# American Institute of Mathematical Sciences

• Previous Article
On decomposition of ambient surfaces admitting $A$-diffeomorphisms with non-trivial attractors and repellers
• DCDS Home
• This Issue
• Next Article
A combinatorial approach to Rauzy-type dynamics II: The labelling method and a second proof of the KZB classification theorem
July  2022, 42(7): 3539-3556. doi: 10.3934/dcds.2022023

## On a class of singularly perturbed elliptic systems with asymptotic phase segregation

 1 CAMGSD, Instituto Superior Técnico, University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal 2 Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg, Erlangen (FAU), Germany 3 Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

*Corresponding author: Morteza Fotouhi

Received  July 2021 Revised  January 2022 Published  July 2022 Early access  March 2022

This work is devoted to study a class of singular perturbed elliptic systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behavior of limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain at least one of the components is zero, which implies that all components can not coexist simultaneously. We present a novel method, which provides an explicit solution of the limiting problem for a special choice of parameters. Moreover, we present some numerical simulations of the asymptotic problem.

Citation: Farid Bozorgnia, Martin Burger, Morteza Fotouhi. On a class of singularly perturbed elliptic systems with asymptotic phase segregation. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3539-3556. doi: 10.3934/dcds.2022023
##### References:
 [1] A. Arakelyan, Convergence of the finite difference scheme for a general class of the spatial segregation of reaction-diffusion systems, Comput. Math. Appl., 75 (2018), 4232-4240.  doi: 10.1016/j.camwa.2018.03.025. [2] A. Arakelyan and F. Bozorgnia, On the uniqueness of the limiting solution to a strongly competing system, Electronic J. Differential Equations, 96 (2017), 1-8. [3] F. Bozorgnia, Numerical algorithms for the spatial segregation of competitive systems, SIAM J. Sci. Comput., 31 (2009), 3946-3958.  doi: 10.1137/080722588. [4] F. Bozorgnia and A. Arakelyan, Numerical algorithms for a variational problem of the spatial segregation of reaction-diffusion systems, Appl. Math. Comput., 219 (2013), 8863-8875.  doi: 10.1016/j.amc.2013.03.074. [5] L. A. Caffarelli and A. Friedman, Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations, J. Differential Equations, 60 (1985), 420-433.  doi: 10.1016/0022-0396(85)90133-0. [6] L. A. Caffarelli and F.-H. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc., 21 (2008), 847-862.  doi: 10.1090/S0894-0347-08-00593-6. [7] L. A. Caffarelli and J.-M. Roquejoffre, Uniform Hölder estimate in a class of elliptic systems and applications to singular limits in models for diffusion flames, Arch. Ration. Mech. Anal., 183 (2007), 457-487.  doi: 10.1007/s00205-006-0013-9. [8] M. Conti, S. Terracini and G. Verzini, Uniqueness and least energy property for solutions to strongly competing systems, Interfaces Free Boundaries, 8 (2006), 437-446.  doi: 10.4171/IFB/150. [9] E. C. M. Crooks, E. N. Dancer and D. Hilhorst, On long-term dynamics for competition-diffusion system with inhomogeneous dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36. [10] E. N. Dancer and Y. H. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations, 114 (1994), 434-475.  doi: 10.1006/jdeq.1994.1156. [11] J. W. Dold, Flame propagation in a nonuniform mixture: Analysis of a slowly varying triple flame, Comb. Flame., 76 (1989), 71-88.  doi: 10.1016/0010-2180(89)90079-5. [12] S-I. Ei and E. Yanagida, Dynamics of interfaces in competition-diffusion systems, SIAM J. Appl. Math., 54 (1994), 1355-1373.  doi: 10.1137/S0036139993247343. [13] E. Giusti and M. Giaquinta, Global $C^{1, \alpha}$-regularity for second order quasilinear elliptic euqations in divergence form, Journal für die Reine und Angewandte Mathematik, 351 (1984), 55–65. [14] P. W. Schaefer, Some maximum principles in semilinear elliptic equations, Proceedings of the American Mathematical Society, 98 (1986), 97-102.  doi: 10.1090/S0002-9939-1986-0848884-X. [15] K. Wang and Z. Zhang, Some new results in competing systems with many species, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 739-776.  doi: 10.1016/j.anihpc.2009.11.004. [16] F. A. Williams, Combustion Theory: The Fundamental Theory of Chemically Reacting Flow Systems, Benjamin-Cummings, 1985. doi: 10.1201/9780429494055.

show all references

##### References:
 [1] A. Arakelyan, Convergence of the finite difference scheme for a general class of the spatial segregation of reaction-diffusion systems, Comput. Math. Appl., 75 (2018), 4232-4240.  doi: 10.1016/j.camwa.2018.03.025. [2] A. Arakelyan and F. Bozorgnia, On the uniqueness of the limiting solution to a strongly competing system, Electronic J. Differential Equations, 96 (2017), 1-8. [3] F. Bozorgnia, Numerical algorithms for the spatial segregation of competitive systems, SIAM J. Sci. Comput., 31 (2009), 3946-3958.  doi: 10.1137/080722588. [4] F. Bozorgnia and A. Arakelyan, Numerical algorithms for a variational problem of the spatial segregation of reaction-diffusion systems, Appl. Math. Comput., 219 (2013), 8863-8875.  doi: 10.1016/j.amc.2013.03.074. [5] L. A. Caffarelli and A. Friedman, Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations, J. Differential Equations, 60 (1985), 420-433.  doi: 10.1016/0022-0396(85)90133-0. [6] L. A. Caffarelli and F.-H. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc., 21 (2008), 847-862.  doi: 10.1090/S0894-0347-08-00593-6. [7] L. A. Caffarelli and J.-M. Roquejoffre, Uniform Hölder estimate in a class of elliptic systems and applications to singular limits in models for diffusion flames, Arch. Ration. Mech. Anal., 183 (2007), 457-487.  doi: 10.1007/s00205-006-0013-9. [8] M. Conti, S. Terracini and G. Verzini, Uniqueness and least energy property for solutions to strongly competing systems, Interfaces Free Boundaries, 8 (2006), 437-446.  doi: 10.4171/IFB/150. [9] E. C. M. Crooks, E. N. Dancer and D. Hilhorst, On long-term dynamics for competition-diffusion system with inhomogeneous dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36. [10] E. N. Dancer and Y. H. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations, 114 (1994), 434-475.  doi: 10.1006/jdeq.1994.1156. [11] J. W. Dold, Flame propagation in a nonuniform mixture: Analysis of a slowly varying triple flame, Comb. Flame., 76 (1989), 71-88.  doi: 10.1016/0010-2180(89)90079-5. [12] S-I. Ei and E. Yanagida, Dynamics of interfaces in competition-diffusion systems, SIAM J. Appl. Math., 54 (1994), 1355-1373.  doi: 10.1137/S0036139993247343. [13] E. Giusti and M. Giaquinta, Global $C^{1, \alpha}$-regularity for second order quasilinear elliptic euqations in divergence form, Journal für die Reine und Angewandte Mathematik, 351 (1984), 55–65. [14] P. W. Schaefer, Some maximum principles in semilinear elliptic equations, Proceedings of the American Mathematical Society, 98 (1986), 97-102.  doi: 10.1090/S0002-9939-1986-0848884-X. [15] K. Wang and Z. Zhang, Some new results in competing systems with many species, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 739-776.  doi: 10.1016/j.anihpc.2009.11.004. [16] F. A. Williams, Combustion Theory: The Fundamental Theory of Chemically Reacting Flow Systems, Benjamin-Cummings, 1985. doi: 10.1201/9780429494055.
$u_1$, $u_2$ and $u_3$
In the left picture coefficients are $A_{1}(x) = 1+x, A_2 = A_3 = 1.$ In the right picture, $A_{1}(x) = 1+100x, \, A_2 = A_3 = 1$. The free boundary point is $x_f = 0.09.$ At this point the free boundary condition is $u'_{1}(.09^-) = -A_{1}(x_{f}) \, u'_{2}(.09^+) = u'_{3}(.09^-)-A_{1}(x_{f}) \, u'_{3}(.09^+).$
The left picture shows the graph of $u_1$ while the right one depicts the graph of $u_1, u_2, u_3$ together
Picture (A) depicts the graph of $u_1 + u_2$. In (B), the plot indicate $h\times \Delta u_3$ on the free boundary. We note that $h\times \Delta u_3$ is fixed for every $h$
Free boundary and supports of the components
Laplacian of $u_1$ as measure(scaled) on the interfaces. The mesh size is $\triangle x = \triangle y = 10^{-3}$
 [1] Avner Friedman. Free boundary problems arising in biology. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013 [2] R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497 [3] Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44 [4] Wenlei Li, Shaoyun Shi. Singular perturbed renormalization group theory and its application to highly oscillatory problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1819-1833. doi: 10.3934/dcdsb.2018089 [5] Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006 [6] Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025 [7] Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386 [8] Xinfu Chen, Bei Hu, Jin Liang, Yajing Zhang. Convergence rate of free boundary of numerical scheme for American option. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1435-1444. doi: 10.3934/dcdsb.2016004 [9] Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763 [10] Arnaud Münch, Ademir Fernando Pazoto. Boundary stabilization of a nonlinear shallow beam: theory and numerical approximation. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 197-219. doi: 10.3934/dcdsb.2008.10.197 [11] G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279 [12] Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101 [13] Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Dynamic boundary conditions as limit of singularly perturbed parabolic problems. Conference Publications, 2011, 2011 (Special) : 737-746. doi: 10.3934/proc.2011.2011.737 [14] Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092 [15] Hirotoshi Kuroda, Noriaki Yamazaki. Approximating problems of vectorial singular diffusion equations with inhomogeneous terms and numerical simulations. Conference Publications, 2009, 2009 (Special) : 486-495. doi: 10.3934/proc.2009.2009.486 [16] Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5125-5147. doi: 10.3934/dcds.2019208 [17] Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6825-6842. doi: 10.3934/dcds.2019233 [18] Ugur G. Abdulla, Evan Cosgrove, Jonathan Goldfarb. On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems. Evolution Equations and Control Theory, 2017, 6 (3) : 319-344. doi: 10.3934/eect.2017017 [19] Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673 [20] Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6961-6978. doi: 10.3934/dcds.2019239

2021 Impact Factor: 1.588