
Previous Article
On the retention of the interfaces in some elliptic and parabolic nonlinear problems
 DCDS Home
 This Issue

Next Article
Long time convergence for a class of variational phasefield models
Reactiondiffusion equations for population dynamics with forced speed II  cylindricaltype domains
1.  EHESS, CAMS, 54 Boulevard Raspail, F75006, Paris 
$\partial_t u=\Delta u+f(xcte,u),\qquad t>0,\quad x\in\R^N,$
where $e\in S^{N1}$ and $c>0$ are given and $f(x,s)$ satisfies
some usual assumptions in population dynamics, together with
$f_s(x,0)<0$ for $x$ large. The interest for such equation comes
from an ecological model introduced in [1]
describing the effects of global
warming on biological species. In [6],we proved that
existence and uniqueness of travelling wave solutions of the type
$u(x,t)=U(xcte)$ and the large time behaviour of solutions with
arbitrary nonnegative bounded initial datum depend on the sign of
the generalized principal in $\R^N$ of an associated linear operator.
Here, we establish analogous results for the Neumann problem in
domains which are asymptotically cylindrical, as well as for the problem in
the whole space with $f$ periodic in some space variables,
orthogonal to the direction of the shift $e$.
The $L^1$ convergence of solution $u(t,x)$ as $t\to\infty$ is established
next. In this paper, we also show
that a bifurcation from the zero solution takes place as the principal crosses $0$. We are
able to describe the shape of solutions close to extinction
thus answering a question raised by M.~Mimura.
These two results are new even in the framework
considered in [6].
Another type of problem is obtained by adding to the previous one a term
$g(xc'te,u)$ periodic in $x$ in the direction $e$.
Such a model arises when considering
environmental change on two different scales.
Lastly, we also solve the case of an equation
$\partial_t u=\Delta u+f(t,xcte,u),$
when $f(t,x,s)$ is periodic in $t$. This for instance represents the seasonal dependence of $f$. In both cases, we obtain a necessary and sufficient condition for the existence, uniqueness and stability of pulsating travelling waves, which are solutions with a profile which is periodic in time.
[1] 
Yong Jung Kim, WeiMing Ni, Masaharu Taniguchi. Nonexistence of localized travelling waves with nonzero speed in single reactiondiffusion equations. Discrete & Continuous Dynamical Systems  A, 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 & Continuous Dynamical Systems  A, 2015, 35 (5) : 18431872. doi: 10.3934/dcds.2015.35.1843 
[3] 
Henri Berestycki, Luca Rossi. Reactiondiffusion equations for population dynamics with forced speed I  The case of the whole space. Discrete & Continuous Dynamical Systems  A, 2008, 21 (1) : 4167. doi: 10.3934/dcds.2008.21.41 
[4] 
Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete & Continuous Dynamical Systems  A, 2014, 34 (5) : 17751791. doi: 10.3934/dcds.2014.34.1775 
[5] 
Manjun Ma, XiaoQiang Zhao. Monostable waves and spreading speed for a reactiondiffusion model with seasonal succession. Discrete & Continuous Dynamical Systems  B, 2016, 21 (2) : 591606. doi: 10.3934/dcdsb.2016.21.591 
[6] 
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 
[7] 
Martino Prizzi. A remark on reactiondiffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems  A, 2003, 9 (2) : 281286. doi: 10.3934/dcds.2003.9.281 
[8] 
ShengChen Fu. Travelling waves of a reactiondiffusion model for the acidic nitrateferroin reaction. Discrete & Continuous Dynamical Systems  B, 2011, 16 (1) : 189196. doi: 10.3934/dcdsb.2011.16.189 
[9] 
ShengChen Fu, JeChiang Tsai. Stability of travelling waves of a reactiondiffusion system for the acidic nitrateferroin reaction. Discrete & Continuous Dynamical Systems  A, 2013, 33 (9) : 40414069. doi: 10.3934/dcds.2013.33.4041 
[10] 
Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a nonlocal delayed reactiondiffusion system without quasimonotonicity. Discrete & Continuous Dynamical Systems  B, 2018, 23 (10) : 40634085. doi: 10.3934/dcdsb.2018126 
[11] 
Yuzo Hosono. Phase plane analysis of travelling waves for higher order autocatalytic reactiondiffusion systems. Discrete & Continuous Dynamical Systems  B, 2007, 8 (1) : 115125. doi: 10.3934/dcdsb.2007.8.115 
[12] 
Peter E. Kloeden, Meihua Yang. Forward attracting sets of reactiondiffusion equations on variable domains. Discrete & Continuous Dynamical Systems  B, 2019, 24 (3) : 12591271. doi: 10.3934/dcdsb.2019015 
[13] 
Michal Fečkan, Vassilis M. Rothos. Travelling waves of forced discrete nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems  S, 2011, 4 (5) : 11291145. doi: 10.3934/dcdss.2011.4.1129 
[14] 
Ciprian G. Gal, Mahamadi Warma. Reactiondiffusion equations with fractional diffusion on nonsmooth domains with various boundary conditions. Discrete & Continuous Dynamical Systems  A, 2016, 36 (3) : 12791319. doi: 10.3934/dcds.2016.36.1279 
[15] 
Narcisa Apreutesei, Vitaly Volpert. Reactiondiffusion waves with nonlinear boundary conditions. Networks & Heterogeneous Media, 2013, 8 (1) : 2335. doi: 10.3934/nhm.2013.8.23 
[16] 
Zhaosheng Feng. Traveling waves to a reactiondiffusion equation. Conference Publications, 2007, 2007 (Special) : 382390. doi: 10.3934/proc.2007.2007.382 
[17] 
Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting behavior of dynamics for stochastic reactiondiffusion equations with additive noise on thin domains. Discrete & Continuous Dynamical Systems  A, 2018, 38 (1) : 187208. doi: 10.3934/dcds.2018009 
[18] 
Fuzhi Li, Yangrong Li, Renhai Wang. Regular measurable dynamics for reactiondiffusion equations on narrow domains with rough noise. Discrete & Continuous Dynamical Systems  A, 2018, 38 (7) : 36633685. doi: 10.3934/dcds.2018158 
[19] 
Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting dynamics for nonautonomous stochastic retarded reactiondiffusion equations on thin domains. Discrete & Continuous Dynamical Systems  A, 2019, 39 (7) : 37173747. doi: 10.3934/dcds.2019151 
[20] 
ChiuYen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315335. doi: 10.3934/mbe.2008.5.315 
2017 Impact Factor: 1.179
Tools
Metrics
Other articles
by authors
[Back to Top]