-
Previous Article
Nonlinear heat equation: the radial case
- DCDS Home
- This Issue
-
Next Article
Inertial manifolds with and without delay
Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties
| 1. | Fachbereich Mathematik, Universität Rostock, Universitätsplatz 1, D-18055 Rostock, F. R., Germany |
| 2. | Fachbereich Mathematik, Universität Rostock, Universitätsplatz 1, D-18055 Rostock |
$ \partial_t u = (1 + i\nu)\Delta u + Ru- (1 + i\mu) |u|^2 u; \quad 0\le t < \infty, x\in\Omega $,
is investigated in a bounded domain $\Omega\subset \mathbb R^n$ with suffciently smooth boundary. Standard boundary conditions are considered: Dirichlet, Neumann or periodic. Existence and uniqueness of global smooth solutions is established for all real parameter values $\mu$ and $\nu$ if $n\le 2$, and for certain parameter values $\mu$ and $\nu$ if $n\ge 3$. Furthermore, dynamical properties of the CGL equation, such as existence of determining nodes, are shown. The proof of existence of smooth solutions hinges on the following inequality using the $L^2(\Omega)$-duality,
$|\mathfrak Im$ $<\Delta u ,\ |u|^{p-2}u>\le (|p-2|)/(2\sqrt{p-1})\mathfrak Re$ $< -\Delta u ,\ |u|^{p-2}u >.$
| [1] |
Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205 |
| [2] |
Yirong Jiang, Nanjing Huang, Zhouchao Wei. Existence of a global attractor for fractional differential hemivariational inequalities. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1193-1212. doi: 10.3934/dcdsb.2019216 |
| [3] |
Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 |
| [4] |
Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001 |
| [5] |
Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763 |
| [6] |
Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151 |
| [7] |
Hongyu Liu, Jun Zou. Uniqueness in determining multiple polygonal scatterers of mixed type. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 375-396. doi: 10.3934/dcdsb.2008.9.375 |
| [8] |
Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179 |
| [9] |
Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026 |
| [10] |
Hiroshi Matano, Yoichiro Mori. Global existence and uniqueness of a three-dimensional model of cellular electrophysiology. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1573-1636. doi: 10.3934/dcds.2011.29.1573 |
| [11] |
Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078 |
| [12] |
Aslihan Demirkaya. The existence of a global attractor for a Kuramoto-Sivashinsky type equation in 2D. Conference Publications, 2009, 2009 (Special) : 198-207. doi: 10.3934/proc.2009.2009.198 |
| [13] |
Kotaro Tsugawa. Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index. Communications on Pure and Applied Analysis, 2004, 3 (2) : 301-318. doi: 10.3934/cpaa.2004.3.301 |
| [14] |
Chunxiao Guo, Fan Cui, Yongqian Han. Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1687-1699. doi: 10.3934/dcdss.2016070 |
| [15] |
Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3943-3988. doi: 10.3934/dcdsb.2020270 |
| [16] |
Zijuan Wen, Meng Fan, Asim M. Asiri, Ebraheem O. Alzahrani, Mohamed M. El-Dessoky, Yang Kuang. Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 407-420. doi: 10.3934/mbe.2017025 |
| [17] |
Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326 |
| [18] |
Jan Giesselmann, Niklas Kolbe, Mária Lukáčová-Medvi${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over d} }}$ová, Nikolaos Sfakianakis. Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4397-4431. doi: 10.3934/dcdsb.2018169 |
| [19] |
Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55 |
| [20] |
F. R. Guarguaglini, R. Natalini. Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology. Communications on Pure and Applied Analysis, 2007, 6 (1) : 287-309. doi: 10.3934/cpaa.2007.6.287 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]