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

• *Corresponding author: Morteza Fotouhi
• 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: • • 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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