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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 |
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.
Mathematics Subject Classification:
Primary: 35R35, 35B40; Secondary: 35A01.
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
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References:
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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. |
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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. |
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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