually the case, but the system is not dissipative and is considered in the whole space. We prove global existence and nonexistence results. In the case of global existence, we show that solutions go to zero in a polynomial manner as time goes to infinity under some conditions on the source.
References
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- Nasser-eddine Tatar, On a problem arising in isothermal viscoelasticity, Int. J. Pure Appl. Math. 8 (2003), no. 1, 1–12. MR 1993399
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- William J. Hrusa and Michael Renardy, A model equation for viscoelasticity with a strongly singular kernel, SIAM J. Math. Anal. 19 (1988), no. 2, 257–269. MR 930025, DOI https://doi.org/10.1137/0519019
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- M. M. Cavalcanti and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, Differential Integral Equations 18 (2005), no. 5, 583–600. MR 2136980
- Mohammed Aassila and Marcelo M. Cavalcanti, On nonlinear hyperbolic problems with nonlinear boundary feedback, Bull. Belg. Math. Soc. Simon Stevin 7 (2000), no. 4, 521–540. MR 1806933
- Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
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References
- Wei Zhuang and Guitong Yang, Propagation of solitary waves in the nonlinear rods, Appl. Math. Mech. 7 (1986), 571-581.
- Xiangying Chen and Guowang Chen, Asymptotic behavior and blow-up of solutions to a nonlinear evolution equation of fourth order, Nonlinear Anal. 68 (2008), no. 4, 892–904. MR 2382305 (2009c:35393), DOI https://doi.org/10.1016/j.na.2006.11.045
- Mauro Fabrizio and Angelo Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1153021 (93a:73034)
- Michael Renardy, William J. Hrusa, and John A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 919738 (89b:35134)
- Vladimir Georgiev and Grozdena Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations 109 (1994), no. 2, 295–308. MR 1273304 (95b:35141), DOI https://doi.org/10.1006/jdeq.1994.1051
- Ryo Ikehata, Yasuaki Miyaoka, and Takashi Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbf {R}^N$ with lower power nonlinearities, J. Math. Soc. Japan 56 (2004), no. 2, 365–373. MR 2048464 (2005b:35190), DOI https://doi.org/10.2969/jmsj/1191418635
- Salim A. Messaoudi, Blow up in the Cauchy problem for a nonlinearly damped wave equation, Commun. Appl. Anal. 7 (2003), no. 2-3, 379–386. MR 1986245 (2004e:35154)
- Grozdena Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 2, 191–196 (English, with English and French summaries). MR 1646932 (99e:35154), DOI https://doi.org/10.1016/S0764-4442%2897%2989469-4
- Grozdena Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, J. Math. Anal. Appl. 239 (1999), no. 2, 213–226. MR 1723057 (2000k:35204), DOI https://doi.org/10.1006/jmaa.1999.6528
- Grozdena Todorova and Borislav Yordanov, The energy decay problem for wave equations with nonlinear dissipative terms in $\mathbb {R}^n$, Indiana Univ. Math. J. 56 (2007), no. 1, 389–416. MR 2305940 (2008c:35205), DOI https://doi.org/10.1512/iumj.2007.56.2963
- W. J. Hrusa and J. A. Nohel, The Cauchy problem in one-dimensional nonlinear viscoelasticity, J. Differential Equations 59 (1985), no. 3, 388–412. MR 807854 (87a:73030), DOI https://doi.org/10.1016/0022-0396%2885%2990147-0
- George Dassios and Filareti Zafiropoulos, Equipartition of energy in linearized $3$-D viscoelasticity, Quart. Appl. Math. 48 (1990), no. 4, 715–730. MR 1079915 (92e:73021)
- Jaime E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math. 52 (1994), no. 4, 628–648. MR 1306041 (95j:73052)
- C. M. Dafermos, Development of singularities in the motion of materials with fading memory, Arch. Rational Mech. Anal. 91 (1985), no. 3, 193–205. MR 806001 (87a:73033), DOI https://doi.org/10.1007/BF00250741
- Mohammad Kafini and Salim A. Messaoudi, A blow-up result in a Cauchy viscoelastic problem, Appl. Math. Lett. 21 (2008), no. 6, 549–553. MR 2412376 (2009e:35186), DOI https://doi.org/10.1016/j.aml.2007.07.004
- Mohammad Kafini and Salim A. Messaoudi, A blow-up result for a viscoelastic system in $\mathbb {R}^n$, Electron. J. Differential Equations (2007), No. 113, 7 pp. (electronic). MR 2349941 (2008f:35391)
- Mohammad Kafini and Salim A. Messaoudi, On the uniform decay in viscoelastic problems in $\mathbb {R}^n$, Appl. Math. Comput. 215 (2009), no. 3, 1161–1169. MR 2568974, DOI https://doi.org/10.1016/j.amc.2009.06.058
- M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho, and J. A. Soriano, Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential Integral Equations 14 (2001), no. 1, 85–116. MR 1797934 (2001m:35201)
- Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, and Juan A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations (2002), No. 44, 14. MR 1907720 (2003c:35116)
- Marcelo Moreira Cavalcanti and Higidio Portillo Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 42 (2003), no. 4, 1310–1324 (electronic). MR 2044797 (2005d:35257), DOI https://doi.org/10.1137/S0363012902408010
- Khaled M. Furati and Nasser-eddine Tatar, Uniform boundedness and stability for a viscoelastic problem, Appl. Math. Comput. 167 (2005), no. 2, 1211–1220. MR 2169762 (2006d:35284), DOI https://doi.org/10.1016/j.amc.2004.08.036
- Mohamed Medjden and Nasser-eddine Tatar, On the wave equation with a temporal non-local term, Dynam. Systems Appl. 16 (2007), no. 4, 665–671. MR 2370149 (2008k:35295)
- Nasser-eddine Tatar, On a problem arising in isothermal viscoelasticity, Int. J. Pure Appl. Math. 8 (2003), no. 1, 1–12. MR 1993399 (2004d:35243)
- Salim A. Messaoudi and Nasser-eddine Tatar, Exponential decay for a quasilinear viscoelastic equation, Math. Nachr. 282 (2009), no. 10, 1443–1450. MR 2571706 (2011a:35354), DOI https://doi.org/10.1002/mana.200610800
- Nasser-eddine Tatar, Exponential decay for a viscoelastic problem with a singular kernel, Z. Angew. Math. Phys. 60 (2009), no. 4, 640–650. MR 2520604 (2010m:35519), DOI https://doi.org/10.1007/s00033-008-8030-1
- W. J. Hrusa and M. Renardy, On a class of quasilinear partial integro-differential equations with singular kernels, J. Differential Equations 64 (1986), no. 2, 195–220. MR 851911 (88c:45010), DOI https://doi.org/10.1016/0022-0396%2886%2990087-2
- William J. Hrusa and Michael Renardy, A model equation for viscoelasticity with a strongly singular kernel, SIAM J. Math. Anal. 19 (1988), no. 2, 257–269. MR 930025 (89d:35159), DOI https://doi.org/10.1137/0519019
- M. Aassila, M. M. Cavalcanti, and V. N. Domingos Cavalcanti, Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term, Calc. Var. Partial Differential Equations 15 (2002), no. 2, 155–180. MR 1930245 (2003j:35216), DOI https://doi.org/10.1007/s005260100096
- M. M. Cavalcanti and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, Differential Integral Equations 18 (2005), no. 5, 583–600. MR 2136980 (2006b:35224)
- Mohammed Aassila and Marcelo M. Cavalcanti, On nonlinear hyperbolic problems with nonlinear boundary feedback, Bull. Belg. Math. Soc. Simon Stevin 7 (2000), no. 4, 521–540. MR 1806933 (2002c:35184)
- Haïm Brezis, Analyse fonctionnelle, Théorie et applications. [Theory and applications], Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). MR 697382 (85a:46001)
- J. A. Oguntuase, On integral inequalities of Gronwall-Bellman-Bihari type in several variables, JIPAM. J. Inequal. Pure Appl. Math. 1 (2000), no. 2, Article 20, 7 pp. (electronic). MR 1786407 (2001g:26023)
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Additional Information
Mohammad Kafini
Affiliation:
Department of Mathematics and Statistics, KFUPM, Dhahran 31261, Saudi Arabia
Email:
mkafini@kfupm.edu.sa
Muhammad I. Mustafa
Affiliation:
Department of Mathematics and Statistics, KFUPM, Dhahran 31261, Saudi Arabia
Email:
mmustafa@kfupm.edu.sa
Keywords:
Polynomial decay,
Cauchy problem,
global existence,
nondissipative viscoelastic problem
Received by editor(s):
April 3, 2014
Received by editor(s) in revised form:
October 14, 2014
Published electronically:
September 11, 2015
Article copyright:
© Copyright 2015
Brown University