Abstract
We investigate the effects of a Heaviside cutoff on the dynamics of traveling fronts in a family of scalar reaction-diffusion equations with degenerate polynomial potential that includes the classical Zeldovich equation. We prove the existence and uniqueness of front solutions in the presence of the cutoff, and we derive the leading-order asymptotics of the corresponding propagation speed in terms of the cutoff parameter. For the Zeldovich equation, an explicit solution to the equation without cutoff is known, which allows us to calculate higher-order terms in the resulting expansion for the front speed; in particular, we prove the occurrence of logarithmic (switchback) terms in that case. Our analysis relies on geometric methods from dynamical systems theory and, in particular, on the desingularization technique known as ‘blow-up.’
Similar content being viewed by others
References
Abramowitz, M.A., Stegun, I.A. (eds): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Ninth printing. Dover Publications, New York (1974)
Amann, H.: Ordinary differential equations. An introduction to nonlinear analysis. Translated from the German by Gerhard Metzen. de Gruyter Studies in Mathematics, vol. 13. Walter de Gruyter, Berlin (1990)
Aronson D.G.: Density-dependent Interaction Systems. In: Stewart, W.E., Ray, W.H., Conley, C.C. (eds) Dynamics and Modeling of Reactive Systems, pp. 161–176. Academic Press, New York (1980)
Aronson, D.G., Weinberger, H.F.: Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Pulse Propagation. In: Partial Differential Equations and Related Topics, Tulane University, New Orleans, LA, 1974, Lecture Notes in Mathematics, vol. 446, pp. 5–49. Springer, Berlin (1975)
Benguria R.D., Depassier M.C.: Variational principle for the asymptotic speed of fronts of the density-dependent diffusion-reaction equation. Phys. Rev. E 52(3), 3285–3287 (1995)
Benguria R.D., Depassier M.C.: Speed of Fronts of the Reaction-Diffusion Equation. Phys. Rev. Lett. 77(6), 1171–1173 (1996)
Benguria R.D., Depassier M.C.: Speed of pulled fronts with cutoff. Phys. Rev. E 75(5), 051106 (2007)
Benguria R.D., Depassier M.C., Haikala V.: Effect of a cutoff on pushed and bistable fronts of the reaction-diffusion equation. Phys. Rev. E 76(5), 051101 (2007)
Billingham J., Needham D.J.: A note on the properties of a family of travelling-wave solutions arising in cubic autocatalysis. Dyn. Stab. Syst. 6(1), 33–49 (1991)
Billingham J.: Phase plane analysis of one-dimensional reaction-diffusion waves with degenerate reaction terms. Dyn. Stab. Syst. 15(1), 23–33 (2000)
Britton N.F.: Reaction-Diffusion Equations and Their Applications to Biology. Academic Press, London (1986)
Brunet E., Derrida B.: Shift in the velocity of a front due to a cutoff. Phys. Rev. E 56(3), 2597–2604 (1997)
De Maesschalck, P.: Personal communication (2009)
Dumortier F.: Techniques in the Theory of Local Bifurcations: Blow-Up, Normal Forms, Nilpotent Bifurcations, Singular Perturbations. In: Schlomiuk, D. (ed.) Bifurcations and Periodic Orbits of Vector Fields, NATO ASI Series C vol. 408, pp. 19–73. Kluwer, Dordrecht, The Netherlands (1993)
Dumortier F., Llibre J., Artés J.C.: Qualitative theory of planar differential systems. Universitext, Springer, Berlin (2006)
Dumortier F., Popović N., Kaper T.J.: The asymptotic critical wave speed in a family of scalar reaction-diffusion equations. J. Math. Anal. Appl. 326(2), 1007–1023 (2007)
Dumortier F., Popović N., Kaper T.J.: The critical wave speed for the FKPP equation with cut-off. Nonlinearity 20(4), 855–877 (2007)
Dumortier F., Popović N., Kaper T.J.: A geometric approach to bistable front propagation in scalar reaction-diffusion equations with cut-off. Phys. D 239(20), 1984–1999 (2010)
Dumortier, F., Popović, N., Kaper, T.J.: Wave speeds for pushed fronts in reaction-diffusion equations with cut-offs, in preparation (2010)
Fisher R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937)
Gilding, B.H., Kersner, R.: Travelling waves in nonlinear diffusion-convection reaction. Progress in Nonlinear Differential Equations and their Applications, vol. 60. Birkhäuser, Basel (2004)
van Gils S., Krupa M., Szmolyan P.: Asymptotic expansions using blow-up. Z. Angew. Math. Phys. 56(3), 369–397 (2005)
Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds. Lecture Notes in Mathematics, vol. 583. Springer, Berlin (1977)
Kessler D.A., Ner Z., Sander L.M.: Front propagation: precursors, cutoffs, and structural stability. Phys. Rev. E 58(1), 107–114 (1998)
King A.C., Needham D.J.: The effects of variable diffusivity on the development of travelling waves in a class of reaction-diffusion equations. Philos. Trans. Roy. Soc. Lond. Ser. A 348(1687), 229–260 (1994)
Kolmogorov A.N., Petrowskii I.G., Piscounov N.: Etude de l’équation de la diffusion avec croissance de la quantité de matiére et son application à un problème biologique. Moscow Univ. Math. Bull. 1, 1–25 (1937)
Krupa M., Szmolyan P.: Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions. SIAM J. Math. Anal. 33(2), 286–314 (2001)
Lagerstrom P.A.: Matched Asymptotic Expansions. Ideas and Techniques. Applied Mathematical Sciences vol. 76. Springer, New York (1988)
Méndez V., Campos D., Zemskov E.P.: Variational principles and the shift in the front speed due to a cutoff. Phys. Rev. E 72(5), 056113 (2005)
Merkin J.H., Needham D.J.: Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension. Z. Angew. Math. Phys. 44(4), 707–721 (1993)
Needham D.J., Barnes A.N.: Reaction-diffusion and phase waves occurring in a class of scalar reaction-diffusion equations. Nonlinearity 12(1), 41–58 (1999)
Panja D.: Effects of fluctuations on propagating fronts. Phys. Rep. 393, 87–174 (2004)
Popović, N.: A geometric analysis of logarithmic switchback phenomena. In: HAMSA 2004: Proceedings of the International Workshop on Hysteresis and Multi-Scale Asymptotics, Cork 2004, J. Phys. Conference Series, vol. 22, pp. 164–173 (2005)
Popović, N.: Front speeds, cut-offs, and desingularization: a brief case study. In: Fluids and waves, Contemp. Math. vol. 440, pp. 187–195, Am. Math. Soc., Providence, RI (2007)
Popović N., Kaper T.J.: Rigorous asymptotic expansions for critical wave speeds in a family of scalar reaction-diffusion equations. J. Dyn. Differ. Equ. 18(1), 103–139 (2006)
Popović N., Szmolyan P.: Rigorous asymptotic expansions for Lagerstrom’s model equation—a geometric approach. Nonlinear Anal. 59(4), 531–565 (2004)
van Saarlos W.: Front propagation into unstable states. Phys. Rep. 386, 29–222 (2003)
Sánchez-Garduño F., Maini P.K.: Travelling wave phenomena in some degenerate reaction-diffusion equations. J. Differ. Equ. 117(2), 281–319 (1995)
Sherratt J.A., Marchant B.P.: Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation. IMA J. Appl. Math. 56(3), 289–302 (1996)
Witelski T.P., Ono K., Kaper T.J.: Critical wave speeds for a family of scalar reaction-diffusion equations. Appl. Math. Lett. 14(1), 65–73 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Popović, N. A geometric analysis of front propagation in a family of degenerate reaction-diffusion equations with cutoff. Z. Angew. Math. Phys. 62, 405–437 (2011). https://doi.org/10.1007/s00033-011-0115-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-011-0115-6