April  1997, 3(2): 207-216. doi: 10.3934/dcds.1997.3.207

Existence of stable and unstable periodic solutions for semilinear parabolic problems

1. 

School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia

2. 

Department of Mathematics, Yokohama National University, 156 Tokiwadai Hodogaya-ku - Yokohama

Received  October 1996 Published  January 1997

In this paper, we show the existence of stable and unstable $T-$periodic solutions for a semilinear parabolic equation

$\frac{\partial u}{\partial t} - \Delta u = g(x,u) + h( t, x ),\quad \text{in} \quad (0,T) \times \Omega$

$u=0 ,\quad \text{on}\quad (0,T) \times \partial \Omega$

$u(0) = u(T),\quad \text{in} \quad \overline \Omega$

where $\Omega \subset R^N$ is a bounded domain with a smooth boundary, $g:\overline{\Omega} \times R \rightarrow R$ is a continuous function such that $g(x,\cdot )$ has a superlinear growth for each $x \in \overline{\Omega} $ and $h:(0,T) \times \Omega \to R$ is a continuous function.

Citation: E. N. Dancer, Norimichi Hirano. Existence of stable and unstable periodic solutions for semilinear parabolic problems. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 207-216. doi: 10.3934/dcds.1997.3.207
[1]

Mourad Choulli, El Maati Ouhabaz, Masahiro Yamamoto. Stable determination of a semilinear term in a parabolic equation. Communications on Pure and Applied Analysis, 2006, 5 (3) : 447-462. doi: 10.3934/cpaa.2006.5.447

[2]

Charles A. Stuart. Stability analysis for a family of degenerate semilinear parabolic problems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5297-5337. doi: 10.3934/dcds.2018234

[3]

Flank D. M. Bezerra, Alexandre N. Carvalho, Marcelo J. D. Nascimento. Fractional approximations of abstract semilinear parabolic problems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4221-4255. doi: 10.3934/dcdsb.2020095

[4]

Eric Benoît. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934/dcdss.2009.2.911

[5]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313

[6]

Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451

[7]

Ruediger Landes. Stable and unstable initial configuration in the theory wave fronts. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 797-808. doi: 10.3934/dcdss.2012.5.797

[8]

G. Métivier, K. Zumbrun. Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 205-220. doi: 10.3934/dcds.2004.11.205

[9]

Mickaël D. Chekroun. Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3723-3753. doi: 10.3934/dcdsb.2018075

[10]

Alexandre Nolasco de Carvalho, Marcelo J. D. Nascimento. Singularly non-autonomous semilinear parabolic problems with critical exponents. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 449-471. doi: 10.3934/dcdss.2009.2.449

[11]

Sébastien Court, Karl Kunisch, Laurent Pfeiffer. Hybrid optimal control problems for a class of semilinear parabolic equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1031-1060. doi: 10.3934/dcdss.2018060

[12]

J.-P. Raymond. Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 341-370. doi: 10.3934/dcds.1997.3.341

[13]

V. Carmona, E. Freire, E. Ponce, F. Torres. The continuous matching of two stable linear systems can be unstable. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 689-703. doi: 10.3934/dcds.2006.16.689

[14]

Raoul-Martin Memmesheimer, Marc Timme. Stable and unstable periodic orbits in complex networks of spiking neurons with delays. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1555-1588. doi: 10.3934/dcds.2010.28.1555

[15]

Thierry Gallay, Guido Schneider, Hannes Uecker. Stable transport of information near essentially unstable localized structures. Discrete and Continuous Dynamical Systems - B, 2004, 4 (2) : 349-390. doi: 10.3934/dcdsb.2004.4.349

[16]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225

[17]

Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems and Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139

[18]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems and Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267

[19]

Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control and Related Fields, 2019, 9 (2) : 385-409. doi: 10.3934/mcrf.2019018

[20]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems and Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]