-
Previous Article
Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition
- DCDS Home
- This Issue
-
Next Article
Asymptotic properties and classification of bistable fronts with Lipschitz level sets
Multiple stable patterns for some reaction-diffusion equation in disrupted environments
1. | Aisin AW. Co. Ltd., Q21 Promotion Department, Engineering Division, 0 Takane, Fujii-Cho, Anjo, Aichi, 444-1192, Japan |
2. | Department of Mathematics and Infomation Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi,Tokyo 192-0397, Japan |
3. | Department of Mathematics, Waseda University, 3-4-1 Okubo, Shinjyuku-ku, Tokyo 169-8555, Japan |
$ -\epsilon^2\Delta u(x) = u(x)^2(b(x)-u(x)) \ \mbox{in}\ \Omega, \quad$ $ \frac{\partial u}{\partial n}(x) = 0 \ \mbox{on}\ \partial\Omega.$
Here $\epsilon>0$ is a small parameter and $b(x)$ is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.
[1] |
Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293 |
[2] |
Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133 |
[3] |
Wei Wang, Yan Lv. Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 175-193. doi: 10.3934/dcdsb.2010.13.175 |
[4] |
Zhaoli Liu, Jiabao Su. Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 617-634. doi: 10.3934/dcds.2004.10.617 |
[5] |
Tomas Godoy, Alfredo Guerin. Existence of nonnegative solutions to singular elliptic problems, a variational approach. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1505-1525. doi: 10.3934/dcds.2018062 |
[6] |
R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339 |
[7] |
Haiyun Deng, Hairong Liu, Long Tian. Classification of singular sets of solutions to elliptic equations. Communications on Pure and Applied Analysis, 2020, 19 (6) : 2949-2964. doi: 10.3934/cpaa.2020129 |
[8] |
L. Ke. Boundary behaviors for solutions of singular elliptic equations. Conference Publications, 1998, 1998 (Special) : 388-396. doi: 10.3934/proc.1998.1998.388 |
[9] |
Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121 |
[10] |
Xavier Cabré. A new proof of the boundedness results for stable solutions to semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7249-7264. doi: 10.3934/dcds.2019302 |
[11] |
Mostafa Fazly, Yuan Li. Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4185-4206. doi: 10.3934/dcds.2021033 |
[12] |
Kelei Wang. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 3805-3816. doi: 10.3934/era.2021062 |
[13] |
Hyung Ju Hwang, Thomas P. Witelski. Short-time pattern formation in thin film equations. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 867-885. doi: 10.3934/dcds.2009.23.867 |
[14] |
Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2505-2518. doi: 10.3934/cpaa.2020272 |
[15] |
Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure and Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527 |
[16] |
J. Chen, K. Murillo, E. M. Rocha. Two nontrivial solutions of a class of elliptic equations with singular term. Conference Publications, 2011, 2011 (Special) : 272-281. doi: 10.3934/proc.2011.2011.272 |
[17] |
Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Singular solutions of some elliptic equations involving mixed absorption-reaction. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022036 |
[18] |
Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541 |
[19] |
Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108 |
[20] |
Necdet Bildik, Sinan Deniz. New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 503-518. doi: 10.3934/dcdss.2020028 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]