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

Grzegorz Karch; Kanako  Suzuki; Jacek  Zienkiewicz

doi:10.3934/dcds.2016016

Article Contents

2016, Volume 36, Issue 9: 4997-5010. Doi: 10.3934/dcds.2016016

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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

Received: June 30, 2015

Revised: January 31, 2016

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 35B40, 35K50.

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