- Previous Article
- CPAA Home
- This Issue
-
Next Article
Asymptotic behavior in a general diffusive three-species predator-prey model
Families of periodic orbits for some PDE’s in higher dimensions
| 1. | Dipartimento di Matematica “F. Enriques”, Universita di Milano, Via Saldini 50, 20133 Milano, Italy, Italy |
$u_{t t} +\Delta \Delta u + m u = \psi(x, u) ,$
$\psi(x, u) = \pm u^3 + O(u^5),\quad \psi(-x,-u)=-\psi(x,u), \qquad\qquad\qquad $(1)
with Navier boundary conditions in a $n$–dimensional cube, here $\psi$ is a $C^\infty$ function, and $m$ is a positive parameter. For this equation we construct some Cantor families of periodic orbits. Our proof is very simple and is based on contraction mapping principle and on a suitable correspondence between Lyapunov Schmidt decomposition and averaging theory.
| [1] |
Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739 |
| [2] |
Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201 |
| [3] |
N. Kamran, K. Tenenblat. Periodic systems for the higher-dimensional Laplace transformation. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 359-378. doi: 10.3934/dcds.1998.4.359 |
| [4] |
Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071 |
| [5] |
Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048 |
| [6] |
Tomoharu Suda. Construction of Lyapunov functions using Helmholtz–Hodge decomposition. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2437-2454. doi: 10.3934/dcds.2019103 |
| [7] |
Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185 |
| [8] |
George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure & Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035 |
| [9] |
Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691 |
| [10] |
Jiashan Zheng. Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 627-643. doi: 10.3934/dcds.2017026 |
| [11] |
Xinru Cao. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1891-1904. doi: 10.3934/dcds.2015.35.1891 |
| [12] |
Hong Cai, Zhong Tan. Time periodic solutions to the three--dimensional equations of compressible magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1847-1868. doi: 10.3934/dcds.2016.36.1847 |
| [13] |
Małgorzata Migda, Ewa Schmeidel, Małgorzata Zdanowicz. Periodic solutions of a $2$-dimensional system of neutral difference equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 359-367. doi: 10.3934/dcdsb.2018024 |
| [14] |
Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 |
| [15] |
Julien Barral, Yan-Hui Qu. On the higher-dimensional multifractal analysis. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 1977-1995. doi: 10.3934/dcds.2012.32.1977 |
| [16] |
Marianne Beringhier, Adrien Leygue, Francisco Chinesta. Parametric nonlinear PDEs with multiple solutions: A PGD approach. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 383-392. doi: 10.3934/dcdss.2016002 |
| [17] |
Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control & Related Fields, 2020, 10 (1) : 113-140. doi: 10.3934/mcrf.2019032 |
| [18] |
Antonio Vitolo. On the growth of positive entire solutions of elliptic PDEs and their gradients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1335-1346. doi: 10.3934/dcdss.2014.7.1335 |
| [19] |
Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9 |
| [20] |
Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 111-122. doi: 10.3934/dcds.2013.33.111 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]