An Explicitly Solvable Nonlocal Eigenvalue Problem and the Stability of a Spike for a Sub-Diffusive Reaction-Diffusion System

Y. Nec¹ ; M. J. Ward¹

¹ Department of Mathematics, University of British Columbia 1984 Mathematics Road, Vancouver, V6T1Z2, BC, Canada

Mathematical modelling of natural phenomena, Tome 8 (2013) no. 2, pp. 55-87

Résumé

The stability of a one-spike solution to a general class of reaction-diffusion (RD) system with both regular and anomalous diffusion is analyzed. The method of matched asymptotic expansions is used to construct a one-spike equilibrium solution and to derive a nonlocal eigenvalue problem (NLEP) that determines the stability of this solution on an O(1) time-scale. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined in terms of the roots of certain simple transcendental equations that involve two key parameters related to the choice of the nonlinear kinetics. From a rigorous analysis of these transcendental equations by using a winding number approach and explicit calculations, sufficient conditions are given to predict the occurrence of Hopf bifurcations of the one-spike solution. Our analysis determines explicitly the number of possible Hopf bifurcation points as well as providing analytical formulae for them. The analysis is implemented for the shadow limit of the RD system defined on a finite domain and for a one-spike solution of the RD system on the infinite line. The theory is illustrated for two specific RD systems. Finally, in parameter ranges for which the Hopf bifurcation is unique, it is shown that the effect of sub-diffusion is to delay the onset of the Hopf bifurcation.

Détail

DOI : 10.1051/mmnp/20138205

Bibliographie

[1] W. Chen, M. J. Ward Europ. J. Appl. Math. 2009 187 214

[2] W. Chen, M. J. Ward SIAM J. Appl. Dyn. Sys. 2011 582 666

[3] A. Doelman, W. Eckhaus, T. J. Kaper SIAM J. Appl. Math. 2000 1080 1102

[4] A. Doelman, W. Eckhaus, T. J. Kaper SIAM J. Appl. Math. 2000 2036 2061

[5] A. Doelman, R. A. Gardner, T. Kaper Indiana U. Math. Journ. 2001 443 507

[6] A. Doelman, T. Kaper SIAM J. Appl. Dyn. Sys. 2003 53 96

[7] A. Doelman, T. Kaper, K. Promislow SIAM J. Math. Anal. 2007 1760 1789

[8] A. Doelman, H. Van Der Ploeg SIAM J. Appl. Dyn. Sys. 2002 65 104

[9] J. Ehrt, J. D. Rademacher, M. Wolfrum. First and second order semi-strong interaction of pulses in the Schnakenberg model. preprint, (2012).

[10] A. A. Golovin, B. J. Matkowsky, V. A. Volpert SIAM J. Appl. Math. 2008 251 272

[11] B. I. Henry, S. L. Wearne SIAM J. Appl. Math. 2002 870 887

[12] D. Iron, M. J. Ward SIAM J. Appl. Math. 2000 778 802

[13] D. Iron, M. J. Ward, J. Wei Physica D 2001 25 62

[14] D. Iron, M. J. Ward SIAM J. Appl. Math. 2002 1924 1951

[15] T. Kolokolnikov, M. J. Ward, J. Wei Studies in Appl. Math. 2005 21 71

[16] T. Koloklonikov, M. J. Ward, J. Wei Interfaces and Free Boundaries 2006 185 222

[17] T. Kolokolnikov, M. Ward, J. Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Disc. Cont. Dyn. Sys Series B., to appear, (2013), (34 pages).

[18] C. S. Lin, W. M. Ni, I. Takagi J. Diff. Eq. 1988 1 27

[19] R. Metzler, J. Klafter Phys. Rep. 2000 1 77

[20] C. Muratov, V. V. Osipov SIAM J. Appl. Math. 2002 1463 1487

[21] Y. Nec, V. A. Volpert, A. A. Nepomnyashchy Discr. Cont. Dyn. Sys. Series A 2010 827 846

[22] Y. Nec, A. A. Nepomnyashchy Math. Model. Nat. Phenom. 2007 77 105

[23] Y. Nec, A. A. Nepomnyashchy J. Physics A: Math. Theor. 2007 14687 14702

[24] Y. Nec, M. J. Ward Physica D 2012 947 963

[25] Y. Nec, M. J. Ward. The stability and slow dynamics of two-spike patterns for a class of reaction-diffusion system. submitted, (2013), (28 pages)

[26] K. B. Oldham, J. Spanier. The fractional calculus. Academic Press, New York, 1974.

[27] I. Podlubny. Fractional differential equations. Academic Press, San Diego, 1999.

[28] J. D. Rademacher. First and second order semi-strong interface interaction in reaction-diffusion systems. SIAM J. App. Dyn. Syst., (2012), to appear.

[29] R. K. Saxena, A. M. Mathai, H. J. Haubold Astrophys. Space Sci. 2006 289 296

[30] W. Sun, T. Tang, M. J. Ward, J. Wei Studies in Appl. Math. 2003 41 84

[31] W. Sun, M. J. Ward, R. Russell SIAM J. App. Dyn. Sys. 2005 904 953

[32] W. H. Tse, M. J. Ward. On explicitly solvable nonlocal eigenvalue problems and the stability of localized pulses. to be submitted, Applied Math Letters, (2013).

[33] J. C. Tzou, A. Bayliss, B. J. Matkowsky, V. A. Volpert Math. Model. Nat. Phenom. 2011 87 118

[34] J. C. Tzou, A. Bayliss, B. J. Matkowsky, V. A. Volpert Europ. J. Appl. Math. 2011 423 453

[35] J. C. Tzou, Y. Nec, M. J. Ward, The Stability of Localized Spikes for the 1-D Brusselator Reaction-Diffusion Model. Europ. J. Appl. Math., (2012), under review.

[36] H. Van Der Ploeg, A. Doelman Indiana U. Math. J. 2005 1219 1301

[37] M. J. Ward, J. Wei J. Nonlinear Science 2003 209 264

[38] M. J. Ward, J. Wei Europ. J. Appl. Math. 2003 677 711

[39] J. Wei Europ. J. Appl. Math. 1999 353 378

[40] J. Wei. Existence and stability of spikes for the Gierer-Meinhardt system. book chapter in Handbook of Differential Equations, Stationary Partial Differential Equations. Vol. 5 (M. Chipot ed.), Elsevier, (2008), pp. 489–581.

[41] M. Wolfrum, J. Ehrt WIAS Preprint 2007