| Title:
|
Invariant sets and connecting orbits for nonlinear evolution equations at resonance (English) |
| Author:
|
Kokocki, Piotr |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
140 |
| Issue:
|
4 |
| Year:
|
2015 |
| Pages:
|
447-455 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
semigroup |
| Keyword:
|
evolution equation |
| Keyword:
|
invariant set |
| Keyword:
|
Conley index |
| Keyword:
|
resonance |
| MSC:
|
35L10 |
| MSC:
|
35P05 |
| MSC:
|
37B30 |
| idZBL:
|
Zbl 06537676 |
| idMR:
|
MR3432545 |
| DOI:
|
10.21136/MB.2015.144462 |
| . |
| Date available:
|
2015-11-17T20:50:12Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144462 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[4] Kokocki, P.: The averaging principle and periodic solutions for nonlinear evolution equations at resonance.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 85 (2013), 253-278. Zbl 1292.34059, MR 3040364, 10.1016/j.na.2013.02.030 |
| Reference:
|
[5] Kokocki, P.: Connecting orbits for nonlinear differential equations at resonance.J. Differ. Equations 255 (2013), 1554-1575. Zbl 1302.34098, MR 3072663, 10.1016/j.jde.2013.05.012 |
| Reference:
|
[6] Kokocki, P.: Dynamics of Nonlinear Evolution Equations at Resonance, PhD dissertation.Nicolaus Copernicus University Toruń (2012). |
| Reference:
|
[7] Kokocki, P.: Effect of resonance on the existence of peridic solutions for strongly damped wave equation.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 125 (2015), Article ID 10526, 167-200. MR 3373579, 10.1016/j.na.2015.05.012 |
| Reference:
|
[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. MR 0267269 |
| Reference:
|
[9] Massatt, P.: Limiting behavior for strongly damped nonlinear wave equations. Nonlinear phenomena in mathematical sciences, Proc. Int. Conf., Arlington/Tex., 1980.J. Differential Equations 48 (1982), 334-349. MR 0702424, 10.1016/0022-0396(83)90098-0 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
