Hostname: page-component-76c49bb84f-hwk5l Total loading time: 0 Render date: 2025-07-09T05:56:23.080Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Information

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.

References

[1] Baer, S. M., Erneux, T. & Rinzel, J. (1989) The slow passage through a Hopf bifurcation: Delay, memory effects, and resonance. SIAM J. Appl. Math. 49 (1), 55–71.Google Scholar
[2] Bernoff, A. J. & Witelski, T. P. (2010) Stability and dynamics of self-similarity in evolution equations. J. Eng. Math. 66 (1), 11–31.Google Scholar
[3] Budd, C., Dold, B. & Stuart, A. (1993) Blowup in a partial differential equation with conserved first integral. SIAM J. Appl. Math. 53 (3), 718–742.Google Scholar
[4] Budd, C., Rottschäfer, V. & Williams, J. F. (2008) Multi-bump, blow-up, self-similar solutions of the complex Ginzburg-Landau equation. Physica D 237 (4), 510–539.Google Scholar
[5] Doelman, A., Gardner, R. A. & Kaper, T. J. (1998) Stability analysis of singular patterns in the 1D Gray-Scott model: A matched asymptotics approach. Physica D 122, 1–36.Google Scholar
[6] Doelman, A., Gardner, R. A. & Kaper, T. J. (2001) Large stable pulse solutions in reaction-diffusion equations. Indiana U. Math. J. 50 (1), 443–507.CrossRefGoogle Scholar
[7] Doelman, A. & Kaper, T. (2003) Semistrong pulse interactions in a class of coupled reaction-diffusion equations. SIAM J. Appl. Dyn. Sys. 2 (1), 53–96.Google Scholar
[8] Ei, S., Nishiura, Y. & Ueda, K. (2001) 2 n splitting or edge splitting?: A manner of splitting in dissipative systems. Japan. J. Indust. Appl. Math. 18 (2), 181–205.Google Scholar
[9] Erneux, T., Reiss, E. L., Holden, L. J. & Georgiou, M. (1991) Slow passage through bifurcation and limit points. Asymptotic theory and applications. In: Dynamic Bifurcations (Luminy 1990), Lecture Notes in Math., Vol. 1493, Springer, Berlin, pp. 14–28.Google Scholar
[10] Filippas, S. & Guo, J. S. (1993) Quenching profiles for one-dimensional semilinear heat equations. Quart. Appl. Math. 51 (4), 713–729.Google Scholar
[11] Filippas, S. & Kohn, R. V. (1992) Refined asymptotics for the blow up of u t -Δu=u p . Comm. Pure Appl. Math. 45 (7), 821–869.Google Scholar
[12] FlexPDE6, PDE Solutions Inc. URL: http://www.pdesolutions.com Google Scholar
[13] Gao, W. J. & Han, Y. Z. (2011) Blow-up of a nonlocal semilinear parabolic equation with positive initial energy. Appl. Math. Lett. 24 (5), 784–788.Google Scholar
[14] Gierer, A. & Meinhardt, H. (1972) A theory of biological pattern formation. Kybernetik 12, 30–39.CrossRefGoogle ScholarPubMed
[15] Iron, D., Ward, M. J. & Wei, J. (2001) The stability of spike solutions to the one-dimensional Gierer-Meinhardt model. Physica D 150 (1–2), 25–62.Google Scholar
[16] Iron, D. & Ward, M. J. (2002) The dynamics of multi-spike solutions for the one-dimensional Gierer-Meinhardt model. SIAM J. Appl. Math. 62 (6), 1924–1951.CrossRefGoogle Scholar
[17] Jazar, M. & Kiwan, R. (2008) Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. Inst. H. Poincar Anal. Non Linaire 25, 215–218.Google Scholar
[18] Kolokolnikov, T., Ward, M. & Wei, J. (2005) The stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime. Physica D 202 (3–4), 258–293.Google Scholar
[19] Kolokolnikov, T., Ward, M. J. & Wei, J. (2005) Pulse-splitting for some reaction-diffusion systems in one-space dimension. Stud. Appl. Math. 114 (2), 115–165.Google Scholar
[20] Kolokonikov, T., Ward, M. J. & Wei, J. (2014) The stability of hot-spot patterns for a reaction-diffusion system of Urban crime. DCDS-B 19 (5), 1373–1410.CrossRefGoogle Scholar
[21] Kuehn, C. (2011) A mathematical framework for critical transitions: Bifurcations, fast/slow systems and stochastic dynamics. Physica D 240 (12), 1020–1035.Google Scholar
[22] Mandel, P. & Erneux, T. (1987) The slow passage through a steady bifurcation: Delay and memory effects. J. Stat. Phys. 48 (5–6), 1059–1070.Google Scholar
[23] Matthews, P. C. & Cox, S. M. (2000) Pattern formation with a conservation law. Nonlinearity 13 (4), 1293–1320.Google Scholar
[24] Moyles, I., Tse, W. H. & Ward, M. J. (2016) On explicitly solvable nonlocal eigenvalue problems and the stability of localized stripes in reaction-diffusion systems. Stud. Appl. Math. 136 (1), 89–136.Google Scholar
[25] Nec, Y. & Ward, M. J. (2013) An explicitly solvable nonlocal eigenvalue problem and the stability of a spike for a sub-diffusive reaction-diffusion system. Math. Modeling Natur. Phen. 8 (2), 55–87.Google Scholar
[26] Norbury, J., Wei, J. & Winter, M. (2002) Existence and stability of singular patterns in a Ginzburg-Landau equation coupled with a mean field. Nonlinearity 15 (6), 2077–2096.Google Scholar
[27] Nishiura, Y. & Ueyama, D. (1999) A skeleton structure of self-replicating dynamics. Physica D 130 (1–2), 73–104.Google Scholar
[28] Riecke, H. & Rappel, W. J. (1995) Coexisting pulses in a model for binary-mixture convection. Phys. Rev. Lett. 75, 4035–4038.Google Scholar
[29] Rottschäfer, V. (2013) Asymptotic analysis of a new type of multi-bump, self-similar, blowup solutions of the Ginzbrug Landau equation. Europ. J. Appl. Math. 24 (91), 103–129.CrossRefGoogle Scholar
[30] Soufi, E., Jazar, M. & Monneau, R. (2007) A gamma-convergence argument for the blow-up of a nonlocal semilinear parabolic equation with Neumann boundary conditions. Ann. Inst. H. Poincaré-Anal. Non Lineaire 24, 17–39.Google Scholar
[31] Souplet, P. (1999) Uniform blowup profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. J. Diff. Eq. 153 (2), 374–406.Google Scholar
[32] Souplet, P. (1998) Blowup in nonlocal reaction-diffusion equations. SIAM J. Math. Anal. 29 (6), 1301–1334.Google Scholar
[33] Tzou, J. C., Ward, M. J. & Kolokolnikov, T. (2015) Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction-diffusion systems. Physica D 290 (1), 24–43.Google Scholar
[34] Wang, X., Tian, F. & Li, G. (2015) Nonlocal parabolic equation with conserved spatial integral. Arch. Math. 105 (1), 93–100.Google Scholar
[35] Ward, M. J. & Wei, J. (2003) Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model. J. Nonlinear Sci. 13 (2), 209–264.Google Scholar
[36] Wei, J. & Winter, M. (2005) On a Cubic-Quintic Ginzburg-Landau equation with global coupling. Proc. Amer. Math. Soc. 133 (6), 1787–1796.Google Scholar