-
Previous Article
Long time convergence for a class of variational phase-field models
- DCDS Home
- This Issue
-
Next Article
On the retention of the interfaces in some elliptic and parabolic nonlinear problems
Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains
| 1. | EHESS, CAMS, 54 Boulevard Raspail, F-75006, Paris |
$\partial_t u=\Delta u+f(x-cte,u),\qquad t>0,\quad x\in\R^N,$
where $e\in S^{N-1}$ 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
$u(x,t)=U(x-cte)$ 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
$g(x-c'te,u)$ 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
$\partial_t u=\Delta u+f(t,x-cte,u),$
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, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707 |
| [2] |
Juliette Bouhours, Grégroie Nadin. A variational approach to reaction-diffusion equations with forced speed in dimension 1. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1843-1872. doi: 10.3934/dcds.2015.35.1843 |
| [3] |
Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 41-67. 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) : 1775-1791. doi: 10.3934/dcds.2014.34.1775 |
| [5] |
Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591 |
| [6] |
C. van der Mee, Stella Vernier Piro. Travelling waves for solid-gas reaction-diffusion systems. Conference Publications, 2003, 2003 (Special) : 872-879. doi: 10.3934/proc.2003.2003.872 |
| [7] |
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 |
| [8] |
Sheng-Chen Fu. Travelling waves of a reaction-diffusion model for the acidic nitrate-ferroin reaction. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 189-196. doi: 10.3934/dcdsb.2011.16.189 |
| [9] |
Sheng-Chen Fu, Je-Chiang Tsai. Stability of travelling waves of a reaction-diffusion system for the acidic nitrate-ferroin reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4041-4069. doi: 10.3934/dcds.2013.33.4041 |
| [10] |
Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126 |
| [11] |
Yuzo Hosono. Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 115-125. doi: 10.3934/dcdsb.2007.8.115 |
| [12] |
Peter E. Kloeden, Meihua Yang. Forward attracting sets of reaction-diffusion equations on variable domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1259-1271. 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) : 1129-1145. doi: 10.3934/dcdss.2011.4.1129 |
| [14] |
Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279 |
| [15] |
Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks & Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23 |
| [16] |
Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382 |
| [17] |
Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 187-208. doi: 10.3934/dcds.2018009 |
| [18] |
Fuzhi Li, Yangrong Li, Renhai Wang. Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3663-3685. doi: 10.3934/dcds.2018158 |
| [19] |
Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3717-3747. doi: 10.3934/dcds.2019151 |
| [20] |
Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]