Abstract
The long time dynamics for a semilinear system of reaction and diffusion equations with nonlinear boundary conditions in which large diffusion is assumed on all parts of the domain is studied. We show in both local and global dynamics of the system that flows on attractors are essentially close to the ones of a finite dimensional system of equations, which turn out to be the natural limit of the process for large diffusivity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Alikakos, N.D. (1981). Regularity and asymptotic behaviour for second order parabolic equation with nonlinear boundary condition in L p. J.Diff.Eqns 39, 311–344.
Alikakos, N.D. (1979). An application of the invariance principle to reaction-diffusion Equations. J.Diff.Eqns 33, 201–225.
Arrieta, J., Rodríguez-Bernal, A.,and de Carvalho, A.N. (1999). Parabolic problems with nonlinear boundary conditions and critical nonlinearities. J.Diff.Eqns 165, 376–406.
Aarrieta, J., Rodríguez-Bernal, A.,and de Carvalho, A.N. (1990). Critical Nonlinear at Boundary, C.R.Acad.Sci. Paris, t.327,Serie I, pp.353–358.
Arrieta, J., Rodríguez-Bernal, A.,and de Carvalho, A.N. (2000). Attractors for parabolic problems with nonlinear boundary conditions.Uniform bounds. Commun.Partial Diff. Eqns 25(1/2), 1–33.
Arrieta, J., Rodríguez-Bernal, A.,and de Carvalho, A.N. (2000). Upper semicontinuity of attractors for parabolic problems with localized large diffusion and nonlinear boundary conditions. J.Diff.Eqns 168, 33–59.
Brézis, H. (1983). Anális Funcional, Alianza Universidad Textos, Masson Editeur de Paris.
Carvalho, A.N. (1995). In nite dimensional dynamics described by ordinary differential equations. J.Diff.Eqns 116(2), 338–404.
Carvalho, A.N.,and Pereira, A.L. (1994). A scalar parabolic equation whose asymptotic behavior is dictated by a system of ordinary differential equations. J.Diff.Eqns 112, 81–130.
Carvalho, A.N. (1997). Parabolic problems with nonlinear boundary conditions in cell tissues. Resenhas IME-USP 3(1), 123–138.
Carvalho, A.N.,and Hale, J.K. (1991). Large diffusion with dispersion. Nonlinear Anal. Theory,Meth.Appl. 17(12), 1139–1151.
Carvalho, A.N., Oliva, S.M., Pereira, A.L.,and Rodríguez-Bernal, A. (1997). Attractors for parabolic problems with nonlinear boundary conditions. J.Math.Anal.Appl. 207(2), 409–461.
Conway, E., Hoff, D.,and Smoller, J. (1978). Large time behavior of solutions of systems of nonlinear reaction–diffusion equations. SIAM J.Appl.Math. 35(1), 1–16
Cosner, C. (1981). Pointwise a prior bounds for strongly coupled semilinear system of parabolic partial differential equations. Indiana Univ.Math.J. 30(4), 607–620.
Foias, C., Sell, G.R.,and Temam, R. (1988). Inertial manifold for nonlinear evolutionary equations. J.Diff.Eqns 73, 309–353.
Friedman, A. (1964). Partial differential equations of parabolic type, Prentice Hall, Eaglewood IC,NJ.
Hale, J.K. (1986). Large diffusivity and asymptotic behavior in parabolic systems. J.Math.Anal.Appl. 118, 455–466.
Hale, J.K. (1988). Asymptotic behavior of dissipative systems. Math.Surv.Monogr. Vol.25, AMS.
Hale, J.K. (1969). Ordinary Differential Equations, Wiley-Interscience, New York.
Hale, J.K., Magalhães, L.,and Oliva, W. (1984). An introduction to in nite dimensional dynamical systems-geometric theory. Applied Math.Sciences, Vol. 47, Springer-Verlag.
Hale, J.K.,and Rocha, C. (1987). Varying boundary conditions with large diffusivity. J.Pures et Appl. 66, 139–158.
Hale, J.K.,and Rocha, C. (1987). Interaction of diffusion and boundary conditions. Nonlinear Anal.Theory,Meth.Appl. 11(5), 633–649.
Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations, Lecture notes in mathematics, Vol. 840, Springer-Verlag.
Lunardi, A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications, Vol.16, Birkhäuser, Basel.
Mora, X. (1983). Semilinear parabolic problemas define flows on Ck spaces. Trans.AMS 278(1), 21–55.
Nee Chow S.,and Lu, K. (1988). Invariant manifolds for flows in banach spaces. J.Diff. Eqns 74, 283–317.
Protter, M.H.,and Weinberger, H.F. (1967). Maximum Principles in Differential Equations. Englewood Cliffs, N.J.Prentice-Hall.
Rodríguez-Bernal, A. (1991). Sistemas Dinámicos In nito Dimensionales.Aplicaciones a la Ecuación de Kuramoto-Velarde, Tesis Doctoral,Universidad Complutense de Madrid.
Rodríguez-Bernal, A. (1998). Localized spatial homogenization and large difusion. SIAM J.Math.Anal. 29(6), 1361–1380.
Rodríguez-Bernal, A. (1990). Inertial manifolds for dissipative semi flows in banach spaces. Appl.Anal. 37, 95–141.
Rodríguez-Bernal, A.,and Willie, R. (2004). Singular large diffusivity and spatial homogenization in non homogeneous linear parabolic problems. Preprint,Universidad Complutense Madrid,Matemática Aplicada,28040 Madrid.
Renardy, M.,and Rogers, R.C. (1993). An Introduction to Partial Differential Equations. Texts in Applied Mathematics, Vol. 13, Springer-Verlag.
Sell, G.R.,and Yungcheng, Y. (1992). Inertial manifolds:the non self adjoint case. J.Diff.Eqns 96, 203–255.
Willie, R. (1999). Ecuaciones de Reacción y Diffusion con Diffusion Alta. Tesis Doctoral, U.C.M.Depto de Mat.Aplicada,Madrid,España.
Willie, R. (2003). The inertial manifold for a nonlinear reaction and diffusion system with large diffusion. Submitted to J.Diff.Eqns.
Willie, R. (2004). Upper semi-continuity of attractors for a semilinear reaction–diffusion system of equations in sup norm. To appear in Journal des Math.Pures et Appl.
Yoshizawa, T. (1966). Stability Theory by Liapunov's Second Method. Univ.Tokyo Press.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Willie, R. A Semilinear Reaction–Diffusion System of Equations and Large Diffusion. Journal of Dynamics and Differential Equations 16, 35–64 (2004). https://doi.org/10.1023/B:JODY.0000041280.69325.a4
Issue Date:
DOI: https://doi.org/10.1023/B:JODY.0000041280.69325.a4