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Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition


Author: Marius Ghergu
Journal: Trans. Amer. Math. Soc. 361 (2009), 3953-3976
MSC (2000): Primary 35J55; Secondary 35B40, 35J60
DOI: https://doi.org/10.1090/S0002-9947-09-04670-4
Published electronically: March 12, 2009
MathSciNet review: 2500874
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Abstract: This paper is concerned with the following Gierer-Meinhardt type systems subject to Dirichlet boundary conditions:

$\displaystyle \left\{\begin{tabular}{ll} $\displaystyle\Delta u- \alpha u +\fra... ...yle u=0,\; v=0$ \quad & $\mbox{\rm on } \partial\Omega,$ \end{tabular} \right. $

where $ \Omega\subset\mathbb{R}^N$ ($ N\geq 1$) is a smooth bounded domain, $ \rho(x)\geq 0$ in $ \Omega$ and $ \alpha,\beta\geq 0$. We are mainly interested in the case of different source terms, that is, $ (p,q)\neq (r,s)$. Under appropriate conditions on the exponents $ p,q,r$ and $ s$ we establish various results of existence, regularity and boundary behavior. In the one dimensional case a uniqueness result is also presented.


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Additional Information

Marius Ghergu
Affiliation: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
Email: marius.ghergu@imar.ro

DOI: https://doi.org/10.1090/S0002-9947-09-04670-4
Keywords: Gierer-Meinhardt system, singular nonlinearities, asymptotic behavior
Received by editor(s): March 12, 2007
Published electronically: March 12, 2009
Article copyright: © Copyright 2009 American Mathematical Society

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