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September  2016, 36(9): 4997-5010. doi: 10.3934/dcds.2016016

Finite-time blowup of solutions to some activator-inhibitor systems

 1 Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław 2 College of Science, Ibaraki University, 2-1-1 Bunkyo, Mito 310-8512, Japan

Received  July 2015 Revised  February 2016 Published  May 2016

We study a dynamics of solutions to a system of reaction-diffusion equations modeling a biological pattern formation. This model has activator-inhibitor type nonlinearities and we show that it has solutions blowing up in a finite time. More precisely, in the case of absence of a diffusion of an activator, we show that there are solutions which blow up in a finite time at one point, only. This result holds true for the whole range of nonlinearity exponents in the considered activator-inhibitor system. Next, we consider a range of nonlinearities, where some space-homogeneous solutions blow up in a finite time and we show an analogous result for space-inhomogeneous solutions.
Citation: Grzegorz Karch, Kanako Suzuki, Jacek Zienkiewicz. Finite-time blowup of solutions to some activator-inhibitor systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4997-5010. doi: 10.3934/dcds.2016016
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