Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping

Electronic Journal of Differential Equations

Volumn 2004, Issue , 2004, Pages 1-10

  Berrimi, Said ^a   Messaoudi, Salim A ^b  

^a FERHAT ABBAS UNIVERSITY   (Algeria)

^b KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS   (Saudi Arabia)

Author keywords

Exponential decay; Global existence; Nonlinear localized damping

Indexed keywords

EID: 4644248319     PISSN: 10726691     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (19)

1. Ball J., Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Quart. J. Math. Oxford 28 (1977), 473 - 486.
2. Cavalcanti M. M., V. N. Domingos Cavalcanti and J. Ferreira; Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci. 24 (2001), 1043 - 1053.
3. Cavalcanti M. M., V. N. Domingos Cavalcanti and J. A. Soriano; Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Elect. J. Diff. Eqs. Vol. 2002 (2002) 44, 1 - 14.
4. Cacalcanti M. M. and Oquendo H. P.; Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. vol 42 # 4 (2003), 1310 - 1324.
5. Cavalcanti M. M., V. N. Domingos Cavalcanti, J. S. Prates Filho and J. A. Soriano; Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Diff. Integ. Eqs. 14 (2001) 1, 85 - 116.
6. Georgiev, V. and G. Todorova, Existence of solutions of the wave equation with nonlinear damping and source terms, J. Diff. Eqs. 109 (1994), 295 - 308.
7. Haraux, A. and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal. 150 (1988), 191 - 206.
8. Kalantarov V. K. and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Soviet Math. 10 (1978), 53 -70.
9. Kawashima S. and Shibata Y., Global Existence and exponential stability of small solutions to nonlinear viscoelasticity, Comm. Math. Physics, Vol. 148 # 1(1992), 189 - 208.
10. Kopackova M., Remarks on bounded solutions of a semilinear dissipative hyperbolic equation, Comment. Math. Univ. Carolin. 30 (1989), 713 - 719.
11. tt = Au + F(u), Trans. Amer. Math. Soc. 192 (1974), 1 - 21.
12. Levine H. A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equation, SIAM J. Math. Anal. 5 (1974), 138 - 146.
13. Levine H. A. and J. Serrin, A global nonexistence theorem for quasilinear evolution equation with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341 - 361.
14. Levine, H. A. and S. Ro Park, Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl. 228 (1998), 181 - 205.
15. Messaoudi S. A., Blow up in a nonlinearly damped wave equation, Mathematische Nachrichten, 231 (2001), 1 - 7.
16. Messaoudi S. A. and Tatar N-E., Global existence asymptotic behavior for a Nonlinear Viscoelastic Problem, Math. Meth. Sci. Research J., Vol. 7 # 4 (2003), 136-149.
17. Messaoudi S. A., Blow up and global existence in a nonlinear viscoelastic wave equation, Mathematische Nachrichten, vol. 260 (2003), 5 8 - 66.
18. Vitillaro E., Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Rational Mech. Anal., 149 (1999), 155 - 182.
19. Zuazua E., Exponential decay for the semilinear wave equation with locally distributed damping, Comm. PDE. 15 (1990), 205 - 235.