Study of ODE limit problems for reaction-diffusion equations

• Autorzy: Simsen J. , Simsen M. S. , Zimmermann A.

Identyfikatory

DOI

10.7494/OpMath.2018.38.1.117

• Języki publikacji: EN

Abstrakty

EN

In this work we study ODE limit problems for reaction-diffusion equations for large diffusion and we study the sensitivity of nonlinear ODEs with respect to initial conditions and exponent parameters. Moreover, we prove continuity of the flow and weak upper semicontinuity of a family of global attractors for reaction-diffusion equations with spatially variable exponents when the exponents go to 2 in [formula] and the diffusion coefficients go to infinity.

Słowa kluczowe

EN

ODE limit problems; shadow systems; reaction-diffusion equations; parabolic problems; variable exponents; attractors; upper semicontinuity

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2018
• Tom: Vol. 38, no. 1
• Strony: 117--131
• Opis fizyczny: Bibliogr. 28 poz.

Twórcy

autor

Simsen J.

• Universidade Federal de Itajuba Instituto de Matematica e Computacao Av. BPS n. 1303, Bairro Pinheirinho 37500-903, Itajuba - MG - Brazil
• Universitat of Duisburg-Essen Fakultat fur Mathematik Thea-Leymann-Str. 9, 45127 Essen, Germany

autor

Simsen M. S.

• Universidade Federal de Itajuba Instituto de Matematica e Computacao Av. BPS n. 1303, Bairro Pinheirinho 37500-903, Itajuba - MG - Brazil
• Universitat of Duisburg-Essen Fakultat fur Mathematik Thea-Leymann-Str. 9, 45127 Essen, Germany

autor

Zimmermann A.

• Universitat of Duisburg-Essen Fakultat fur Mathematik Thea-Leymann-Str. 9, 45127 Essen, Germany

Bibliografia

• [1] L. Arnold, I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dynamics and Stability of Systems 13 (1998) 3, 265-280.
• [2] J.M. Arrieta, A.N. Carvalho, A. Rodriguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations 168 (2000), 33-59.
• [3] T. Caraballo, J.A. Langa, J. Valero, Asymptotic behaviour of monotone multi-valued dynamical systems, Dyn. Syst. 20 (2005) 3, 301-321.
• [4] A.N. Carvalho, Infinite dimensional dynamics described by ordinary differential equations, J. Differential Equations 116 (1995), 338-404.
• [5] A.N. Carvalho, J.K. Hale, Large diffusion with dispersion, Nonlinear Anal. 17 (1991) 12, 1139-1151.
• [6] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Math. 66 (2006), 1383-1406.
• [7] J.W. Cholewa, A. Rodriguez-Bernal, Extremal equilibria for monotone semigroups in ordered spaces with applications to evolutionary equations, J. Differential Equations 249 (2010), 485-525.
• [8] E. Conway, D. Hoff, J. Smoller, Large time behavior of solutions of systems of non-linear reaction-diffusion equations, SIAM J. Appl. Math. 35 (1978) 1, 1-16.
• [9] L. Diening, P. Harjulehto, P. Hasto, M. Rużićka, Le.be.sgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011.
• [10] F. Ettwein, M. Rużićka, Existence of local strong solutions for motions of electrorheological fluids in three dimensions, Computers and Mathematics with Applications 53 (2007), 595-604.
• [11] Z. Guo, Q. Liu, J. Sun, B. Wu, Reaction-diffusion systems with p{x)-growth for image denoising, Nonlinear Anal. Real World Appl. 12 (2011), 2904-2918.
• [12] J.K. Hale, Large diffusivity and asymptotic behavior in parabolic systems, J. Math. Anal. Appl. 118 (1986), 455-466.
• [13] J.K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988.
• [14] J.K. Hale, C. Rocha, Varying boundary conditions with large diffusivity, J. Math. Pures Appl. 66 (1987), 139-158.
• [15] J.K. Hale, K. Sakamoto, Shadow systems and attractors in reaction-diffusion equations, Appl. Anal. 32 (1989), 287-303.
• [16] Ph. Hartman, Ordinary Differential Equations, Classics Appl. Math., vol. 38, SIAM, Philadelphia, 2002.
• [17] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Lezioni Lincee, 1991.
• [18] De. Liu, The critical forms of the attractors for semigroups and the existence of critical attractors, Proc. Roy Soc. Lond. Ser. A Math. Phys. Eng. Sci. 454 (1998), 2157-2171.
• [19] K. Rajagopal, M. Rużićka, Mathematical modelling of electrorheological materials, Contin. Mech. Thermodyn. 13 (2001), 59-78.
• [20] M. Rużićka, Flow of shear dependent electrorheological fluids, C.R. Acad. Sci. Paris Ser. I 329 (1999), 393-398.
• [21] M. Rużićka, Electrorheological Fluids: Modeling and Mathematical Theory, Lectures Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000.
• [22] J. Simsen, C.B. Gentile, Well-posed p-Lapladan problems with large diffusion, Nonlinear Anal. 71 (2009), 4609-4617.
• [23] J. Simsen, M.S. Simsen, PDE and ODE limit problems for p{x)-Laplacian parabolic equations, J. Math. Anal. Appl. 383 (2011), 71-81.
• [24] J. Simsen, M.S. Simsen, M.R.T. Primo, Continuity of the flows and upper semicontinuity of global attractors for ps{x)-Laplacian parabolic problems, J. Math. Anal. Appl. 398 (2013), 138-150.
• [25] J. Simsen, M.S. Simsen, M.R.T. Primo, On ps(x)-Laplacian parabolic problems with non-globally Lipschitz forcing term, Z. Anal. Anwend. 33 (2014), 447-462.
• [26] J. Simsen, M.S. Simsen, M.R.T. Primo, Reaction-diffusion equations with spatially variable exponents and large diffusion, Commun. Pure Appl. Anal. 15 (2016) 2, 495-506.
• [27] H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, 1995.
• [28] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer--Verlag, New York, 1988.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).