American Institute of Mathematical Sciences

Advanced
January  2008, 20(1): 81-113. doi: 10.3934/dcds.2008.20.81

Exponential separation and principal Floquet bundles for linear parabolic equations on $R^N$

J. Húska 1, and  Peter Poláčik 1,

1. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455

Received  January 2007 Revised  June 2007 Published  October 2007

We consider linear nonautonomous second order parabolic equations on $\R^N$. Under an instability condition, we prove the existence of two complementary Floquet bundles, one spanned by a positive entire solution - the principal Floquet bundle, the other one consisting of sign-changing solutions. We establish an exponential separation between the two bundles, showing in particular that a class of sign-changing solutions are exponentially dominated by positive solutions.
Keywords: principal Floquet bundle., exponential separation, Parabolic equations on $R^N$, positive entire solutions.
Mathematics Subject Classification: 35K15, 35B05, 35B4.
Citation: J. Húska, Peter Poláčik. Exponential separation and principal Floquet bundles for linear parabolic equations on $R^N$. Discrete & Continuous Dynamical Systems, 2008, 20 (1) : 81-113. doi: 10.3934/dcds.2008.20.81
[1]

J. Húska, Peter Poláčik, M.V. Safonov. Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations. Conference Publications, 2005, 2005 (Special) : 427-435. doi: 10.3934/proc.2005.2005.427
[2]

Peter Poláčik. On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 13-26. doi: 10.3934/dcds.2005.12.13
[3]

Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6163-6193. doi: 10.3934/dcds.2018265
[4]

Xiang-Dong Fang. Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1603-1615. doi: 10.3934/cpaa.2017077
[5]

Dongyan Li, Yongzhong Wang. Nonexistence of positive solutions for a system of integral equations on $R^n_+$ and applications. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2601-2613. doi: 10.3934/cpaa.2013.12.2601
[6]

Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198
[7]

Qingfang Wang. Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1671-1688. doi: 10.3934/cpaa.2016008
[8]

Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113
[9]

Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N $. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 449-472. doi: 10.3934/dcdss.2012.5.449
[10]

Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50
[11]

Jiguang Bao, Nguyen Lam, Guozhen Lu. Polyharmonic equations with critical exponential growth in the whole space $ \mathbb{R}^{n}$. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 577-600. doi: 10.3934/dcds.2016.36.577
[12]

Fang-Fang Liao, Chun-Lei Tang. Four positive solutions of a quasilinear elliptic equation in $ R^N$. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2577-2600. doi: 10.3934/cpaa.2013.12.2577
[13]

Yuxia Guo, Bo Li. Nonexistence of positive solutions for polyharmonic systems in $\mathbb{R}^N_+$. Communications on Pure & Applied Analysis, 2016, 15 (3) : 701-713. doi: 10.3934/cpaa.2016.15.701
[14]

Alberto Farina. Some symmetry results for entire solutions of an elliptic system arising in phase separation. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2505-2511. doi: 10.3934/dcds.2014.34.2505
[15]

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
[16]

Tianhong Li. Some special solutions of the multidimensional Euler equations in $R^N$. Communications on Pure & Applied Analysis, 2005, 4 (4) : 757-762. doi: 10.3934/cpaa.2005.4.757
[17]

E. N. Dancer, Shusen Yan. On the existence of multipeak solutions for nonlinear field equations on $R^N$. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 39-50. doi: 10.3934/dcds.2000.6.39
[18]

Kazuhiro Ishige, Tatsuki Kawakami. Asymptotic behavior of solutions for some semilinear heat equations in $R^N$. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1351-1371. doi: 10.3934/cpaa.2009.8.1351
[19]

Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211
[20]

Ziqing Yuan, Jianshe Yu. Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3285-3303. doi: 10.3934/dcdss.2020281

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences