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Transition to blow-up in a reaction–diffusion model with localized spike solutions†

Published online by Cambridge University Press:  07 March 2017

V. ROTTSCHÄFER ,
J. C. TZOU  and
M.J. WARD
V. ROTTSCHÄFER
Affiliation:
Mathematical Institute, Leiden University, P.O. Box 9512, Leiden, the Netherlands email: vivi@math.leidenuniv.nl
J. C. TZOU
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada emails: tzou.justin@gmail.com, ward@math.ubc.ca
M.J. WARD
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada emails: tzou.justin@gmail.com, ward@math.ubc.ca
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Abstract

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For certain singularly perturbed two-component reaction–diffusion systems, the bifurcation diagram of steady-state spike solutions is characterized by a saddle-node behaviour in terms of some parameter in the system. For some such systems, such as the Gray–Scott model, a spike self-replication behaviour is observed as the parameter varies across the saddle-node point. We demonstrate and analyse a qualitatively new type of transition as a parameter is slowly decreased below the saddle node value, which is characterized by a finite-time blow-up of the spike solution. More specifically, we use a blend of asymptotic analysis, linear stability theory, and full numerical computations to analyse a wide variety of dynamical instabilities, and ultimately finite-time blow-up behaviour, for localized spike solutions that occur as a parameter β is slowly ramped in time below various linear stability and existence thresholds associated with steady-state spike solutions. The transition or route to an ultimate finite-time blow-up can include spike nucleation, spike annihilation, or spike amplitude oscillation, depending on the specific parameter regime. Our detailed analysis of the existence and linear stability of multi-spike patterns, through the analysis of an explicitly solvable non-local eigenvalue problem, provides a theoretical guide for predicting which transition will be realized. Finally, we analyse the blow-up profile for a shadow limit of the reaction–diffusion system. For the resulting non-local scalar parabolic problem, we derive an explicit expression for the blow-up rate near the parameter range where blow-up is predicted. This blow-up rate is confirmed with full numerical simulations of the full PDE. Moreover, we analyse the linear stability of this solution that blows up in finite time.

Keywords

Nonlocal eigenvalue problemdelayed bifurcationsaddle nodeHopf bifurcationfinite-time blow-up
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Special Issue 6: Big Data and Partial Differential Equations , December 2017 , pp. 1015 - 1055
Copyright
Copyright © Cambridge University Press 2017 

Footnotes

†

M. J. Ward was supported by an NSERC Discovery Grant 81541 (Canada). J. C. Tzou was supported by a PIMS postdoctoral fellowship.

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