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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. ContiS. 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. CrooksE. 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. ContiS. 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. CrooksE. 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.

Figure 1.  $ u_1 $, $ u_2 $ and $ u_3 $
Figure 2.  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^+). $
Figure 3.  The left picture shows the graph of $ u_1 $ while the right one depicts the graph of $ u_1, u_2, u_3 $ together
Figure 4.  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 $
Figure 6.  Free boundary and supports of the components
Figure 7.  Laplacian of $ u_1 $ as measure(scaled) on the interfaces. The mesh size is $ \triangle x = \triangle y = 10^{-3} $
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