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

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


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