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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 and Continuous Dynamical Systems, 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.

[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.

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