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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:
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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

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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

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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

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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

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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

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

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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

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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

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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

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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

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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

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