Global existence and energy decay of a nondissipative Cauchy viscoelastic problem

Kafini, Mohammad; Mustafa, Muhammad

doi:10.1090/qam/1420

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
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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

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, 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

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

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, 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, 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

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, 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, 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 $\Bbb R^n$, Indiana Univ. Math. J. 56 (2007), no. 1, 389–416. MR 2305940, 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, 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, DOI https://doi.org/10.1090/qam/1079915

Jaime E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math. 52 (1994), no. 4, 629–648. MR 1306041, DOI https://doi.org/10.1090/qam/1306041

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, 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, 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 $\Bbb R^n$, Electron. J. Differential Equations (2007), No. 113, 7. MR 2349941

Mohammad Kafini and Salim A. Messaoudi, On the uniform decay in viscoelastic problems in $\Bbb R^n$, 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

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

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. MR 2044797, 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, 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

Nasser-eddine Tatar, On a problem arising in isothermal viscoelasticity, Int. J. Pure Appl. Math. 8 (2003), no. 1, 1–12. MR 1993399

Salim A. Messaoudi and Nasser-eddine Tatar, Exponential decay for a quasilinear viscoelastic equation, Math. Nachr. 282 (2009), no. 10, 1443–1450. MR 2571706, 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, 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, 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, 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, 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

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

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. MR 1786407