Large‐time behaviour and blow up of solutions for Gierer–Meinhardt systems

The purpose of this paper is to give a proof of global existence of solutions for Gierer–Meinhardt systems with homogeneous Neumann boundary conditions. Our technique is based on Lyapunov functional argument that yields the uniform boundedness of solutions. The asymptotic behaviour of the solutions under suitable conditions is also studied. Moreover, under reasonable conditions on the exponents of the nonlinear term, we show the blow up in finite time of the solutions for the considered system. These results are valid for any positive initial data in , without any differentiability conditions. Copyright © 2016 John Wiley & Sons, Ltd.