Well-posedness and long-time behavior for a class of doubly nonlinear equations

Discrete and Continuous Dynamical Systems

Volumn 18, Issue 1, 2007, Pages 15-38

  Schimperna, Giulio ^a   Segatti, Antonio ^b   Stefanelli, Ulisse ^c  

^a UNIVERSITY OF PAVIA   (Italy)

^b WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^c CNR   (Italy)

Author keywords

Doubly nonlinear equation; Maximal monotone operator; Singular potential; ojasiewicz Simon inequality; limit set

Indexed keywords

EID: 34250897419     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2007.18.15     Document Type: Article

References (68)

1. S. Aizicovici, E. Feireisl, Long-time stabilization of solutions to a phase-field model with memory, J. Evol. Equ., 1 (2001), 69-84.
2. S. Aizicovici, E. Feireisl, F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci., 24 (2001), 277-287.
3. G. Akagi, Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Danach spaces, J. Differential Equations, 231 (2006), 32-56.
4. H.W. Alt, S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341.
5. T. Arai, On the existence of the solution for ∂φ(u′(t)) + ∂Ψ(u(t)) ∋ f(t), J. Fac. Sci. Univ. Tokyo Sect. IA Math., 26 (1979), 75-96.
6. M. Aso, T. Fukao, N. Kenmochi, A new class of doubly nonlinear evolution equations, in "Proceedings of Third East Asia Partial Differential Equation Conference", Taiwanese J. Math., 8 (2004), 103-124.
7. H. Attouch, "Variational Convergence For Functions And Operators," Applicable Mathematics Series, Pitman, Boston, 1984.
8. L. Ambrosio, N. Gigli, G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Birkhäuser Verlag AG, 2005.
9. J.M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.
10. J.M. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.
11. V. Barbu, Existence theorems for a class of two point boundary problems, J. Differential Equations, 17 (1975), 236-257.
12. V. Barbu, "Nonlinear Semigroups And Differential Equations In Banach Spaces," Noordhoff, Leyden, 1976.
13. V. Barbu, Existence for nonlinear Volterra equations in Hubert spaces, SIAM J. Math. Anal., 10 (1979), 552-569.
14. D. Blanchard, A. Damlamian, H. Ghidouche, A nonlinear system for phase change with dissipation, Differential Integral Equations, 2 (1989), 344-362.
15. B. Bonetti, G. Schimperna, Local existence for Frémond's model of damage in elastic materials, Contin. Mech. Thermodyn., 16 (2004), 319-335.
16. E. Bonetti, G. Schimperna, A. Segatti, On a doubly non linear model for the evolution of damaging in viscoelastic materials, J. Differential Equations, 218 (2005), 91-116.
17. G. Bonfanti, M. Frémond, F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24.
18. H. Brezis, "Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hubert," North-Holland Math. Studies n. 5, North-Holland, Amsterdam, 1973.
19. R. Chill, M.A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal., 53 (2003), 1017-1039.
20. R. Chill, A. Fiorenza, Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic problems, J. Differential Equations, 228 (2006), 611-632.
21. F.H. Clarke, "Optimization and Nonsmooth Analysis," Classics in Applied Mathematics, vol. 5, SIAM, Philadelphia, PA, 1990.
22. P. Colli, On some doubly nonlinear evolution equations in Banach spaces, Japan J. Indust. Appl. Math., 9 (1992), 181-203.
23. P. Colli, F. Luterotti, G. Schimperna, U. Stefanelli, Global existence for a class of generalized systems for irreversible phase changes, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 255-276.
24. P. Colli, A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations, 15 (1990), 737-756.
25. G. Dal Maso, A. DeSimone, M. G. Mora, Quasistatic evolution problems for linearly elastic - perfectly plastic materials, Arch. Rational Mech. Anal., 180 (2006), 237-291.
26. G. Dal Maso, G. Francfort, R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225.
27. E. DiBenedetto, R. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal., 12 (1981), 731-751.
28. E. Feireisl, F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Differential Equations, 12 (2000), 647-673.
29. G. Francfort, A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math., 595 (2006), 55-91.
30. M. Frémond, K. Kuttler, M. Shillor, Existence and uniqueness of solutions for a dynamic one-dimensional damage model, J. Math. Anal. Appl., 229 (1999), 271-294.
31. M. Frémond, K. Kuttler, B. Nedjar, M. Shillor, One-dimensional models of damage, Adv. Math. Sci. Appl., 8 (1998), 541-570.
32. M. Frémond, A. Visintin, Dissipation dans le changement de phase. Surfusion. Changement de phase irréversible, C. R. Acad. Sci. Paris Sér. II Mec. Phys. Chim. Sei. Univers Sci. Terre, 301 (1985), 1265-1268.
33. M. Giaquinta, "Multiple Integrals in The Calculus of Variations and Nonlinear Elliptic Systems," Annals of Mathematics Studies, 105, Princeton University Press, Princeton, 1983.
34. G. Gilardi, U. Stefanelli, Time-discretization and global solution for a doubly nonlinear Volterra equation, J. Differential Equations, 228 (2006), 707-736.
35. D. Gilbarg, N.S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, second edition, 1983.
36. O. Grange, F. Mignot. Sur la résolution d'une équation et d'une inéquation paraboliques non linéaires, J. Functional Analysis, 11 (1972), 77-92.
37. M. Grasselli, H. Petzeltová, G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72.
38. M. Grasselli, H. Petzeltová, G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system, Commun. Pure Appl. Anal., 5 (2006), 827-838.
39. A. Haraux, Systèmes dynamiques dissipatifs et applications, in "Recherches en Mathematiques Appliquees," vol. 17, Massen, Paris, 1991.
40. A. Haraux, M.A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (1999), 95-124.
41. M.A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differential Equations, 144 (1998), 302-312.
42. J.W. Jerome, "Approximation of Nonlinear Evolution Systems," Math. Sci. Engrg., vol. 164, Academic Press, Orlando, 1983.
43. N. Kenmochi, I. Pawlow, A class of nonlinear elliptic-parabolic equations with time-dependent constraints, Nonlinear Anal., 10 (1986), 1181-1202.
44. S. Łojasiewicz, Une propriété topologique des sous ensembles analytiques réels, in "Colloques Internationaux du C.N.R.S. n. 117: Les équations Aux dérivées Partielles" (Eds: C.N.R.S.), Paris Paris, (1962), 87-89.
45. S. Łojasiewicz, "Ensembles Semi-analytiques," notes, I.H.E.S., Bures-sur-Yvette, 1965.
46. F. Luterotti, G. Schimperna, U. Stefanelli, Existence result for a nonlinear model related to irreversible phase changes, Math. Models Methods Appl. Sci., 11 (2001), 809-825.
47. F. Luterotti, G. Schimperna, U. Stefanelli, Global solution to a phase-field model with irreversible and constrained phase evolution, Quart. Appl. Math., 60 (2002), 301-316.
48. A. Mainik, A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.
49. A. Mielke, Evolution of rate-independent systems, in "Handbook of Differential Equations, Evolutionary Equations" (eds: C. Dafermos and E. Feireisl), 461-559, Elsevier, 2005.
50. A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn., 15 (2003), 351-382.
51. A. Mielke, Evolution of rate-independent inelasticity with microstructure using relaxation and Young measures, in "IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains" (Stuttgart, 2001), Solid Mech. Appl., vol. 108, 33-44, Kluwer Acad. Publ., Dordrecht, 2003.
52. A. Mielke, Existence of minimizers in incremental elasto-plasticity with finite strains, SIAM J. Math. Anal., 36 (2004), 384-404.
53. A. Mielke, R. Rossi, Existence and uniqueness results for general rate-independent hysteresis problems, Math. Models Methods Appl. Sci., (2006), to appear.
54. A. Mielke, T. Roubíček, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci., 16 (2006), 177-209.
55. A. Mielke, F. Theil, V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177.
56. A. Mielke, A.M. Timofte, An energetic material model for time-dependent ferroelectric behavior: existence and uniqueness, Math. Meth. Appl. Sci., 29 (2005), 1399-1410.
57. A. Miranville, Finite dimensional global attractor for a class of doubly nonlinear parabilic equations, Cent. Eur. J. Math, 4 (2006), 163-182.
58. A. Miranville, S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.
59. R.H. Nochetto, G. Savaré, C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math., 53 (2000), 525-589.
60. G. Savaré, Regularity results for elliptic equations in Lipschitz domains, J. Funct. Anal., 152 (1998), 176-201.
61. A. Segatti, Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete Contin. Dyn. Syst., 14 (2006), 801-820.
62. T. Senba, On some nonlinear evolution equation, Funkcial. Ekvac., 29 (1986), 243-257.
63. K. Shirakawa, Large time behaviour or doubly nonlinear systems generated by subdifferentials, Adv. Math. Sci. Appl., 10 (2000), 417-442
64. P(0,T;B), Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
65. L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math. (2), 118 (1983), 525-571.
66. U. Stefanelli, On a class of doubly nonlinear nonlocal evolution equations, Differential Integral Equations, 15 (2002), 897-922.
67. U. Stefanelli, On some nonlocal evolution equations in Banach spaces, J. Evol. Equ., 4 (2004), 1-26.
68. A. Visintin, Models of phase transitions, in "Progress in Nonlinear Differential Equations and Their Applications,", vol. 28, Birkhäuser, 1996.