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Variants and Additional Results

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Abstract

In the applications to semilinear PDE of parabolic or hyperbolic type, some conditions, necessary for the results of the previous chapter to be applicable, can in fact be relaxed. This has been the object of several research papers which we recall in this last chapter. Several other questions (rate of convergence, asymptotically autonomous case, infinite dimensional non convergence results) are also discussed with reference to the specialized literature.

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References

  1. S.-Z. Huang, Gradient inequalities. With applications to asymptotic behavior and stability of gradient-like systems. Mathematical surveys and monographs, vol. 126. American Mathematical Society, Providence, RI, 2006. viii+184 pp

    Google Scholar 

  2. R. Chill, On the Łojasiewicz-Simon gradient inequality. J. Funct. Anal. 201, 572–601 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Haraux, M.A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity. Asymptot. Anal. 26(1), 21–36 (2001)

    MathSciNet  MATH  Google Scholar 

  4. M.A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon. J. Funct. Anal. 153, 187–202 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. Math. (2) 118(3), 525–571 (1983)

    Google Scholar 

  6. O. Kavian, Introduction la théorie des points critiques et applications aux problèmes elliptiques. Mathématiques & Applications (Berlin), vol. 13. (Springer, Paris, 1993)

    Google Scholar 

  7. S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Chergui, Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity. J. Evol. Equ. 9, 405–418 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Aloui, I. Ben Hassen, A. Haraux, Compactness of trajectories to some nonlinear second order evolution equations and applications. J. Math. Pures Appl. (9) 100, 295–326 (2013)

    Google Scholar 

  10. A. Haraux, M.A. Jendoubi, O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations. Dedicated to Philippe Bnilan. J. Evol. Equ. 3(3), 463–484 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. I. Ben Hassen, A. Haraux, Convergence and decay estimates for a class of second order dissipative equations involving a non-negative potential energy. J. Funct. Anal. 260, 2933–2963 (2011)

    Google Scholar 

  12. A. Haraux, Slow and fast decay of solutions to some second order evolution equations. J. Anal. Math. 95, 297–321 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Haraux, Sharp decay estimates of the solutions to a class of nonlinear second order ODE’s. Anal. Appl. (Singap.) 9(1), 49–69 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. I. Ben Arbi, Rate of decay to 0 of the solutions to a nonlinear parabolic equation. Port. Math. 69(1), 2339 (2012)

    Google Scholar 

  15. I. Ben Arbi, A. Haraux, A sufficient condition for slow decay of a solution to a semilinear parabolic equation. Anal. Appl. (Singap.) 10(4), 363371 (2012)

    Google Scholar 

  16. M. Ghisi, M. Gobbino, A. Haraux, Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations, JEMS, in press

    Google Scholar 

  17. M. Ghisi, M. Gobbino, A. Haraux, A description of all possible decay rates for solutions of some semilinear parabolic equations, JMPA, in press

    Google Scholar 

  18. M. Ghisi, M. Gobbino, A. Haraux, Finding the exact decay rate of all solutions to some second order evolution equations with dissipation, to appear

    Google Scholar 

  19. S.-Z. Huang, P. Takač, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal., Ser. A, Theory. Methods 46, 675–698 (2001)

    MATH  Google Scholar 

  20. R. Chill, M.A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations. Nonlinear Anal. 53(7–8), 1017–1039 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Chill, M.A. Jendoubi, Convergence to steady states of solutions of non-autonomous heat equations in \(\mathbb{R}^N\). J. Dynam. Diff. Equat. 19(3), 777–788 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations. Asymptot. Anal. 69, 31–44 (2010)

    MathSciNet  MATH  Google Scholar 

  23. I. Ben Hassen, L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation. J. Dynam. Diff. Equat. 23, 315–332 (2011)

    Google Scholar 

  24. P. Poláčik, K.P. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations. J. Diff. Equat. 124, 472–494 (1996)

    Article  MATH  Google Scholar 

  25. P. Poláčik, F. Simondon, Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains. J. Diff. Equat. 186, 586–610 (2002)

    Article  MATH  Google Scholar 

  26. M.A. Jendoubi, P. Poláčik, Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping. Proc. Roy. Soc. Edinburgh Sect. A 133(5), 1137–1153 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Alain Haraux .

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Haraux, A., Jendoubi, M. (2015). Variants and Additional Results. In: The Convergence Problem for Dissipative Autonomous Systems. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-23407-6_12

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