Invariant sets and connecting orbits for nonlinear evolution equations at resonance

Piotr Kokocki

Mathematica Bohemica (2015)

  • Volume: 140, Issue: 4, page 447-455
  • ISSN: 0862-7959

Abstract

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We study the problem of existence of orbits connecting stationary points for the nonlinear heat and strongly damped wave equations being at resonance at infinity. The main difficulty lies in the fact that the problems may have no solutions for general nonlinearity. To address this question we introduce geometrical assumptions for the nonlinear term and use them to prove index formulas expressing the Conley index of associated semiflows. We also prove that the geometrical assumptions are generalizations of the well known Landesman-Lazer and strong resonance conditions. Obtained index formulas are used to derive criteria determining the existence of orbits connecting stationary points.

How to cite

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Kokocki, Piotr. "Invariant sets and connecting orbits for nonlinear evolution equations at resonance." Mathematica Bohemica 140.4 (2015): 447-455. <http://eudml.org/doc/271822>.

@article{Kokocki2015,
abstract = {We study the problem of existence of orbits connecting stationary points for the nonlinear heat and strongly damped wave equations being at resonance at infinity. The main difficulty lies in the fact that the problems may have no solutions for general nonlinearity. To address this question we introduce geometrical assumptions for the nonlinear term and use them to prove index formulas expressing the Conley index of associated semiflows. We also prove that the geometrical assumptions are generalizations of the well known Landesman-Lazer and strong resonance conditions. Obtained index formulas are used to derive criteria determining the existence of orbits connecting stationary points.},
author = {Kokocki, Piotr},
journal = {Mathematica Bohemica},
keywords = {semigroup; evolution equation; invariant set; Conley index; resonance},
language = {eng},
number = {4},
pages = {447-455},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Invariant sets and connecting orbits for nonlinear evolution equations at resonance},
url = {http://eudml.org/doc/271822},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Kokocki, Piotr
TI - Invariant sets and connecting orbits for nonlinear evolution equations at resonance
JO - Mathematica Bohemica
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 140
IS - 4
SP - 447
EP - 455
AB - We study the problem of existence of orbits connecting stationary points for the nonlinear heat and strongly damped wave equations being at resonance at infinity. The main difficulty lies in the fact that the problems may have no solutions for general nonlinearity. To address this question we introduce geometrical assumptions for the nonlinear term and use them to prove index formulas expressing the Conley index of associated semiflows. We also prove that the geometrical assumptions are generalizations of the well known Landesman-Lazer and strong resonance conditions. Obtained index formulas are used to derive criteria determining the existence of orbits connecting stationary points.
LA - eng
KW - semigroup; evolution equation; invariant set; Conley index; resonance
UR - http://eudml.org/doc/271822
ER -

References

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  1. Bartolo, P., Benci, V., Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with ``strong'' resonance at infinity, Nonlinear Anal., Theory Methods Appl. 7 (1983), 981-1012. (1983) Zbl0522.58012MR0713209
  2. Ćwiszewski, A., Rybakowski, K. P., 10.1016/j.jde.2009.09.006, J. Differ. Equations 247 (2009), 3202-3233. (2009) Zbl1187.35002MR2571574DOI10.1016/j.jde.2009.09.006
  3. Henry, D., 10.1007/BFb0089647, Lecture Notes in Mathematics 840 Springer, Berlin (1981). (1981) Zbl0456.35001MR0610244DOI10.1007/BFb0089647
  4. Kokocki, P., 10.1016/j.na.2013.02.030, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 85 (2013), 253-278. (2013) Zbl1292.34059MR3040364DOI10.1016/j.na.2013.02.030
  5. Kokocki, P., 10.1016/j.jde.2013.05.012, J. Differ. Equations 255 (2013), 1554-1575. (2013) Zbl1302.34098MR3072663DOI10.1016/j.jde.2013.05.012
  6. Kokocki, P., Dynamics of Nonlinear Evolution Equations at Resonance, PhD dissertation, Nicolaus Copernicus University Toruń (2012). (2012) 
  7. Kokocki, P., 10.1016/j.na.2015.05.012, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 125 (2015), Article ID 10526, 167-200. (2015) MR3373579DOI10.1016/j.na.2015.05.012
  8. Landesman, E. M., Lazer, A. C., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1969/1970), 609-623. (1969) MR0267269
  9. Massatt, P., 10.1016/0022-0396(83)90098-0, J. Differential Equations 48 (1982), 334-349. (1982) MR0702424DOI10.1016/0022-0396(83)90098-0
  10. Prizzi, M., 10.4064/fm176-3-5, Fundam. Math. 176 (2003), 261-275. (2003) MR1992823DOI10.4064/fm176-3-5
  11. Rybakowski, K. P., The Homotopy Index and Partial Differential Equations, Universitext Springer, Berlin (1987). (1987) Zbl0628.58006MR0910097
  12. Rybakowski, K. P., Nontrivial solutions of elliptic boundary value problems with resonance at zero, Ann. Mat. Pura Appl. (4) 139 (1985), 237-277. (1985) Zbl0572.35037MR0798176

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