American Institute of Mathematical Sciences

Advanced
February  2009, 25(1): 133-157. doi: 10.3934/dcds.2009.25.133

On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry

J. Földes 1, and  Peter Poláčik 1,

1. 

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

Received  August 2007 Revised  December 2007 Published  June 2009

We consider fully nonlinear weakly coupled systems of parabolic equations on a bounded reflectionally symmetric domain. Assuming the system is cooperative we prove the asymptotic symmetry of positive bounded solutions. To facilitate an application of the method of moving hyperplanes, we derive Harnack type estimates for linear cooperative parabolic systems.
Keywords: positive solutions, Harnack inequality., asymptotic symmetry, Parabolic cooperative systems.
Mathematics Subject Classification: 35K50, 35B40, 35B4.
Citation: J. Földes, Peter Poláčik. On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 133-157. doi: 10.3934/dcds.2009.25.133
[1]

Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051
[2]

Yan Deng, Junfang Zhao, Baozeng Chu. Symmetry of positive solutions for systems of fractional Hartree equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3085-3096. doi: 10.3934/dcdss.2021079
[3]

Francesco Esposito. Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 549-577. doi: 10.3934/dcds.2020022
[4]

Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043
[5]

Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035
[6]

Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri. Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electronic Research Announcements, 2006, 12: 95-99.

[7]

Zhitao Zhang, Haijun Luo. Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction. Communications on Pure & Applied Analysis, 2018, 17 (3) : 787-806. doi: 10.3934/cpaa.2018040
[8]

Daniela De Silva, Ovidiu Savin. A note on higher regularity boundary Harnack inequality. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 6155-6163. doi: 10.3934/dcds.2015.35.6155
[9]

Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153
[10]

Giuseppe Di Fazio, Maria Stella Fanciullo, Pietro Zamboni. Harnack inequality for degenerate elliptic equations and sum operators. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2363-2376. doi: 10.3934/cpaa.2015.14.2363
[11]

Changbing Hu, Yang Kuang, Bingtuan Li, Hao Liu. Spreading speeds and traveling wave solutions in cooperative integral-differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1663-1684. doi: 10.3934/dcdsb.2015.20.1663
[12]

Marcello Lucia, Guido Sweers. Nondegeneracy of solutions for a class of cooperative systems on $ \mathbb{R}^n $. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021152
[13]

Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685
[14]

Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
[15]

Yunyun Hu. Symmetry of positive solutions to fractional equations in bounded domains and unbounded cylinders. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3723-3734. doi: 10.3934/cpaa.2020164
[16]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045
[17]

Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975
[18]

Pablo Álvarez-Caudevilla, Julián López-Gómez. The dynamics of a class of cooperative systems. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 397-415. doi: 10.3934/dcds.2010.26.397
[19]

Minkyu Kwak, Kyong Yu. The asymptotic behavior of solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 1996, 2 (4) : 483-496. doi: 10.3934/dcds.1996.2.483
[20]

Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (23)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences