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Non-genericity of initial data with punctual \(\omega \)-limit set

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Abstract

We show the existence of a non-linear differential equation of gradient type for which the set of initial values for which the corresponding solution converges to a point is non-generic.

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References

  1. Absil, P.A., Mahony, R., Andrews, B.: Convergence of the iterates of descent methods for analytic cost functions. SIAM J. Optim. 1(6), 531–547 (2005)

    Article  MathSciNet  Google Scholar 

  2. Jendoubi, M.A., Poláčik, P.: Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping. Proc. R. Soc. Edinb. Sect. A 133, 1137–1153 (2003)

    Article  MathSciNet  Google Scholar 

  3. Lions, P.L.: Structure of the set of the steady-state solutions and asymptotic behavior of semilinear heat equations. J. Differ. Equ. 53, 362–386 (1984)

    Article  Google Scholar 

  4. Łojasiewicz, S.: Ensembles semi-analytiques. Preprint, I.H.E.S. Bures-sur-Yvette (1965)

  5. Łojasiewicz, S.: Une propriété topologique des sous ensembles analytiques réels. Colloques internationaux du C.N.R.S # 117. Les équations aux dérivées partielles (1963)

  6. Palis, J., de Melo, W.: Geometric Theory of Dynamical Systems. Springer, New York (1982)

    Book  Google Scholar 

  7. Poláčik, P., Rybakowski, K.: Nonconvergent bounded trajectories in semilinear heat equations. J. Differ. Equ. 124, 472–494 (1996)

    Article  MathSciNet  Google Scholar 

  8. Poláčik, P., Simondon, F.: Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains. J. Differ. Equ. 186, 586–610 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The second author wishes to thank the department of mathematics and statistics and the research department M2N of the CNAM where this work has been initiated. The first author wishes to thank the Tunisian Mathematical Society (SMT) for its kind invitation to its annual congress during which this work has been completed.

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Authors and Affiliations

  1. Laboratoire M2N, EA7340, CNAM, 292 rue Saint-Martin, 75003, Paris, France

    Thierry Horsin

  2. Institut Préparatoire aux Etudes Scientifiques et Techniques, Université de Carthage, B.P. 51 2070, La Marsa, Tunisia

    Mohamed Ali Jendoubi

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  1. Thierry Horsin
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  2. Mohamed Ali Jendoubi
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Correspondence to Thierry Horsin.

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Horsin, T., Jendoubi, M.A. Non-genericity of initial data with punctual \(\omega \)-limit set. Arch. Math. 114, 185–193 (2020). https://doi.org/10.1007/s00013-019-01377-8

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  • DOI: https://doi.org/10.1007/s00013-019-01377-8

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