
Previous Article
Supercritical elliptic problems from a perturbation viewpoint
 DCDS Home
 This Issue

Next Article
Varying domains: Stability of the Dirichlet and the Poisson problem
Reactiondiffusion equations for population dynamics with forced speed I  The case of the whole space
1.  EHESS, CAMS, 54 Boulevard Raspail, F75006, Paris, France, France 
$\partial_t u=$Δ$u+f(xcte,u),t>0,x\in\R^N.$
These kind of equations have been introduced in [1] in
the case $N=1$ for studying the impact of a climate shift on the
dynamics of a biological species.
In the present paper, we first extend the results of
[1] to arbitrary dimension $N$ and to a greater
generality in the assumptions on $f$. We establish a necessary
and sufficient condition for the existence of travelling wave
solutions, that is, solutions of the type $u(t,x)=U(xcte)$. This
is expressed in terms of the sign of the generalized principal eigenvalue $\l$ of
an associated linear elliptic operator in $\R^N$. With this
criterion, we then completely describe the large time dynamics for
this equation. In particular, we characterize situations in which
there is either extinction or persistence.
Moreover, we consider the problem obtained by adding a term
$g(x,u)$ periodic in $x$ in the direction $e$:
$\partial_t u=$Δ$u+f(xcte,u)+g(x,u),t>0,x\in\R^N.$
Here, $g$ can be viewed as representing geographical characteristics of the territory which are not subject to shift. We derive analogous results as before, with $\l$ replaced by the generalized principal eigenvalue of the parabolic operator obtained by linearization about $u\equiv0$ in the whole space. In this framework, travelling waves are replaced by pulsating travelling waves, which are solutions of the form $U(t,xcte)$, with $U(t,x)$ periodic in $t$. These results still hold if the term $g$ is also subject to the shift, but on a different time scale, that is, if $g(x,u)$ is replaced by $g(xc'te,u)$, with $c'\in\R$.
[1] 
Yong Jung Kim, WeiMing Ni, Masaharu Taniguchi. Nonexistence of localized travelling waves with nonzero speed in single reactiondiffusion equations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 37073718. doi: 10.3934/dcds.2013.33.3707 
[2] 
Juliette Bouhours, Grégroie Nadin. A variational approach to reactiondiffusion equations with forced speed in dimension 1. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 18431872. doi: 10.3934/dcds.2015.35.1843 
[3] 
Henri Berestycki, Luca Rossi. Reactiondiffusion equations for population dynamics with forced speed II  cylindricaltype domains. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 1961. doi: 10.3934/dcds.2009.25.19 
[4] 
Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 17751791. doi: 10.3934/dcds.2014.34.1775 
[5] 
ZhaoXing Yang, GuoBao Zhang, Ge Tian, Zhaosheng Feng. Stability of nonmonotone noncritical traveling waves in discrete reactiondiffusion equations with time delay. Discrete and Continuous Dynamical Systems  S, 2017, 10 (3) : 581603. doi: 10.3934/dcdss.2017029 
[6] 
Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal timedelayed reactiondiffusion equations. Kinetic and Related Models, 2018, 11 (5) : 12351253. doi: 10.3934/krm.2018048 
[7] 
C. van der Mee, Stella Vernier Piro. Travelling waves for solidgas reactiondiffusion systems. Conference Publications, 2003, 2003 (Special) : 872879. doi: 10.3934/proc.2003.2003.872 
[8] 
Manjun Ma, XiaoQiang Zhao. Monostable waves and spreading speed for a reactiondiffusion model with seasonal succession. Discrete and Continuous Dynamical Systems  B, 2016, 21 (2) : 591606. doi: 10.3934/dcdsb.2016.21.591 
[9] 
Michal Fečkan, Vassilis M. Rothos. Travelling waves of forced discrete nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems  S, 2011, 4 (5) : 11291145. doi: 10.3934/dcdss.2011.4.1129 
[10] 
WeiJie Sheng, WanTong Li. Multidimensional stability of timeperiodic planar traveling fronts in bistable reactiondiffusion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 26812704. doi: 10.3934/dcds.2017115 
[11] 
Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reactiondiffusion equations on a periodic domain. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 253273. doi: 10.3934/dcds.2011.31.253 
[12] 
Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reactiondiffusion equations in periodic media. Networks and Heterogeneous Media, 2016, 11 (3) : 369393. doi: 10.3934/nhm.2016001 
[13] 
Ming Mei. Stability of traveling wavefronts for timedelayed reactiondiffusion equations. Conference Publications, 2009, 2009 (Special) : 526535. doi: 10.3934/proc.2009.2009.526 
[14] 
ShengChen Fu. Travelling waves of a reactiondiffusion model for the acidic nitrateferroin reaction. Discrete and Continuous Dynamical Systems  B, 2011, 16 (1) : 189196. doi: 10.3934/dcdsb.2011.16.189 
[15] 
ShengChen Fu, JeChiang Tsai. Stability of travelling waves of a reactiondiffusion system for the acidic nitrateferroin reaction. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 40414069. doi: 10.3934/dcds.2013.33.4041 
[16] 
Yuzo Hosono. Phase plane analysis of travelling waves for higher order autocatalytic reactiondiffusion systems. Discrete and Continuous Dynamical Systems  B, 2007, 8 (1) : 115125. doi: 10.3934/dcdsb.2007.8.115 
[17] 
Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a nonlocal delayed reactiondiffusion system without quasimonotonicity. Discrete and Continuous Dynamical Systems  B, 2018, 23 (10) : 40634085. doi: 10.3934/dcdsb.2018126 
[18] 
J. Húska, Peter Poláčik, M.V. Safonov. Principal eigenvalues, spectral gaps and exponential separation between positive and signchanging solutions of parabolic equations. Conference Publications, 2005, 2005 (Special) : 427435. doi: 10.3934/proc.2005.2005.427 
[19] 
Abraham Solar. Stability of nonmonotone and backward waves for delay nonlocal reactiondiffusion equations. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 57995823. doi: 10.3934/dcds.2019255 
[20] 
Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reactiondiffusion equations. Evolution Equations and Control Theory, 2012, 1 (1) : 4356. doi: 10.3934/eect.2012.1.43 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]