-
Previous Article
Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity
- DCDS Home
- This Issue
-
Next Article
Normalization in Banach scale Lie algebras via mould calculus and applications
Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system
1. | School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China |
2. | Department of Mathematics, School of Mathematics, Tianjin University, Tianjin 300072, China |
3. | School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China |
$\left\{\begin{array}{ll} d_1\Delta u-a_1 u+\frac{u^p}{v^q}+\rho_1(x)=0, \ \ & x\in\Omega, \\ d_2\Delta v-a_2 v+\frac{u^r}{v^s}+\rho_2(x)=0,\ \ & x\in\Omega,\\ u(x)>0,\ \ v(x)>0,\ \ & x\in \Omega,\\ \displaystyle u(x)=v(x)=0,\ \ & x\in\partial\Omega, \end{array}\right.$ |
References:
[1] |
S. Chen, Y. Salmaniw and R. Xu,
Global existence for a singular Gierer-Meinhardt system, J. Differential Equations, 262 (2017), 2940-2960.
doi: 10.1016/j.jde.2016.11.022. |
[2] |
S. Chen,
Steady state solutions for a general activator-inhibitor model, Nonlinear Anal., 135 (2016), 84-96.
doi: 10.1016/j.na.2016.01.013. |
[3] |
Y. S. Choi and P. J. McKenna,
A singular Gierer-Meinhardt system of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 503-522.
doi: 10.1016/S0294-1449(00)00115-3. |
[4] |
Y. S. Choi and P. J. McKenna,
A singular Gierer-Meinhardt system of elliptic equations: The classical case, Nonlinear Anal., 55 (2003), 521-541.
doi: 10.1016/j.na.2003.07.003. |
[5] |
M. Ghergu,
Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition, Trans. Amer. Math. Soc., 361 (2009), 3953-3976.
doi: 10.1090/S0002-9947-09-04670-4. |
[6] |
M. Ghergu,
Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318.
doi: 10.1016/j.jfa.2010.02.003. |
[7] |
M. Ghergu and V. Rădulescu,
On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl., 311 (2005), 635-646.
doi: 10.1016/j.jmaa.2005.03.012. |
[8] |
M. Ghergu and V. Rădulescu,
On a class of singular Gierer-Meinhardt systems arising in morphogenesis, C. R. Math. Acad. Sci. Paris, 344 (2007), 163-168.
doi: 10.1016/j.crma.2006.12.008. |
[9] |
M. Ghergu and V. Rădulescu,
A singular Gierer-Meinhardt system with different source terms, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1215-1234.
doi: 10.1017/S0308210507000637. |
[10] |
M. Ghergu and V. Rădulescu, The influence of the distance function in some singular elliptic problems, Potential theory and stochastics in Albac, 125-137, Theta Ser. Adv. Math. , 11, Theta, Bucharest, 2009. |
[11] |
M. Ghergu and V. Rădulescu, Nonlinear PDEs. Mathematical Models in Biology, Chemistry and Population Genetics, With a foreword by Viorel Barbu. Springer Monographs in Mathematics. Springer, Heidelberg, 2012. xviii+391 pp. ISBN: 978-3-642-22663-2.
doi: 10.1007/978-3-642-22664-9. |
[12] |
A. Gierer and H. Meinhardt,
A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[13] |
H. Jiang,
Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751.
doi: 10.3934/dcds.2006.14.737. |
[14] |
H. Jiang and W. -M. Ni,
A priori estimates of stationary solutions of an activator-inhibitor system, Indiana Univ. Math. J., 56 (2007), 681-732.
doi: 10.1512/iumj.2007.56.2982. |
[15] |
E. H. Kim,
Singular Gierer-Meinhardt systems of elliptic boundary value problems, J. Math. Anal. Appl., 308 (2005), 1-10.
doi: 10.1016/j.jmaa.2004.10.039. |
[16] |
E. H. Kim,
A class of singular Gierer-Meinhardt systems of elliptic boundary value problems, Nonlinear Anal., 59 (2004), 305-318.
doi: 10.1016/S0362-546X(04)00260-3. |
[17] |
A. Lazer and J. P. McKenna,
On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.
doi: 10.1090/S0002-9939-1991-1037213-9. |
[18] |
F. Li, R. Peng and X. F. Song,
Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system, J. Differential Equations, 262 (2017), 559-589.
doi: 10.1016/j.jde.2016.09.040. |
[19] |
W. -M. Ni,
Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.
|
[20] |
W. -M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math. 82, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611971972. |
[21] |
W. -M. Ni, K. Suzuki and I. Takagi,
The dynamics of a kynetics activator-inhibitor system, J. Differential Equations, 229 (2006), 426-465.
doi: 10.1016/j.jde.2006.03.011. |
[22] |
W. -M. Ni, I. Takagi and E. Yanagida,
Stability of least energy patterns of the shadow system for an activator-inhibitor model, Japan J. Indust. Appl. Math., 18 (2001), 259-272.
doi: 10.1007/BF03168574. |
[23] |
W. -M. Ni and I. Takagi,
On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc., 297 (1986), 351-368.
doi: 10.1090/S0002-9947-1986-0849484-2. |
[24] |
W. -M. Ni and I. Takagi,
Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math., 12 (1995), 327-365.
doi: 10.1007/BF03167294. |
[25] |
W. -M. Ni and J. Wei,
On positive solutions concentrating on spheres for the Gierer-Meinhardt system, J. Differential Equations, 221 (2006), 158-189.
doi: 10.1016/j.jde.2005.03.004. |
[26] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
![]() ![]() |
[27] |
V. Rădulescu, Bifurcation and asymptotics for elliptic problems with singular nonlinearity. Elliptic and Parabolic Problems, 389-401, Progr. Nonlinear Differential Equations Appl. , 63, Birkhäuser, Basel, 2005.
doi: 10.1007/3-7643-7384-9_38. |
[28] |
A. Trembley, M'emoires pour servir à l'histoire dun genre de polype d'eau douce, à bras en forme de corne, Verbeek, Leiden, Netherland, 1744. |
[29] |
A. M. Turing,
The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society (B), 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[30] |
J. C. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied Mathematical Sciences, 189. Springer, London, 2014. xii+319 pp. ISBN: 978-1-4471-5525-6; 978-1-4471-5526-3.
doi: 10.1007/978-1-4471-5526-3. |
[31] |
Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, Introduction to Reaction-Diffusion Equations, Second Edition, Science Press, Beijing, 2011.
![]() |
[32] |
Y. Zhang,
Positive solutions of singular sublinear Dirichlet boundary value problems, SIAM J. Math. Anal., 26 (1995), 329-339.
doi: 10.1137/S0036141093246087. |
show all references
References:
[1] |
S. Chen, Y. Salmaniw and R. Xu,
Global existence for a singular Gierer-Meinhardt system, J. Differential Equations, 262 (2017), 2940-2960.
doi: 10.1016/j.jde.2016.11.022. |
[2] |
S. Chen,
Steady state solutions for a general activator-inhibitor model, Nonlinear Anal., 135 (2016), 84-96.
doi: 10.1016/j.na.2016.01.013. |
[3] |
Y. S. Choi and P. J. McKenna,
A singular Gierer-Meinhardt system of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 503-522.
doi: 10.1016/S0294-1449(00)00115-3. |
[4] |
Y. S. Choi and P. J. McKenna,
A singular Gierer-Meinhardt system of elliptic equations: The classical case, Nonlinear Anal., 55 (2003), 521-541.
doi: 10.1016/j.na.2003.07.003. |
[5] |
M. Ghergu,
Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition, Trans. Amer. Math. Soc., 361 (2009), 3953-3976.
doi: 10.1090/S0002-9947-09-04670-4. |
[6] |
M. Ghergu,
Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318.
doi: 10.1016/j.jfa.2010.02.003. |
[7] |
M. Ghergu and V. Rădulescu,
On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl., 311 (2005), 635-646.
doi: 10.1016/j.jmaa.2005.03.012. |
[8] |
M. Ghergu and V. Rădulescu,
On a class of singular Gierer-Meinhardt systems arising in morphogenesis, C. R. Math. Acad. Sci. Paris, 344 (2007), 163-168.
doi: 10.1016/j.crma.2006.12.008. |
[9] |
M. Ghergu and V. Rădulescu,
A singular Gierer-Meinhardt system with different source terms, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1215-1234.
doi: 10.1017/S0308210507000637. |
[10] |
M. Ghergu and V. Rădulescu, The influence of the distance function in some singular elliptic problems, Potential theory and stochastics in Albac, 125-137, Theta Ser. Adv. Math. , 11, Theta, Bucharest, 2009. |
[11] |
M. Ghergu and V. Rădulescu, Nonlinear PDEs. Mathematical Models in Biology, Chemistry and Population Genetics, With a foreword by Viorel Barbu. Springer Monographs in Mathematics. Springer, Heidelberg, 2012. xviii+391 pp. ISBN: 978-3-642-22663-2.
doi: 10.1007/978-3-642-22664-9. |
[12] |
A. Gierer and H. Meinhardt,
A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[13] |
H. Jiang,
Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751.
doi: 10.3934/dcds.2006.14.737. |
[14] |
H. Jiang and W. -M. Ni,
A priori estimates of stationary solutions of an activator-inhibitor system, Indiana Univ. Math. J., 56 (2007), 681-732.
doi: 10.1512/iumj.2007.56.2982. |
[15] |
E. H. Kim,
Singular Gierer-Meinhardt systems of elliptic boundary value problems, J. Math. Anal. Appl., 308 (2005), 1-10.
doi: 10.1016/j.jmaa.2004.10.039. |
[16] |
E. H. Kim,
A class of singular Gierer-Meinhardt systems of elliptic boundary value problems, Nonlinear Anal., 59 (2004), 305-318.
doi: 10.1016/S0362-546X(04)00260-3. |
[17] |
A. Lazer and J. P. McKenna,
On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.
doi: 10.1090/S0002-9939-1991-1037213-9. |
[18] |
F. Li, R. Peng and X. F. Song,
Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system, J. Differential Equations, 262 (2017), 559-589.
doi: 10.1016/j.jde.2016.09.040. |
[19] |
W. -M. Ni,
Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.
|
[20] |
W. -M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math. 82, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611971972. |
[21] |
W. -M. Ni, K. Suzuki and I. Takagi,
The dynamics of a kynetics activator-inhibitor system, J. Differential Equations, 229 (2006), 426-465.
doi: 10.1016/j.jde.2006.03.011. |
[22] |
W. -M. Ni, I. Takagi and E. Yanagida,
Stability of least energy patterns of the shadow system for an activator-inhibitor model, Japan J. Indust. Appl. Math., 18 (2001), 259-272.
doi: 10.1007/BF03168574. |
[23] |
W. -M. Ni and I. Takagi,
On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc., 297 (1986), 351-368.
doi: 10.1090/S0002-9947-1986-0849484-2. |
[24] |
W. -M. Ni and I. Takagi,
Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math., 12 (1995), 327-365.
doi: 10.1007/BF03167294. |
[25] |
W. -M. Ni and J. Wei,
On positive solutions concentrating on spheres for the Gierer-Meinhardt system, J. Differential Equations, 221 (2006), 158-189.
doi: 10.1016/j.jde.2005.03.004. |
[26] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
![]() ![]() |
[27] |
V. Rădulescu, Bifurcation and asymptotics for elliptic problems with singular nonlinearity. Elliptic and Parabolic Problems, 389-401, Progr. Nonlinear Differential Equations Appl. , 63, Birkhäuser, Basel, 2005.
doi: 10.1007/3-7643-7384-9_38. |
[28] |
A. Trembley, M'emoires pour servir à l'histoire dun genre de polype d'eau douce, à bras en forme de corne, Verbeek, Leiden, Netherland, 1744. |
[29] |
A. M. Turing,
The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society (B), 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[30] |
J. C. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied Mathematical Sciences, 189. Springer, London, 2014. xii+319 pp. ISBN: 978-1-4471-5525-6; 978-1-4471-5526-3.
doi: 10.1007/978-1-4471-5526-3. |
[31] |
Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, Introduction to Reaction-Diffusion Equations, Second Edition, Science Press, Beijing, 2011.
![]() |
[32] |
Y. Zhang,
Positive solutions of singular sublinear Dirichlet boundary value problems, SIAM J. Math. Anal., 26 (1995), 329-339.
doi: 10.1137/S0036141093246087. |
[1] |
Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks and Heterogeneous Media, 2013, 8 (1) : 291-325. doi: 10.3934/nhm.2013.8.291 |
[2] |
Henghui Zou. On global existence for the Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 583-591. doi: 10.3934/dcds.2015.35.583 |
[3] |
Manuel del Pino, Patricio Felmer, Michal Kowalczyk. Boundary spikes in the Gierer-Meinhardt system. Communications on Pure and Applied Analysis, 2002, 1 (4) : 437-456. doi: 10.3934/cpaa.2002.1.437 |
[4] |
Juncheng Wei, Matthias Winter. On the Gierer-Meinhardt system with precursors. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 363-398. doi: 10.3934/dcds.2009.25.363 |
[5] |
Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885 |
[6] |
Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto. Dynamics of a boundary spike for the shadow Gierer-Meinhardt system. Communications on Pure and Applied Analysis, 2012, 11 (1) : 115-145. doi: 10.3934/cpaa.2012.11.115 |
[7] |
Siu-Long Lei. Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 651-668. doi: 10.3934/dcdsb.2011.15.651 |
[8] |
Nabil T. Fadai, Michael J. Ward, Juncheng Wei. A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1431-1458. doi: 10.3934/dcdsb.2018158 |
[9] |
Kazuhiro Kurata, Kotaro Morimoto. Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1443-1482. doi: 10.3934/cpaa.2008.7.1443 |
[10] |
Theodore Kolokolnikov, Michael J. Ward. Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1033-1064. doi: 10.3934/dcdsb.2004.4.1033 |
[11] |
Mengxin Chen, Ranchao Wu, Yancong Xu. Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2275-2312. doi: 10.3934/dcdsb.2021132 |
[12] |
Jan-Phillip Bäcker, Matthias Röger. Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1139-1155. doi: 10.3934/cpaa.2022013 |
[13] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[14] |
Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613 |
[15] |
Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic and Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707 |
[16] |
Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213 |
[17] |
Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569 |
[18] |
Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763 |
[19] |
Yong Zeng. Existence and uniqueness of very weak solution of the MHD type system. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5617-5638. doi: 10.3934/dcds.2020240 |
[20] |
Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2189-2212. doi: 10.3934/dcds.2020111 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]