We are interested in the threshold phenomenon for propagation in nonlocal diffusion equations with some compactly supported initial data. In the so-called bistable and ignition cases, we provide the first quantitative estimates for such phenomena. The outcomes dramatically depend on the tails of the dispersal kernel and can take a large variety of forms. The strategy is to combine sharp estimates of the tails of the sum of independent and identically random variables (coming, in particular, from large deviation theory) and the construction of accurate sub- and supersolutions.


  1. extinction
  2. propagation
  3. threshold phenomena
  4. nonlocal diffusion equations

MSC codes

  1. 35B40
  2. 45K05
  3. 35K57

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M. Alfaro, Fujita blow up phenomena and hair trigger effect: The role of dispersal tails, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), pp. 1309–1327.
M. Alfaro and J. Coville, Propagation phenomena in monostable integro-differential equations: Acceleration or not?, J. Differential Equations, 263 (2017), pp. 5727–5758.
M. Alfaro, A. Ducrot, and G. Faye, Quantitative estimates of the threshold phenomena for propagation in reaction-diffusion equations, SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 1291–1311.
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), pp. 33–76, https://doi.org/10.1016/0001-8708(78)90130-5.
P. W. Bates, P. C. Fife, X. Ren, and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), pp. 105–136.
H. Berestycki and N. Rodríguez, A non-local bistable reaction-diffusion equation with a gap, Discrete Contin. Dyn. Syst., 37 (2017), 685.
C. Besse, A. Capel, G. Faye, and G. Fouilhé, Asymptotic Behavior of Nonlocal Bistable Reaction-Diffusion Equations, preprint, https://arxiv.org/abs/2201.06482, 2022.
E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), pp. 271–291.
X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), pp. 125–160.
J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl. (4), 185 (2006), pp. 461–485.
J. Coville, Travelling Fronts in Asymmetric Nonlocal Reaction Diffusion Equations: The Bistable and Ignition Cases, preprint, 2007.
J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), pp. 727–755, https://doi.org/10.1017/S0308210504000721.
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Stoch. Model. Appl. Probab. 38, Springer-Verlag, Berlin, 2010, https://doi.org/10.1007/978-3-642-03311-7.
Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc. (JEMS), 12 (2010), pp. 279–312, https://doi.org/10.4171/JEMS/198.
R. Durrett, Probability, Theory and Examples, 4th ed, Camb. Ser. Stat. Probab. Math. 49, Cambridge University Press, Cambridge, UK, 2010.
G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), pp. 461–478, https://doi.org/10.1017/S030821050002583X.
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, John Wiley & Sons, New York, 1966.
P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), pp. 335–361.
C. Gui and T. Huan, Traveling wave solutions to some reaction diffusion equations with fractional Laplacians, Calc Var. Partial Differential Equations, 54 (2015), pp. 251–273, https://doi.org/10.1007/s00526-014-0785-y.
T. Höglund, A unified formulation of the central limit theorem for small and large deviations from the mean, Z Wahrscheinlichkeitstheorie und Verw. Gebiete, 49 (1979), pp. 105–117.
J. I. Kanel’, Stabilization of the solutions of the equations of combustion theory with finite initial functions, Mat. Sb., 107 (1964), pp. 398–413.
T. S. Lim, Long Time Dynamics for Multi-dimensional Reaction-Diffusion Equations with Non-local Diffusion, preprint, 2019.
A. Mellet, J.-M. Roquejoffre, and Y. Sire, Existence and asymptotics of fronts in non local combustion models, Commun. Math. Sci., 12 (2014), pp. 1–11.
T. Mikosch and A. V. Nagaev, Large deviations of heavy-tailed sums with applications in insurance, Extremes, 1 (1998), pp. 81–110.
C. B. Muratov and X. Zhong, Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), pp. 1519–1552, https://doi.org/10.1007/s00030-013-0220-7.
C. B. Muratov and X. Zhong, Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations, Discrete Contin. Dyn. Syst., 37 (2017), pp. 915–944, https://doi.org/10.3934/dcds.2017038.
S. V. Nagaev, Large deviations of sums of independent random variables, Ann. Probab., (1979), pp. 745–789.
E. J. G. Pitman, On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin, J. Aust. Math. Soc., 8 (1968), pp. 423–443.
P. Poláčik, Threshold solutions and sharp transitions for nonautonomous parabolic equations on \(\Bbb R^N\) , Arch. Ration. Mech. Anal., 199 (2011), pp. 69–97, https://doi.org/10.1007/s00205-010-0316-8.
W.-B. Xu, W.-T. Li, and S. Ruan, Spatial propagation in nonlocal dispersal Fisher-KPP equations, J. Funct. Anal., (2021), 108957.
H. Zhang, Y. Li, and X. Yang, Threshold solutions for nonlocal reaction diffusion equations, Commun. Math. Res., 38 (2022), pp. 389–421.
A. Zlatoš, Sharp transition between extinction and propagation of reaction, J. Amer. Math. Soc., 19 (2006), pp. 251–263.

Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 1596 - 1630
ISSN (online): 1095-7154


Submitted: 18 February 2022
Accepted: 13 October 2022
Published online: 31 May 2023


  1. extinction
  2. propagation
  3. threshold phenomena
  4. nonlocal diffusion equations

MSC codes

  1. 35B40
  2. 45K05
  3. 35K57



Matthieu Alfaro
Université de Rouen Normandie, CNRS, LMRS, Saint-Etienne-du-Rouvray, France.
Arnaud Ducrot Contact the author
Université Le Havre Normandie, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France.
Hao Kang
Center for Applied Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China.

Funding Information

Funding: The first author is supported by the région Normandie project BIOMA-NORMAN 21E04343 and the ANR project DEEV ANR-20-CE40-0011-01. The third author received support for his postdoc from the région Normandie.

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