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Large‐time behaviour and blow up of solutions for Gierer–Meinhardt systems
Author(s) -
Henine Safia,
Youkana Amar
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3502
Subject(s) - mathematics , neumann boundary condition , differentiable function , nonlinear system , homogeneous , mathematical analysis , exponential stability , lyapunov function , term (time) , boundary value problem , physics , quantum mechanics , combinatorics
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 C ( Ω ̄ ) , without any differentiability conditions. Copyright © 2016 John Wiley & Sons, Ltd.