[Sur une classe de systèmes singuliers de Gierer–Meinhardt avec applications en morphogenèse]
We study the existence or the nonexistence of classical solutions to a singular Gierer–Meinhardt system with Dirichlet boundary condition. The main feature of our model is that the activator and the inhibitor have different sources given by general nonlinearities. Additional regularity and uniqueness results are established for the one-dimensional case.
On étudie l'existence ou la non-existence des solutions classiques pour une classe de systèmes singuliers de Gierer–Meinhardt avec condition de Dirichlet sur le bord. La caractéristique de notre modèle réside dans la présence de sources différentes pour l'activateur et aussi pour l'inhibiteur. Des propriétés supplémentaires de régularité et d'unicité sont établies en dimension 1.
Accepté le :
Publié le :
Marius Ghergu 1 ; Vicenţiu Rădulescu 2
@article{CRMATH_2007__344_3_163_0, author = {Marius Ghergu and Vicen\c{t}iu R\u{a}dulescu}, title = {On a class of singular {Gierer{\textendash}Meinhardt} systems arising in morphogenesis}, journal = {Comptes Rendus. Math\'ematique}, pages = {163--168}, publisher = {Elsevier}, volume = {344}, number = {3}, year = {2007}, doi = {10.1016/j.crma.2006.12.008}, language = {en}, }
Marius Ghergu; Vicenţiu Rădulescu. On a class of singular Gierer–Meinhardt systems arising in morphogenesis. Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 163-168. doi : 10.1016/j.crma.2006.12.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.12.008/
[1] A singular Gierer–Meinhardt system of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 17 (2000), pp. 503-522
[2] A singular Gierer–Meinhardt system of elliptic equations: the classical case, Nonlinear Anal., Volume 55 (2003), pp. 521-541
[3] On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl., Volume 311 (2005), pp. 635-646
[4] M. Ghergu, V. Rădulescu, A singular Gierer–Meinhardt system with different source terms, in preparation
[5] A theory of biological pattern formation, Kybernetik, Volume 12 (1972), pp. 30-39
[6] A class of singular Gierer–Meinhardt systems of elliptic boundary value problems, Nonlinear Anal., Volume 59 (2004), pp. 305-318
[7] Singular Gierer–Meinhardt systems of elliptic boundary value problems, J. Math. Anal. Appl., Volume 308 (2005), pp. 1-10
[8] Generation and regeneration of sequence of structures during morphogenesis, J. Theoret. Biol., Volume 85 (1980), pp. 429-450
[9] Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., Volume 45 (1998), pp. 9-18
[10] The dynamics of a kinetics activato-inhibitor system, J. Differential Equations, Volume 229 (2006), pp. 426-465
[11] On positive solutions concentrating on spheres for the Gierer Meinhardt system, J. Differential Equations, Volume 221 (2006), pp. 158-189
[12] The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London B, Volume 237 (1952), pp. 37-72
[13] Existence and stability analysis of asymmetric for the Gierer Meinhardt system, J. Math. Pures Appl., Volume 83 (2004), pp. 433-476
[14] Spikes for the Gierer–Meinhardt system in two dimensions: the strong coupling case, J. Differential Equations, Volume 178 (2002), pp. 478-518
- A Meyer wavelet neural networks procedure for prediction, pantograph and delayed singular models, Intelligent Systems with Applications, Volume 26 (2025), p. 200457 | DOI:10.1016/j.iswa.2024.200457
- Meta-heuristic tuned with Gudermannian neural network for the singular Neumann, Dirichlet and Neumann-Robin boundary conditions, Multiscale and Multidisciplinary Modeling, Experiments and Design, Volume 8 (2025) no. 8 | DOI:10.1007/s41939-025-00961-6
- Constant sign and nodal solutions for singular Gierer-Meinhardt-type system, Applicable Analysis, Volume 103 (2024) no. 15, pp. 2829-2844 | DOI:10.1080/00036811.2024.2319693 | Zbl:1556.35129
- Efficient hypergeometric wavelet approach for solving lane-emden equations, Journal of Computational Science, Volume 82 (2024), p. 102392 | DOI:10.1016/j.jocs.2024.102392
- Bridging machine learning and weighted residual methods for delay differential equations of fractional order, Applied Soft Computing, Volume 149 (2023), p. 110936 | DOI:10.1016/j.asoc.2023.110936
- Multiple solutions to Gierer-Meinhardt systems, Discrete and Continuous Dynamical Systems, Volume 43 (2023) no. 7, pp. 2835-2851 | DOI:10.3934/dcds.2023031 | Zbl:1514.35176
- Some recent results on singular
-Laplacian systems, Discrete and Continuous Dynamical Systems. Series S, Volume 16 (2023) no. 6, pp. 1435-1451 | DOI:10.3934/dcdss.2022170 | Zbl:1519.35004 - A novel design of a sixth-order nonlinear modeling for solving engineering phenomena based on neuro intelligence algorithm, Engineering with Computers, Volume 39 (2023) no. 3, p. 1807 | DOI:10.1007/s00366-021-01596-0
- An advance computational intelligent approach to solve the third kind of nonlinear pantograph Lane–Emden differential system, Evolving Systems, Volume 14 (2023) no. 3, p. 393 | DOI:10.1007/s12530-022-09469-7
- A novel numerical approach and stability analysis for a class of pantograph delay differential equation, Journal of Computational Science, Volume 67 (2023), p. 101976 | DOI:10.1016/j.jocs.2023.101976
- Swarming Computational Efficiency to Solve a Novel Third-Order Delay Differential Emden-Fowler System, Computers, Materials Continua, Volume 73 (2022) no. 3, p. 4833 | DOI:10.32604/cmc.2022.030888
- Dynamics of multi-point singular fifth-order Lane–Emden system with neuro-evolution heuristics, Evolving Systems, Volume 13 (2022) no. 6, p. 795 | DOI:10.1007/s12530-021-09413-1
- Nonlinear fractional and singular systems: nonexistence, existence, uniqueness, and Hölder regularity, Mathematical Methods in the Applied Sciences, Volume 45 (2022) no. 9, pp. 5283-5303 | DOI:10.1002/mma.8106 | Zbl:1529.35548
- Neuro-swarms intelligent computing using Gudermannian kernel for solving a class of second order Lane-Emden singular nonlinear model, AIMS Mathematics, Volume 6 (2021) no. 3, pp. 2468-2485 | DOI:10.3934/math.2021150 | Zbl:1525.65072
- Heuristic computational design of Morlet wavelet for solving the higher order singular nonlinear differential equations, Alexandria Engineering Journal, Volume 60 (2021) no. 6, p. 5935 | DOI:10.1016/j.aej.2021.04.001
- Evolutionary Integrated Heuristic with Gudermannian Neural Networks for Second Kind of Lane–Emden Nonlinear Singular Models, Applied Sciences, Volume 11 (2021) no. 11, p. 4725 | DOI:10.3390/app11114725
- Neuro-swarm heuristic using interior-point algorithm to solve a third kind of multi-singular nonlinear system, Mathematical Biosciences and Engineering, Volume 18 (2021) no. 5, pp. 5285-5308 | DOI:10.3934/mbe.2021268 | Zbl:1505.65234
- A novel design of Gudermannian function as a neural network for the singular nonlinear delayed, prediction and pantograph differential models, Mathematical Biosciences and Engineering, Volume 19 (2021) no. 1, p. 663 | DOI:10.3934/mbe.2022030
- Integrated intelligent computing paradigm for nonlinear multi-singular third-order Emden–Fowler equation, Neural Computing and Applications, Volume 33 (2021) no. 8, p. 3417 | DOI:10.1007/s00521-020-05187-w
- Evolutionary heuristic with Gudermannian neural networks for the nonlinear singular models of third kind, Physica Scripta, Volume 96 (2021) no. 12, p. 125261 | DOI:10.1088/1402-4896/ac3c56
- Intelligence computing approach for solving second order system of Emden–Fowler model, Journal of Intelligent Fuzzy Systems, Volume 38 (2020) no. 6, p. 7391 | DOI:10.3233/jifs-179813
- Design and numerical solutions of a novel third-order nonlinear Emden-Fowler delay differential model, Mathematical Problems in Engineering, Volume 2020 (2020), p. 9 (Id/No 7359242) | DOI:10.1155/2020/7359242 | Zbl:1459.65103
- Global existence and blowup solutions for the Gierer-Meinhardt system, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 196 (2020), p. 9 (Id/No 111785) | DOI:10.1016/j.na.2020.111785 | Zbl:1439.35263
- Numerical investigations of a new singular second-order nonlinear coupled functional Lane–Emden model, Open Physics, Volume 18 (2020) no. 1, p. 770 | DOI:10.1515/phys-2020-0185
- Multiple positive solutions for a class of quasilinear singular elliptic systems, Rendiconti del Circolo Matemàtico di Palermo. Serie II, Volume 69 (2020) no. 3, pp. 977-994 | DOI:10.1007/s12215-019-00449-6 | Zbl:1460.35176
- Solving a new design of nonlinear second-order Lane–Emden pantograph delay differential model via Bernoulli collocation method, The European Physical Journal Plus, Volume 135 (2020) no. 5 | DOI:10.1140/epjp/s13360-020-00449-x
- Design of neuro-swarming-based heuristics to solve the third-order nonlinear multi-singular Emden–Fowler equation, The European Physical Journal Plus, Volume 135 (2020) no. 5 | DOI:10.1140/epjp/s13360-020-00424-6
- Numerical investigations to design a novel model based on the fifth order system of Emden–Fowler equations, Theoretical and Applied Mechanics Letters, Volume 10 (2020) no. 5, p. 333 | DOI:10.1016/j.taml.2020.01.049
- Existence and asymptotic behaviour of positive solutions for semilinear polyharmonic coupled systems, Complex Variables and Elliptic Equations, Volume 62 (2017) no. 11, p. 1665 | DOI:10.1080/17476933.2017.1286332
- Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system, Discrete and Continuous Dynamical Systems, Volume 37 (2017) no. 8, pp. 4489-4505 | DOI:10.3934/dcds.2017192 | Zbl:1370.35146
- Global behavior of positive solutions of a generalized Lane-Emden system of nonlinear differential equations, Mediterranean Journal of Mathematics, Volume 14 (2017) no. 2, p. 14 (Id/No 81) | DOI:10.1007/s00009-017-0889-7 | Zbl:1369.34037
- Bifurcation analysis of a diffusive activator-inhibitor model in vascular mesenchymal cells, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, Volume 25 (2015) no. 10, p. 25 (Id/No 1530026) | DOI:10.1142/s0218127415300268 | Zbl:1326.35028
- On a class of singular elliptic systems, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 119 (2015), pp. 98-105 | DOI:10.1016/j.na.2014.08.013 | Zbl:1359.35060
- Existence and Stability of Spikes for the Gierer-Meinhardt System in Two Dimensions, Mathematical Aspects of Pattern Formation in Biological Systems, Volume 189 (2014), p. 107 | DOI:10.1007/978-1-4471-5526-3_6
- Positive solutions for some competitive fractional systems in bounded domains, Journal of Function Spaces and Applications, Volume 2013 (2013), p. 6 (Id/No 140130) | DOI:10.1155/2013/140130 | Zbl:1302.35397
- A variational semilinear singular system, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 74 (2011) no. 12, pp. 3849-3860 | DOI:10.1016/j.na.2011.01.017 | Zbl:1219.35085
- Lane-Emden systems with negative exponents, Journal of Functional Analysis, Volume 258 (2010) no. 10, pp. 3295-3318 | DOI:10.1016/j.jfa.2010.02.003 | Zbl:1190.35073
- Existence of positive bounded solutions for some nonlinear elliptic systems, Journal of Mathematical Analysis and Applications, Volume 352 (2009) no. 1, pp. 440-448 | DOI:10.1016/j.jmaa.2008.04.029 | Zbl:1162.35024
- A singular Gierer-Meinhardt system of elliptic equations in
, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 71 (2009) no. 3-4, pp. 708-716 | DOI:10.1016/j.na.2008.10.103 | Zbl:1171.35362 - Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition, Transactions of the American Mathematical Society, Volume 361 (2009) no. 8, pp. 3953-3976 | DOI:10.1090/s0002-9947-09-04670-4 | Zbl:1172.35018
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