Global existence for a singular Gierer-Meinhardt system

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• 作者：Shaohua Chen^a ; ^{george_chen@cbu.ca} ; Yurij Salmaniw^b ; ^{yurij_salmaniw@hotmail.com} ; Runzhang Xu^c ; ^d ; ^{xurunzh@163.com}
• 关键词：Global existence ; Singular Gierer&ndash ; Meinhardt systems ; Activator&ndash ; inhibitor model ; Dirichlet boundary conditions
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：262
• 期：3
• 页码：2940-2960
• 全文大小：313 K
• 卷排序：262

文摘

This paper is concerned with existence results for a singular Gierer–Meinhardt system subject to zero Dirichlet boundary conditions, which originally arose in studies of pattern-formation in biology. The mathematical difficulties are that the system becomes singular near the boundary and it lacks a variational structure. We use a functional method to obtain both upper and lower bounds for the perturbed system and then use Sobolev embedding theorem to prove the existence of a pair of positive solutions under suitable conditions. This method is first used in a singular parabolic system and is completely different than the traditional methods of sub and super solutions.