# American Institute of Mathematical Sciences

• Previous Article
An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity
• DCDS Home
• This Issue
• Next Article
Rigidity of real-analytic actions of $SL(n,\Z)$ on $\T^n$: A case of realization of Zimmer program
May  2010, 27(2): 617-641. doi: 10.3934/dcds.2010.27.617

## An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles

 1 Tokyo University of Marine Science and Technology, 4-5-7 Konan, Minato-ku, Tokyo 108-8477 2 School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 3 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States

Received  October 2009 Revised  February 2010 Published  February 2010

We study the following Neumann problem

$d\Delta u+g(x)u^{2}(1-u)=0 \$ in Ω ,
$0\leq u\leq 1$in Ω and $\frac{\partial u}{\partial\nu}=0$ on ∂Ω,

where $\Delta$ is the Laplace operator, $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$ with $\nu$ as its unit outward normal on the boundary $\partial\Omega$, and $g$ changes sign in $\Omega$. This equation models the "complete dominance" case in population genetics of two alleles. We show that the diffusion rate $d$ and the integral $\int_{\Omega}g\ \d x$ play important roles for the existence of stable nontrivial solutions, and the sign of $g(x)$ determines the limiting profile of solutions as $d$ tends to $0$. In particular, a conjecture of Nagylaki and Lou has been largely resolved.
Our results and methods cover a much wider class of nonlinearities than $u^{2}(1-u)$, and similar results have been obtained for Dirichlet and Robin boundary value problems as well.

Citation: Kimie Nakashima, Wei-Ming Ni, Linlin Su. An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 617-641. doi: 10.3934/dcds.2010.27.617
 [1] Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5673-5694. doi: 10.3934/dcdsb.2019101 [2] 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 [3] Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248 [4] Pablo Amster, Manuel Zamora. Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4819-4835. doi: 10.3934/dcds.2018211 [5] 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 [6] Caihong Chang, Qiangchang Ju, Zhengce Zhang. Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5991-6014. doi: 10.3934/dcds.2020256 [7] Mi-Ho Giga, Yoshikazu Giga, Takeshi Ohtsuka, Noriaki Umeda. On behavior of signs for the heat equation and a diffusion method for data separation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2277-2296. doi: 10.3934/cpaa.2013.12.2277 [8] Wenxiong Chen, Congming Li, Jiuyi Zhu. Fractional equations with indefinite nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1257-1268. doi: 10.3934/dcds.2019054 [9] 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 [10] Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59 [11] Yuchi Qiu, Weitao Chen, Qing Nie. A hybrid method for stiff reaction–diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6387-6417. doi: 10.3934/dcdsb.2019144 [12] Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435 [13] Henrik Garde, Stratos Staboulis. The regularized monotonicity method: Detecting irregular indefinite inclusions. Inverse Problems & Imaging, 2019, 13 (1) : 93-116. doi: 10.3934/ipi.2019006 [14] Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic & Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013 [15] Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229 [16] Pascal Bégout, Jesús Ildefonso Díaz. A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3371-3382. doi: 10.3934/dcds.2014.34.3371 [17] Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169 [18] Messoud Efendiev, Alain Miranville. Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 399-424. doi: 10.3934/dcds.1999.5.399 [19] Claudianor O. Alves, Geilson F. Germano. Existence of ground state solution and concentration of maxima for a class of indefinite variational problems. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2887-2906. doi: 10.3934/cpaa.2020126 [20] King-Yeung Lam, Wei-Ming Ni. Limiting profiles of semilinear elliptic equations with large advection in population dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1051-1067. doi: 10.3934/dcds.2010.28.1051

2019 Impact Factor: 1.338