American Institute of Mathematical Sciences

Advanced
  • 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
August  2017, 37(8): 4489-4505. doi: 10.3934/dcds.2017192

Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system

Rui Peng , Xianfa Song and  Lei Wei

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

Received  November 2016 Revised  March 2017 Published  April 2017

This paper is concerned with the stationary Gierer-Meinhardt system with singularity:
$\left\{\begin{array}{ll} d_1\Delta u-a_1 u+\frac{u^p}{v^q}+\rho_1(x)=0, \ \ & x\in\Omega, \\ d_2\Delta v-a_2 v+\frac{u^r}{v^s}+\rho_2(x)=0,\ \ & x\in\Omega,\\ u(x)>0,\ \ v(x)>0,\ \ & x\in \Omega,\\ \displaystyle u(x)=v(x)=0,\ \ & x\in\partial\Omega, \end{array}\right.$
where $-\infty < p < 1$, $-1 < s$, and $q, r, d_1, d_2$ are positive constants, $a_1, \, a_2$ are nonnegative constants, $\rho_1, \, \rho_2$ are smooth nonnegative functions and $\Omega\subset \mathbb{R}^d\, (d\geq1)$ is a bounded smooth domain. New sufficient conditions, some of which are necessary, on the existence of classical solutions are established. A uniqueness result of solutions in any space dimension is also derived. Previous results are substantially improved; moreover, a much simpler mathematical approach with potential application in other problems is developed.
Keywords: Gierer-Meinhardt system, singularity, stationary solution, Dirichlet boundary, existence, uniqueness.
Mathematics Subject Classification: Primary: 35J55, 35J75; Secondary: 35Q92.
Citation: Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318.  doi: 10.1016/j.jfa.2010.02.003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

W. -M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[32]

Y. Zhang, Positive solutions of singular sublinear Dirichlet boundary value problems, SIAM J. Math. Anal., 26 (1995), 329-339.  doi: 10.1137/S0036141093246087.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318.  doi: 10.1016/j.jfa.2010.02.003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

W. -M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[32]

Y. Zhang, Positive solutions of singular sublinear Dirichlet boundary value problems, SIAM J. Math. Anal., 26 (1995), 329-339.  doi: 10.1137/S0036141093246087.  Google Scholar

[1]

Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks & 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 & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - B, 2004, 4 (4) : 1033-1064. doi: 10.3934/dcdsb.2004.4.1033
[11]

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 & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[12]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613
[13]

Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic & Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707
[14]

Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213
[15]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[16]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763
[17]

Yong Zeng. Existence and uniqueness of very weak solution of the MHD type system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5617-5638. doi: 10.3934/dcds.2020240
[18]

Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2189-2212. doi: 10.3934/dcds.2020111
[19]

Chunxiao Guo, Fan Cui, Yongqian Han. Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1687-1699. doi: 10.3934/dcdss.2016070
[20]

Jérôme Coville, Juan Dávila. Existence of radial stationary solutions for a system in combustion theory. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 739-766. doi: 10.3934/dcdsb.2011.16.739

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (50)
  • HTML views (47)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences