Attractors for parabolic problems with nonlinear boundary conditions

Journal of Mathematical Analysis and Applications

Volumn 207, Issue 2, 1997, Pages 409-461

  Carvalho, Alexandre N ^a   Oliva, Sergio M ^a   Pereira, Antônio L ^a   Rodriguez Bernal, Aníbal ^b  

^a UNIVERSITY OF SÃO PAULO   (Brazil)

^b UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; EIGENVALUES AND EIGENFUNCTIONS; LYAPUNOV METHODS; MATHEMATICAL OPERATORS; OPTIMIZATION; PHASE SPACE METHODS; PROBLEM SOLVING;

ATTRACTORS; LA SALLE INVARIANCE PRINCIPLE; LIPSCHITZ FUNCTIONS; NONLINEAR BOUNDARY CONDITIONS; PARABOLIC PROBLEMS; VARIATION OF CONSTANTS;

NONLINEAR EQUATIONS;

EID: 0031084267     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.1997.5282     Document Type: Article

References (41)

1. Alikakos N. D. Regularity and asymptotic behavior for the second order parabolic equation with nonlinear boundary conditions. J. Differential Equations. 39:1981;311-344.
2. Amann H. Parabolic evolution equations and nonlinear boundary conditions. J. Differential Equations. 72:1988;201-269.
3. Amann H. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. Schmeisser H., Triebel H. Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte Math. 133:1993;9-126 Teubner, Leipzig.
4. Carvalho A. N. Contracting sets and dissipation. Proc. Roy. Soc. Edinburgh. 125A:1995;1305-1329.
5. Carvalho A. N. Infinite dimensional dynamics described by ordinary differential equations. J. Differential Equations. 116:1995;338-404.
6. Carvalho A. N., Hale J. K. Large diffusion with dispersion. Nonlinear Anal. 17:1991;1139-1151.
7. Carvalho A. N., Oliveira A. F. Delay-partial differential equations with some large diffusion. Nonlinear Anal. 22:1994;1057-1095.
8. Carvalho A. N., Ruas-Filho J. G. Global attractors for parabolic problems in fractional power spaces. SIAM J. Math. Anal. 26:1995;415-427.
9. A. N. Carvalho, A. Rodriguez Bernal, Upper semi-continuity of attractors for parabolic problems with localized large diffusion and nonlinear boundary condition.
10. Chueh K., Conley C., Smoller J. Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26:1977;373-392.
11. Courant R., Hilbert D. Methods of Mathematical Physics, I. 1989;Wiley, New York.
12. 1 + α. J. Fac. Sci. Univ. Tokyo. 13:1966;109-124.
13. Grisvard P. Caractérisation de quelques espaces d'interpolation. Arch. Rational Mech. Anal. 25:1967;40-63.
14. Grisvard P. Elliptic Problems in Non-smooth Domains. 1985;Pitman, London.
15. Hale J. K. Large diffusivity and asymptotic behavior in parabolic systems. J. Math. Anal. Appl. 118:1986;455-466.
16. Hale J. K. Asymptotic behavior of dissipative systems. Mathematical Surveys and Monographs. 1988;Amer. Math. Soc. Providence.
17. J. K. Hale, G. Raugel, 1992, Attractors for dissipative evolutionary equations.
18. Hale J. K., Rocha C. Varying boundary conditions with large diffusivity. J. Math. Pures Appl. 66:1987;139-158.
19. Hale J. K., Rocha C. Interaction of diffusion and boundary conditions. Nonlinear Anal. 11:1987;633-649.
20. Hale J. K., Sakamoto K. Shadow systems and attractors in reaction-diffusion equations. Appl. Anal. 32:1989;287-303.
21. Henry D. Geometric theory of semilinear parabolic equations. Lectures Notes in Mathematics. 840:1981;Springer-Verlag, New York/Berlin.
22. Henry D. Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations. J. Differential Equations. 59:1985;165-205.
23. Krein M. G., Rutman M. A. Linear operators leaving invariant a cone in a Banach space. Amer. Math. Soc. Transl. Ser. 1. 10:1962;3-95.
24. t = -AuFu. Arch. Rational Mech. Anal. 51:1971;371-386.
25. Levine H., Payne L. Nonexistence theorems for the heat equation with non-linear boundary conditions and for the porus medium equation backward in time. J. Differential Equations. 16:1974;319-334.
26. López-Gomez J., Marquez V., Wolanski N. Blow-up results and localization of blow-up points for the heat equation with a non-linear boundary condition. J. Differential Equations. 92:1991;384-401.
27. Marion M. Attractors for reaction-diffusion equations: Existence and estimate of their dimension. Appl. Anal. 25:1987;101-147.
28. kspaces. Trans. Amer. Math. Soc. 278:1983;21-55.
29. Moser J. A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20:1971.
30. Pao C. V. Asymptotic behavior and non-existence of global solutions for a class on non-linear boundary value problems of parabolic type. J. Math. Anal. Appl. 65:1978;616-637.
31. Pogorzelski W. Étude de la solution fondamental de l'équation parabolique. Ricerche Mat. 5:1957;25-57.
32. Protter M. H., Weinberger H. F. Maximum Principles in Differential Equations. 1967;Prentice Hall, New York.
33. Protter M. H., Weinberger H. F. On the spectrum of general second order operators. Bull. Amer. Math. Soc. 72:1966;251-255.
34. A. Rodriguez-Bernal, 1991, Existence Uniqueness and Regularity of Solutions of Evolution Equations in Extended Scales of Hilbert Spaces.
35. A. Rodriguez-Bernal, 1993, Some remarks on PDEs with nonlinear Newman boundary conditions.
36. Rodriguez-Bernal A., Zuazua E. Parabolic singular limit of a wave equation with localized boundary damping. Discrete Continuous Dynam. Systems. 1:1995;303-346.
37. Smoller J. Shock waves and reaction-diffusion equations. Grundlehren Math. Wiss. 1983;Springer-Verlag, New York/Berlin.
38. Temam R. Infinite dimensional dynamical systems in mechanics and physics. Appl. Math. Sci. 1988;Springer-Verlag, New York/Berlin.
39. Trudinger N. S. On inbedding into Orlicz spaces and some applications. J. Math. Mech. 17:1967;473-483.
40. Walter W. On existence and nonexistence in the large of solutions of parabolic differential equations with a non-linear boundary condition. SIAM J. Math. Anal. 6:1975;85-90.
41. Wang M., Wu Y. Global existence and blow-up problems for quasilinear parabolic equations with non-linear boundary conditions. SIAM J. Math. Anal. 24:1993;1515-1521.