Opuscula Math. 41, no. 4 (2021), 539-570
https://doi.org/10.7494/OpMath.2021.41.4.539

 
Opuscula Mathematica

Reaction-diffusion coupled inclusions with variable exponents and large diffusion

Jacson Simsen
Mariza Stefanello Simsen
Petra Wittbold

Abstract. This work concerns the study of asymptotic behavior of coupled systems of \(p(x)\)-Laplacian differential inclusions. We obtain that the generalized semiflow generated by the coupled system has a global attractor, we prove continuity of the solutions with respect to initial conditions and a triple of parameters and we prove upper semicontinuity of a family of global attractors for reaction-diffusion systems with spatially variable exponents when the exponents go to constants greater than 2 in the topology of \(L^{\infty}(\Omega)\) and the diffusion coefficients go to infinity.

Keywords: reaction-diffusion coupled systems, variable exponents, attractors, upper semicontinuity, large diffusion.

Mathematics Subject Classification: 35K55, 35K92, 35A16, 35B40, 35B41.

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  1. C.O. Alves, S. Shmarev, J. Simsen, M.S. Simsen, The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: existence and asymptotic behavior, J. Math. Anal. Appl. 443 (2016), no. 1, 265-294.
  2. J.M. Arrieta, A.N. Carvalho, A. Rodríguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations 168 (2000), 33-59.
  3. J.P. Aubin, A. Cellina, Differential inclusions: Set-valued maps and viability theory, Springer-Verlag, Berlin, 1984.
  4. J.P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
  5. J.M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci. 7 (1997), no. 5, 475-502.
  6. F.D.M. Bezerra, J. Simsen, M.S. Simsen, Convergence of quasilinear parabolic equations to semilinear equations, DCDS-B 26 (2021), no. 7, 3823-3834.
  7. H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Company, Amsterdam, 1973.
  8. H. Brézis, Analyse fonctionnelle: Théorie et applications, Masson, Paris, 1983.
  9. A.N. Carvalho, Infinite dimensional dynamics described by ordinary differential equations, J. Differential Equations 116 (1995), 338-404.
  10. A.N. Carvalho, J.K. Hale, Large diffusion with dispersion, Nonlinear Anal. 17 (1991), no. 12, 1139-1151.
  11. A.N. Carvalho, S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim. 27 (2006), no. 7-8, 785-829.
  12. 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), no. 1, 1-16.
  13. J.I. Díaz, I.I. Vrabie, Existence for reaction diffusion systems. A compactness method approach, J. Math. Anal. Appl. 188 (1994), 521-540.
  14. L. Diening, P. Harjulehto, P. Hästö, M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011.
  15. X.L. Fan, Q.H. Zhang, Existence of solutions for \(p(x)\)-Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003), 1843-1852.
  16. J.K. Hale, Large diffusivity and asymptotic behavior in parabolic systems, J. Math. Anal. Appl.118 (1986), 455-466.
  17. J.K. Hale, C. Rocha, Varying boundary conditions with large diffusivity, J. Math. Pures Appl. 66 (1987), 139-158.
  18. J.K. Hale, K. Sakamoto, Shadow systems and attractors in reaction-diffusion equations, Applicable Analysis 32 (1989), 287-303.
  19. S. Shmarev, J. Simsen, M.S. Simsen, M.R.T. Primo, Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces, Asymptotic Analysis 111 (2019), 43-68.
  20. J. Simsen, Partial differential inclusions with spatially variable exponents and large diffusion, Mathematics in Engineering, Science and Aerospace (MESA) 7 (2016), no. 3, 479-489.
  21. J. Simsen, C.B. Gentile, On attractors for multivalued semigroups defined by generalized semiflows, Set-Valued Anal. 16 (2008), no. 1, 105-124.
  22. J. Simsen, C.B. Gentile, On \(p\)-Laplacian differential inclusions - global existence, compactness properties and asymptotic behaviour, Nonlinear Anal. 71 (2009), 3488-3500.
  23. J. Simsen, C.B. Gentile, Systems of \(p\)-Laplacian differential inclusions with large diffusion, J. Math. Anal. Appl. 368 (2010), 525-537.
  24. J. Simsen, C.B. Gentile, Well-posed \(p\)-Laplacian problems with large diffusion, Nonlinear Anal. 71 (2009), 4609-4617.
  25. J. Simsen, M.S. Simsen, PDE and ODE limit problems for \(p(x)\)-Laplacian parabolic equations, J. Math. Anal. Appl. 383 (2011), 71-81.
  26. J. Simsen, M.S. Simsen, Existence and upper semicontinuity of global attractors for \(p(x)\)-Laplacian systems, J. Math. Anal. Appl. 388 (2012), 23-38.
  27. J. Simsen, M.S. Simsen, M.R.T. Primo, Continuity of the flows and upper semicontinuity of global attractors for \(p_s(x)\)-Laplacian parabolic problems, J. Math. Anal. Appl. 398 (2013), 138-150.
  28. J. Simsen, M.S. Simsen, M.R.T. Primo, On \(p_s(x)\)-Laplacian parabolic problems with non-globally Lipschitz forcing term, Zeitschrift fur Analysis und Ihre Anwendungen 33 (2014), 447–462.
  29. J. Simsen, M.S. Simsen, M.R.T. Primo, Reaction-diffusion equations with spatially variable exponents and large diffusion, Commun. Pure and Appl. Analysis 15 (2016), 495-506.
  30. J. Simsen, M.S. Simsen, F.B. Rocha, Existence of solutions for some classes of parabolic problems involving variable exponents, Nonlinear Studies 21 (2014), 113-128.
  31. J. Simsen, M.S. Simsen, On \(p(x)\)-Laplacian parabolic problems, Nonlinear Studies 18 (2011), 393-403.
  32. J. Simsen, M.S. Simsen, A. Zimmermann, Study of ODE limit problems for reaction-diffusion equations, Opuscula Math. 38 (2018), no. 1, 117-131.
  33. R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
  • Jacson Simsen (corresponding author)
  • ORCID iD https://orcid.org/0000-0002-6683-1363
  • Universidade Federal de Itajubá, Instituto de Matemática e Computação, Av. BPS n. 1303, Bairro Pinheirinho, 37 500-903, Itajubá - MG - Brazil
  • Mariza Stefanello Simsen
  • ORCID iD https://orcid.org/0000-0002-2378-2442
  • Universidade Federal de Itajubá, Instituto de Matemática e Computação, Av. BPS n. 1303, Bairro Pinheirinho, 37 500-903, Itajubá - MG - Brazil
  • Communicated by J.I. Díaz.
  • Received: 2021-03-23.
  • Revised: 2021-06-24.
  • Accepted: 2021-06-24.
  • Published online: 2021-07-09.
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Cite this article as:
Jacson Simsen, Mariza Stefanello Simsen, Petra Wittbold, Reaction-diffusion coupled inclusions with variable exponents and large diffusion, Opuscula Math. 41, no. 4 (2021), 539-570, https://doi.org/10.7494/OpMath.2021.41.4.539

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