Blow-up of positive solutions to a singular Gierer-Meinhardt system

This paper deals with blow-up of positive solutions for a singular Gierer-Meinhardt system subject to zero Dirichlet boundary conditions. We first prove the existence of a local solution and then show blow-up solutions under certain conditions for parameters. We use a functional method to obtain a local solution which is bounded by the first eigenfunction both from above and from below and then obtain the bound of a ratio of two solutions. This method is completely different from the traditional methods of sub and super solutions. (C) 2019 Elsevier Ltd. All rights reserved.