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

  • *Corresponding author: Morteza Fotouhi

    *Corresponding author: Morteza Fotouhi
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  • 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.


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  • 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 5.   

    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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