Blow up and global existence in a nonlinear viscoelastic wave equation

Mathematische Nachrichten

Volumn 260, Issue , 2003, Pages 58-66

  Messaoudi, Salim A ^a  

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

Author keywords

Blow up; Finite time; Global existence; Negative initial energy; Nonlinear damping; Viscoelastic

Indexed keywords

EID: 0242427516     PISSN: 0025584X     EISSN: None     Source Type: Journal    
DOI: 10.1002/mana.200310104     Document Type: Article

References (20)

1. M. Aassila, M. M. Cavalcanti, and J. A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in star-shaped domains, SIAM J. Control and Optimization 38 (5), 1581-1602 (2000).
2. J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Quart. J. Math. Oxford 28, 473-486 (1977).
3. M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci. 24, 1043-1053 (2001).
4. M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Elect. J. Diffe. Eqns. 44, 1-14 (2002).
5. 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, Diff. and Integral Eqns 14 (1), 85-116 (2001).
6. V. Georgiev and G. Todorova, Existence of solutions of the wave equation with nonlinear damping and source terms, J. Diff. Eqns. 109, 295-308 (1994).
7. A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal. 150, 191-206 (1988).
8. V. K. Kalantarov and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Soviet Math. 10, 53-70 (1978).
9. M. Kopackova, Remarks on bounded solutions of a semilinear dissipative hyperbolic equation, Comment. Math. Univ. Carolin 30, 713-719 (1989).
10. tt = Au + F(u), Trans. Amer. Math. Soc. 192, 1-21 (1974).
11. H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equation, SIAM J. Math. Anal. 5, 138-146 (1974).
12. H. A. Levine and J. Serrin, A global nonexistence theorem for quasilinear evolution equation with dissipation, Arch. Rational Mech. Anal. 137, 341-361 (1997).
13. H. A. Levine 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, 181-205 (1998).
14. W. J. Liu, Partial exact controllablity and exponential stability in higher-dimensional linear thermoelasticity, ESIAM: Control, Optimisation and Calculus of Variations 3, 23-48 (1998).
15. W. J. Liu, The exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity, J. Math. Pures et Appliquees 37, 355-386 (1998).
16. W. J. Liu, Partial exact controllability for the linear thermo-viscoelastic model, Elect. J. Differential Eqns 17, 1-11 (1998).
17. S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math. Nachr. 231, 1-7 (2001).
18. J. E. Munoz Rivera and R. Baretto, Decay rates for viscoelastic plates with memory, J. Elast. 44, 61-87 (1996).
19. E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149, 155-182 (1999).
20. E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. PDE 15, 205-235 (1990).