Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition
HTML articles powered by AMS MathViewer
- by Marius Ghergu PDF
- Trans. Amer. Math. Soc. 361 (2009), 3953-3976 Request permission
Abstract:
This paper is concerned with the following Gierer-Meinhardt type systems subject to Dirichlet boundary conditions: \[ \begin {cases} \Delta u - \alpha u + \frac {u^p}{v^q} + \rho (x) = 0,\; u > 0, & \text {in $\Omega $},\\ \Delta v - \beta v + \frac {u^r}{v^s} = 0,\; v > 0, & \text {in $\Omega $}, \\ u=0,\; v=0 & \text {on $\partial \Omega $}, \end {cases} \] where $\Omega \subset \mathbb {R}^N$ ($N\geq 1$) is a smooth bounded domain, $\rho (x)\geq 0$ in $\Omega$ and $\alpha ,\beta \geq 0$. We are mainly interested in the case of different source terms, that is, $(p,q)\neq (r,s)$. Under appropriate conditions on the exponents $p,q,r$ and $s$ we establish various results of existence, regularity and boundary behavior. In the one dimensional case a uniqueness result is also presented.References
- Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 17 (2000), no. 4, 503–522 (English, with English and French summaries). MR 1782742, DOI 10.1016/S0294-1449(00)00115-3
- Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations: the classical case, Nonlinear Anal. 55 (2003), no. 5, 521–541. MR 2012446, DOI 10.1016/j.na.2003.07.003
- Louis Dupaigne, Marius Ghergu, and Vicenţiu Rădulescu, Lane-Emden-Fowler equations with convection and singular potential, J. Math. Pures Appl. (9) 87 (2007), no. 6, 563–581 (English, with English and French summaries). MR 2335087, DOI 10.1016/j.matpur.2007.03.002
- M. Ghergu and V. Rădulescu, Singular elliptic problems: Bifurcation and asymptotic analysis, Oxford University Press, No. 37, 2008.
- M. Ghergu and V. Rădulescu, A singular Gierer-Meinhardt system with different source terms, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), no. 6, 1215–1234., DOI 10.1017/S0308210507000637
- Marius Ghergu and Vicenţiu Rădulescu, On a class of singular Gierer-Meinhardt systems arising in morphogenesis, C. R. Math. Acad. Sci. Paris 344 (2007), no. 3, 163–168 (English, with English and French summaries). MR 2292281, DOI 10.1016/j.crma.2006.12.008
- Marius Ghergu and Vicenţiu Rădulescu, On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl. 311 (2005), no. 2, 635–646. MR 2168423, DOI 10.1016/j.jmaa.2005.03.012
- A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972), 30–39., DOI 10.1007/BF00289234
- Changfeng Gui and Fang-Hua Lin, Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 6, 1021–1029. MR 1263903, DOI 10.1017/S030821050002970X
- Huiqiang Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst. 14 (2006), no. 4, 737–751. MR 2177095, DOI 10.3934/dcds.2006.14.737
- James P. Keener, Activators and inhibitors in pattern formation, Stud. Appl. Math. 59 (1978), no. 1, 1–23. MR 0479051, DOI 10.1002/sapm19785911
- Eun Heui Kim, A class of singular Gierer-Meinhardt systems of elliptic boundary value problems, Nonlinear Anal. 59 (2004), no. 3, 305–318. MR 2093092, DOI 10.1016/j.na.2004.07.014
- Eun Heui Kim, Singular Gierer-Meinhardt systems of elliptic boundary value problems, J. Math. Anal. Appl. 308 (2005), no. 1, 1–10. MR 2141599, DOI 10.1016/j.jmaa.2004.10.039
- David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
- A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc. 111 (1991), no. 3, 721–730. MR 1037213, DOI 10.1090/S0002-9939-1991-1037213-9
- Wei-Ming Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), no. 1, 9–18. MR 1490535
- Wei-Ming Ni, Diffusion and cross-diffusion in pattern formation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 15 (2004), no. 3-4, 197–214. MR 2148879
- Wei-Ming Ni, Kanako Suzuki, and Izumi Takagi, The dynamics of a kinetic activator-inhibitor system, J. Differential Equations 229 (2006), no. 2, 426–465. MR 2263562, DOI 10.1016/j.jde.2006.03.011
- Wei-Ming Ni and Izumi Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819–851. MR 1115095, DOI 10.1002/cpa.3160440705
- Wei-Ming Ni and Izumi Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), no. 2, 247–281. MR 1219814, DOI 10.1215/S0012-7094-93-07004-4
- Wei-Ming Ni and Juncheng Wei, On positive solutions concentrating on spheres for the Gierer-Meinhardt system, J. Differential Equations 221 (2006), no. 1, 158–189. MR 2193846, DOI 10.1016/j.jde.2005.03.004
- Steven D. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal. 3 (1979), no. 6, 897–904. MR 548961, DOI 10.1016/0362-546X(79)90057-9
- A.M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society (B) 237 (1952), 37–72.
- A. Trembley, Mémoires pour servir à l’histoire d’un genre de polype d’eau douce, à bras en forme de corne, Verbeek, Leiden, Netherland, 1744.
- Juncheng Wei and Matthias Winter, Spikes for the Gierer-Meinhardt system in two dimensions: the strong coupling case, J. Differential Equations 178 (2002), no. 2, 478–518. MR 1879835, DOI 10.1006/jdeq.2001.4019
- Juncheng Wei and Matthias Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl. (9) 83 (2004), no. 4, 433–476. MR 2048385, DOI 10.1016/j.matpur.2003.09.006
- Zhongli Wei, Positive solutions of singular sublinear second order boundary value problems, Systems Sci. Math. Sci. 11 (1998), no. 1, 82–88. MR 1610532
- Zhijun Zhang and Jiangang Cheng, Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems, Nonlinear Anal. 57 (2004), no. 3, 473–484. MR 2064102, DOI 10.1016/j.na.2004.02.025
Additional Information
- Marius Ghergu
- Affiliation: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
- Email: marius.ghergu@imar.ro
- Received by editor(s): March 12, 2007
- Published electronically: March 12, 2009
- © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 3953-3976
- MSC (2000): Primary 35J55; Secondary 35B40, 35J60
- DOI: https://doi.org/10.1090/S0002-9947-09-04670-4
- MathSciNet review: 2500874