American Institute of Mathematical Sciences

Advanced
January  2010, 26(1): 303-312. doi: 10.3934/dcds.2010.26.303

Generalized fronts for one-dimensional reaction-diffusion equations

Antoine Mellet 1, , Jean-Michel Roquejoffre 2, and  Yannick Sire 3,

1. 

Department of Mathematics, University of Maryland, College Park, MD 20742, United States
2. 

Institut de Mathématiques, Université Paul Sabatier, F-31062 Toulouse, France
3. 

LATP, Université Aix-Marseille III, F-13397 Marseille, France

Received  January 2009 Revised  May 2009 Published  October 2009

For a class of one-dimensional reaction-diffusion equations, we establish the existence of generalized fronts, as recently defined by Berestycki and Hamel. We also prove uniform nondegeneracy estimates, such as a lower bound on the time derivative on some level sets, as well as a lower bound on the spatial derivative.
Keywords: Fronts propagation, Reaction-diffusion equations, Heterogeneous media..
Mathematics Subject Classification: Primary: 35K5.
Citation: Antoine Mellet, Jean-Michel Roquejoffre, Yannick Sire. Generalized fronts for one-dimensional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 303-312. doi: 10.3934/dcds.2010.26.303
[1]

Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks & Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001
[2]

Masaharu Taniguchi. Multi-dimensional traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1011-1046. doi: 10.3934/dcds.2012.32.1011
[3]

Masaharu Taniguchi. Traveling fronts in perturbed multistable reaction-diffusion equations. Conference Publications, 2011, 2011 (Special) : 1368-1377. doi: 10.3934/proc.2011.2011.1368
[4]

Henri Berestycki, Guillemette Chapuisat. Traveling fronts guided by the environment for reaction-diffusion equations. Networks & Heterogeneous Media, 2013, 8 (1) : 79-114. doi: 10.3934/nhm.2013.8.79
[5]

Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020126
[6]

Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083
[7]

Zhen-Hui Bu, Zhi-Cheng Wang. Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media. Communications on Pure & Applied Analysis, 2016, 15 (1) : 139-160. doi: 10.3934/cpaa.2016.15.139
[8]

Feng Cao, Wenxian Shen. Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4697-4727. doi: 10.3934/dcds.2017202
[9]

Wenxian Shen, Zhongwei Shen. Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1193-1213. doi: 10.3934/cpaa.2016.15.1193
[10]

Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115
[11]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147
[12]

Maurizio Garrione, Marta Strani. Monotone wave fronts for $(p, q)$-Laplacian driven reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 91-103. doi: 10.3934/dcdss.2019006
[13]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43
[14]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281
[15]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020083
[16]

Shi-Liang Wu, Cheng-Hsiung Hsu. Propagation of monostable traveling fronts in discrete periodic media with delay. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2987-3022. doi: 10.3934/dcds.2018128
[17]

Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921
[18]

Zhi-Xian Yu, Rong Yuan. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 709-728. doi: 10.3934/dcdsb.2010.13.709
[19]

Hiroshi Matsuzawa. On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments. Conference Publications, 2009, 2009 (Special) : 516-525. doi: 10.3934/proc.2009.2009.516
[20]

Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences