November  2019, 18(6): 3011-3034. doi: 10.3934/cpaa.2019135

When fast diffusion and reactive growth both induce accelerating invasions

1. 

IMAG, Univ. Montpellier, CNRS, Montpellier, France

2. 

IECL, Université de Lorraine, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France

* Corresponding author

Received  September 2018 Revised  January 2019 Published  May 2019

Fund Project: M. Alfaro is supported by the ANR I-SITE MUSE, project MICHEL 170544IA (no ANR-IDEX-0006). T. Giletti is supported by the NONLOCAL project (no ANR-14-CE25-0013)

We focus on the spreading properties of solutions of monostable equations with fast diffusion. The nonlinear reaction term involves a weak Allee effect, which tends to slow down the propagation. We complete the picture of [3] by studying the subtle case where acceleration does occur and is induced by a combination of fast diffusion and of reactive growth. This requires the construction of new elaborate sub and supersolutions thanks to some underlying self-similar solutions.

Citation: Matthieu Alfaro, Thomas Giletti. When fast diffusion and reactive growth both induce accelerating invasions. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3011-3034. doi: 10.3934/cpaa.2019135
References:
[1]

M. Alfaro, Slowing Allee effect vs. accelerating heavy tails in monostable reaction diffusion equations, Nonlinearity, 30 (2017), 687-702.  doi: 10.1088/1361-6544/aa53b9.  Google Scholar

[2]

M. Alfaro and J. Coville, Propagation phenomena in monostable integro-differential equations: Acceleration or not?, J. Differential Equations, 263 (2017), 5727-5758.  doi: 10.1016/j.jde.2017.06.035.  Google Scholar

[3]

M. Alfaro and T. Giletti, Interplay of nonlinear diffusion, initial tails and Allee effect on the speed of invasions, preprint arXiv: 1711.10364. Google Scholar

[4]

A. Audrito and J. L. Vázquez, The Fisher–KPP problem with doubly nonlinear diffusion, Nonlinear Analysis, 157 (2017), 212-248.  doi: 10.1016/j.na.2017.03.015.  Google Scholar

[5]

X. Cabré and J.-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722.  doi: 10.1007/s00220-013-1682-5.  Google Scholar

[6] F. CourchampL. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2008.   Google Scholar
[7]

R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.   Google Scholar

[8]

J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.  Google Scholar

[9]

C. Gui and T. Huan, Traveling wave solutions to some reaction diffusion equations with fractional Laplacians, Calc. Var. Partial Differential Equations, 54 (2015), 251-273.  doi: 10.1007/s00526-014-0785-y.  Google Scholar

[10]

J.-S. Guo, Similarity solutions for a quasilinear parabolic equation, J. Austral. Math. Soc. Ser. B, 37 (1995), 253-266.  doi: 10.1017/S0334270000007694.  Google Scholar

[11]

J.-S. Guo and Y.-J. L. Guo, On a fast diffusion equation with source, Tohoku Math. J. (2), 53 (2001), 571-579.  doi: 10.2748/tmj/1113247801.  Google Scholar

[12]

F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Differential Equations, 249 (2010), 1726-1745.  doi: 10.1016/j.jde.2010.06.025.  Google Scholar

[13]

A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189.  doi: 10.1512/iumj.1982.31.31016.  Google Scholar

[14]

M. A. Herrero and M. Pierre, The Cauchy problem for $u_t=\Delta u^m$ when $0 < m < 1$, Trans. Amer. Math. Soc., 291 (1985), 145-158.  doi: 10.2307/1999900.  Google Scholar

[15]

J. R. King and P. M. McCabe, On the Fisher-KPP equation with fast nonlinear diffusion, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2529-2546.  doi: 10.1098/rspa.2003.1134.  Google Scholar

[16]

A. N. KolmogorovI. G. Petrovsky and N. S. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Etat Moscou, Sér. Inter. A, 1 (1937), 1-26.   Google Scholar

[17] J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, vol. 33 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2006.  doi: 10.1093/acprof:oso/9780199202973.001.0001.  Google Scholar

show all references

References:
[1]

M. Alfaro, Slowing Allee effect vs. accelerating heavy tails in monostable reaction diffusion equations, Nonlinearity, 30 (2017), 687-702.  doi: 10.1088/1361-6544/aa53b9.  Google Scholar

[2]

M. Alfaro and J. Coville, Propagation phenomena in monostable integro-differential equations: Acceleration or not?, J. Differential Equations, 263 (2017), 5727-5758.  doi: 10.1016/j.jde.2017.06.035.  Google Scholar

[3]

M. Alfaro and T. Giletti, Interplay of nonlinear diffusion, initial tails and Allee effect on the speed of invasions, preprint arXiv: 1711.10364. Google Scholar

[4]

A. Audrito and J. L. Vázquez, The Fisher–KPP problem with doubly nonlinear diffusion, Nonlinear Analysis, 157 (2017), 212-248.  doi: 10.1016/j.na.2017.03.015.  Google Scholar

[5]

X. Cabré and J.-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722.  doi: 10.1007/s00220-013-1682-5.  Google Scholar

[6] F. CourchampL. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2008.   Google Scholar
[7]

R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.   Google Scholar

[8]

J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.  Google Scholar

[9]

C. Gui and T. Huan, Traveling wave solutions to some reaction diffusion equations with fractional Laplacians, Calc. Var. Partial Differential Equations, 54 (2015), 251-273.  doi: 10.1007/s00526-014-0785-y.  Google Scholar

[10]

J.-S. Guo, Similarity solutions for a quasilinear parabolic equation, J. Austral. Math. Soc. Ser. B, 37 (1995), 253-266.  doi: 10.1017/S0334270000007694.  Google Scholar

[11]

J.-S. Guo and Y.-J. L. Guo, On a fast diffusion equation with source, Tohoku Math. J. (2), 53 (2001), 571-579.  doi: 10.2748/tmj/1113247801.  Google Scholar

[12]

F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Differential Equations, 249 (2010), 1726-1745.  doi: 10.1016/j.jde.2010.06.025.  Google Scholar

[13]

A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189.  doi: 10.1512/iumj.1982.31.31016.  Google Scholar

[14]

M. A. Herrero and M. Pierre, The Cauchy problem for $u_t=\Delta u^m$ when $0 < m < 1$, Trans. Amer. Math. Soc., 291 (1985), 145-158.  doi: 10.2307/1999900.  Google Scholar

[15]

J. R. King and P. M. McCabe, On the Fisher-KPP equation with fast nonlinear diffusion, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2529-2546.  doi: 10.1098/rspa.2003.1134.  Google Scholar

[16]

A. N. KolmogorovI. G. Petrovsky and N. S. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Etat Moscou, Sér. Inter. A, 1 (1937), 1-26.   Google Scholar

[17] J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, vol. 33 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2006.  doi: 10.1093/acprof:oso/9780199202973.001.0001.  Google Scholar
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