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2007 Coincidence sets in semilinear elliptic problems of logistic type
Shingo Takeuchi
Differential Integral Equations 20(9): 1075-1080 (2007). DOI: 10.57262/die/1356039312


This paper concerns the formation of a coincidence set for the positive solution of an equation of the type: $-{\varepsilon} \Delta u=u|a(x)-u|^{\theta} {\operatorname{sgn}} (a(x)-u)$, where ${\varepsilon}$ is a positive parameter, $0 < \theta < 1$ and $a(x)$ is a positive continuous function. Suppose that $\Delta a=0$ in an open subset ${\Omega}_0 \subset {\Omega}$. The positive solution converges to $a(x)$ uniformly on any compact subset of $\Omega$ as ${\varepsilon} \to 0$. It is proved that when ${\varepsilon}$ is sufficiently small, the solution coincides with $a(x)$ somewhere in ${\Omega}_0$ and the coincidence set converges to ${\Omega}_0$ in the Hausdorff distance with the order of $\sqrt{{\varepsilon}}$ as ${\varepsilon} \to 0$. The proof relies on the comparison principle with suitable local comparison functions.


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Shingo Takeuchi. "Coincidence sets in semilinear elliptic problems of logistic type." Differential Integral Equations 20 (9) 1075 - 1080, 2007.


Published: 2007
First available in Project Euclid: 20 December 2012

zbMATH: 1212.35141
MathSciNet: MR2349381
Digital Object Identifier: 10.57262/die/1356039312

Primary: 35J60
Secondary: 35B25 , 35J25

Rights: Copyright © 2007 Khayyam Publishing, Inc.


Vol.20 • No. 9 • 2007
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