Existence of a universal attractor for a fully hyperbolic phase-field system

Journal of Evolution Equations

Volumn 4, Issue 1, 2004, Pages 27-51

  Grasselli, Maurizio ^a   Pata, Vittorino ^a  

^a POLITECNICO DI MILANO   (Italy)

Author keywords

Absorbing sets; Memory effects; Phase field models; Universal attractors

Indexed keywords

EID: 1842843817     PISSN: 14243199     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00028-003-0074-2     Document Type: Article

References (28)

1. BABIN, A. V. and VISHIK, M. I., Attractors of evolution equations, North-Holland, Amsterdam 1992
2. 3 , Discrete Contin. Dynam. Systems 7 (2001), 719-735.
3. BROKATE, M. and SPREKELS, J., Hysteresis and phase transitions, Springer, New York 1996.
4. CAGINALP, G., An analysis ofa phase field model ofa free boundary, Arch. Rational Mech. Anal. 96 (1985), 205-245.
5. CHEPYZHOV, V. V. and VISHIK, M. I., Nonautonomous evolution equations and their attractors, Russian J. Math. Phys. 1 (1993), 165-190.
6. CHEPYZHOV, V. V. and VISHIK, M. I., Attractors of non-autonomous dynamical systems and their dimension, J. Math. Pures Appl. 73 (1994), 279-333.
7. COLLI, P., GILARDI, G. and GRASSELLI, M., Well-posedness of the weak formulation for the phase-field model with memory, Adv. Differential Equations 3 (1997), 487-508.
8. GATTI, S. and SARTORI, E., Well-posedness results for phase-field systems with memory effects in the order parameter dynamics, Discrete Contin. Dynam. Systems (to appear).
9. GIORGI, C., GRASSELLI, M. and PATA, V., Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J. 48 (1999), 1395-1445.
10. GIORGI, C., GRASSELLI, M. and PATA, V., Well-posedness and longtime behavior of the phase-field model with memory in a history space setting, Quart. Appl. Math. 59 (2001), 701-736.
11. GIORGI, C., MUÑOZ RIVERA, J. E. and PATA, V., Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl. 260 (2001), 83-99.
12. GRASSELLI, M., On a phase-field model with memory, in "Free Boundary Problems: Theory and Applications I", (N. Kenmochi, Ed.), Gakuto Int. Ser. Math. Sci. Appl. Gakkōtosho, Tokyo 13 (2000), 89-102.
13. GRASSELLI, M. and PATA, V., Upper semicontinuous attractor for a hyperbolic phase-field model with memory, Indiana Univ. Math. J. 50 (2001), 1189-1216.
14. GRASSELLI, M. and PATA, V., On the dissipativity of a hyperbolic phase-field system with memory, Nonlinear Anal. 47 (2001), 3157-3169.
15. GRASSELLI, M. and PATA, V., Existence of a universal attractor for a parabolic-hyperbolic phase-field system, Adv. Math. Sci. Appl. (to appear).
16. GRASSELLI, M. and PATA, V., Uniform attractors of nonautonomous dynamical systems with memory, in "Evolution Equations, Semigroups and Functional Analysis" (A. Lorenzi and B. Ruf, Eds.), Progr. Nonlinear Differential Equations Appl. Birkhäuser, Boston 50 (2002), 155-178.
17. GRASSELLI, M. and ROTSTEIN, H. G. Hyperbolic phase-field dynamics with memory, J. Math. Anal. Appl. 261 (2001), 205-230.
18. GURTIN, M. E. and PIPKIN, A. C., A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), 113-126.
19. HALE, J. K., Asymptotic behavior of dissipative systems, Math. Surv. Monogr. n. 25, Amer. Math. Soc., Providence 1988.
20. HARAUX, A., Systèmes dynamiques dissipatifs et applications, Coll. RMA n. 17, Masson, Paris 1991.
21. LIONS, J.-L. and MAGENES, E., Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972.
22. NOVICK-COHEN, A., A phase-field system with memory: global existence, J. Integral Equations Appl. (to appear).
23. 3 with linear memory, Math. Methods Appl. Sci. 23 (2000), 633-653.
24. PATA, V., Hyperbolic limit of parabolic semilinear heat equation with fading memory, Z. Anal. Anwendungen 20 (2001), 359-377.
25. PATA, V., PROUSE, G. and VISHIK, M. I., Traveling waves of dissipative nonautonomous hyperbolic equations in a strip, Adv. Differential Equations 3 (1998), 249-270.
26. PATA, V. and ZUCCHI A., Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl. 11 (2001), 505-529.
27. ROTSTEIN, H. G., NEPOMNYASHCHY, A. A., NOVICK-COHEN, A. and BRANDON, S., Phase field equations with memory: the hyperbolic case, SIAM J. Appl. Math. 62 (2001), 264-282.
28. TEMAM, R., Infinite-dimensional dynamical systems in mechanics and physics, Springer, New York, 1997.