Skip to main content
SpringerLink
Log in

Topological Structural Stability of Partial Differential Equations on Projected Spaces

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In this paper we study topological structural stability for a family of nonlinear semigroups \(T_h(\cdot )\) on Banach space \(X_h\) depending on the parameter h. Our results shows the robustness of the internal dynamics and characterization of global attractors for projected Banach spaces, generalizing previous results for small perturbations of partial differential equations. We apply the results to an abstract semilinear equation with Dumbbell type domains and to an abstract evolution problem discretized by the finite element method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Aragão-Costa, E.R., Caraballo, T., Carvalho, A.N., Langa, J.A.: Stability of gradient semigroups under perturbations. Nonlinearity 24(7), 2099–2117 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arrieta, J.M., Bezerra, F.D.M., Carvalho, A.N.: Rate of convergence of attractors for some singularly perturbed parabolic problems. Topol. Methods Nonlinear Anal. Torun 41(2), 229–253 (2013)

    MATH  Google Scholar 

  3. Arrieta, J.M., Carvalho, A.N., Lozada-Cruz, G.J.: Dynamics in dumbbell domains I: continuity of the set of equilibria. J. Diff. Equ. N. Y. 231(2), 551–597 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arrieta, J.M., Carvalho, A.N., Lozada-Cruz, G.J.: Dynamics in dumbbell domains II: the limiting problem. J. Diff. Equ. N. Y. 247(1), 174–202 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arrieta, J.M., Carvalho, A.N., Lozada-Cruz, G.J.: Dynamics in dumbbell domains III: continuity of attractors. J. Diff. Equ. N. Y. 247(1), 225–259 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arrieta, J., Carvalho, A.N., Langa, J.A., Rodríguez-Bernal, A.: Continuity of dynamical structures for non-autonomous evolution equations under singular perturbations. J. Dyn. Diff. Eq. 24(3), 427–481 (2011)

    Article  Google Scholar 

  7. Arrieta, J.M., Carvalho, A.N., Rodríguez-Bernal, A.: Attractors of parabolic problems with nonlinear boundary conditions: uniform bounds. Commun. Partial Diff. Equ. N. Y. 25(1/2), 1–37 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Arrieta, J.M., Carvalho, A.N., Rodríguez-Bernal, A.: Parabolic problems with nonlinear boundary conditions and critical nonlinearities. J. Diff. Equ. N. Y. 156(2), 376–406 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Babin, A.V.; Vishik, M.I.: Attractors in Evolutionary Equations (Studies in mathematics and its applications, v. 25). North-Holland Publishing Company, Amsterdam (1992)

  10. Beyn, W.-J., Piskarev, S.: Shadowing for discrete approximations of abstract parabolic equations. Discrete Contin. Dyn. Syst., Ser. B 10(1), 19–42 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bortolan, M.C., Carvalho, A.N., Langa, J.A.: Structural stability of skew-product semiflows. J. Diff. Equ. N. Y. 257(2), 490–522 (2014)

    Article  MATH  Google Scholar 

  12. Bortolan, M.C., Carvalho, A.N., Langa, J.A., Raugel, G.: Non-autonomous Perturbations of Morse-Smale Semigroups: Stability of the Phase Diagram. In: progress (2015)

  13. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1996)

    MATH  Google Scholar 

  14. Brunovský, P., Poláčik, P.: The Morse–Smale structure of a generic reaction-diffusion equation in higher space dimension. J. Diff. Equ. 135, 129–181 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brunovský, P., Raugel, G.: Genericity of the Morse–Smale property for damped wave equations. J. Dyn. Diff. Equ. 15, 571–658 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Carvalho, A.N., Langa, J.A.: The existence and continuity of stable and unstable manifolds for semilinear problems under non-autonomous perturbation in Banach spaces. J. Diff. Equ. 233, 622–653 (2007)

    Article  MATH  Google Scholar 

  17. Carvalho, A.N., Langa, J.A.: An extension of the concept of gradient semigroup wich is stable under perturbations. J. Diff. Equ. N. Y. 246(7), 2646–2668 (2009)

    Article  MATH  Google Scholar 

  18. Carvalho, A.N., Langa, J.A., Robinson, J.C.: Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182. Springer, New York (2013)

    Book  Google Scholar 

  19. Carvalho, A.N., Piskarev, S.: A general approximations scheme for attractors of abstract parabolic problems. Numer. Funct. Anal. Opt. N. Y. 27(7/8), 785–829 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cholewa, J.W., Dlotko, T.: Global Attractors in Abstract Parabolic Problems. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  21. Conley, C.: Isolated Invariant Sets and the Morse index. CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I. (1978)

  22. Figueroa-López, R.N., Lozada-Cruz, G.: Dynamics of parabolic equations via the finite element method I. Continuity of the set of equilibria. J. Diff. Equ. N. Y. 261(9), 5235–5259 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Figueroa-López, R., Lozada-Cruz, G.: On global attractors for a class of parabolic problems. Appl. Math. Inf. Sci. Kingdom of Bahrain 8(2), 493–500 (2014)

  24. Figueroa-López, R.N., Lozada-Cruz, G.: Some estimates for resolvent operators under the discretization by finite element method. Comput. Appl. Math. 34(3), 1105–1116 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fujita, H., Mizutani, A.: On the finite element method for parabolic equations I: approximation of holomorphic semi-groups. J. Math. Soc. Jpn Tokyo 28(4), 749–771 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Surveys and Monographs, Amer. Math. Soc., Providence, Math (1988)

  27. Hale, J.K., Raugel, G.: A damped hyperbolic equation on thin domains. Trans. Am. Math. Soc. 329(1), 185–219 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hale, J.K., Raugel, G.: A modified Poincaré method for the persistence of periodic orbits and applications. J. Dyn. Diff. Equ. 22(1), 3–68 (2010)

    Article  MATH  Google Scholar 

  29. Hale, J.K., Raugel, G.: Convergence in dynamically gradient systems with applications to PDE. Zeitschrift für angewandte Mathematik und Physik ZAMP 43(1), 63–124 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hale, J.K., Raugel, G.: Lower semi-continuity of attractors of gradient systems and applications. Annal. Math. 154(1), 281–326 (1989)

    Article  MATH  Google Scholar 

  31. Hale, J.K., Lin, X.B., Raugel, G.: Upper semicontinuity of attractors for approximations of semigroups and partial differential equations. Math. Comput. 50(181), 89–123 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hale, J.K., Magalhães, L.T., Oliva, W.M.: An Introduction to Infinite-dimensional Dynamical Systems—Geometric Theory. Applied Mathematical Sciences, Vol. 47. Springer, Berlin (1984)

  33. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (1981)

    Book  MATH  Google Scholar 

  34. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)

    MATH  Google Scholar 

  35. Kloeden, P., Piskarev, S.: Discrete convergence and the equivalence of equi-attraction and the continuous convergence of attractors. Int. J. Dyn. Syst. Diff. Equ. 1(1), 38–43 (2007)

    MathSciNet  MATH  Google Scholar 

  36. Larsson, S.: Numerical analysis of semilinear parabolic problems, in Lecture notes from the 8th EPSRC summer school in numerical analysis (eds. M. Ainsworth et al.). Springer Ser. Comput. Math. 26(1999), 83–117 (1999)

    Article  Google Scholar 

  37. Ladyzhenskaya, O.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  38. Lu, K.: Structural, stability for scalar parabolic equations. J. Diff. Equ. N. Y. 114(1), 253–271 (1994)

  39. Norton, D.E.: The fundamental theorem of dynamical systems. Comment. Math., Univ. Carolinae 36(3), 585–597 (1995)

    MathSciNet  MATH  Google Scholar 

  40. Patrao, M.: Morse decomposition of semiflows on topological spaces. J. Dyn. Diff. Equ. 19(1), 181–198 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Robinson, J.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  42. Rybakowski, K.P.: The Homotopy Index and Partial Differential Equations. Universitext. Springer, New York (1987)

    Book  MATH  Google Scholar 

  43. Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  44. Stummel, F.: Diskrete konvergenz linearer operatoren III. Linear Operators and Approximation, Proceedings of the Conference on Oberwolfach, 1971, Birkhäauser, Basel, pp. 196–216 (1972)

  45. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  46. Vainikko, G.: Funktionalanalysis der diskretisierungsmethoden. BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1976)

    MATH  Google Scholar 

  47. Vainikko, G.: Approximative methods for nonlinear equations (two approaches to the convergence problem). Nonlinear Anal. Theory Methods Appl. Oxford 2(6), 647–687 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  48. Vainikko, G.: Regular convergence of operators and approximate solution of equations. Itogi Nauki i Tehniki: Seriya Matematicheskii Analiz, Moscow 16, 5–53 (1979)

    MathSciNet  Google Scholar 

  49. Vainikko, G.: Multidimensional Weakly Singular Integral Equations. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  50. Yagi, A.: Abstract Parabolic Evolution Equations and their Applications. Monographs in Mathematics. Springer, New York (2010)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee for his/her valuable suggestions. Parts of this work were made when the second author visited the Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Seville, Spain. E.R. Aragão-Costa was partially supported by Grant: 2014/02899-3, São Paulo Research Foundation (FAPESP), Brazil. R.N. Figueroa-López was partially supported by research Grants \(\#\)2014/19915-1 and \(\#\)2013/21155-2, São Paulo Research Foundation (FAPESP). G. Lozada-Cruz was partially supported by research Grants \(\#\)2015/24095-6 and \(\#\)2009/08435-0, São Paulo Research Foundation (FAPESP). J.A. Langa has been partially supported by Junta de Andalucía under Proyecto de Excelencia FQM-1492, Project MTM2015-63723-P, and Brazilian-European partnership in Dynamical Systems (BREUDS) from the FP7-IRSES Grant of the European Union.

Author information

Authors and Affiliations

  1. Instituto de Ciências Matemáticas e de Computação, Campus de São Carlos, Universidade de São Paulo, São Carlos, SP, Brazil

    E. R. Aragão-Costa

  2. Departamento de Matemática, IBILCE, Universidade Estadual Paulista (UNESP), São José do Rio Preto, São Paulo, 15054-000, Brazil

    R. N. Figueroa-López & G. Lozada-Cruz

  3. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla (US), Sevilla, Spain

    J. A. Langa

Authors
  1. E. R. Aragão-Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. N. Figueroa-López
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. A. Langa
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. Lozada-Cruz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. A. Langa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aragão-Costa, E.R., Figueroa-López, R.N., Langa, J.A. et al. Topological Structural Stability of Partial Differential Equations on Projected Spaces. J Dyn Diff Equat 30, 687–718 (2018). https://doi.org/10.1007/s10884-016-9567-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-016-9567-x

Keywords

Search

Navigation