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March  2018, 13(1): 119-153. doi: 10.3934/nhm.2018006

Fisher-KPP equations and applications to a model in medical sciences

Benjamin Contri

Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, UMR 7373, 13453 Marseille, France

Received  August 2016 Revised  January 2018 Published  March 2018

Fund Project: This work has been carried out in the framework of Archimedes LabEx (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir' French Government program managed by the French National Research Agency (ANR). The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 - ReaDi - Reaction-Diffusion Equations, Propagation and Modelling, and from the ANR project NONLOCAL (ANR-14-CE25-0013).

This paper is devoted to a class of reaction-diffusion equations with nonlinearities depending on time modeling a cancerous process with chemotherapy. We begin by considering nonlinearities periodic in time. For these functions, we investigate equilibrium states, and we deduce the large time behavior of the solutions, spreading properties and the existence of pulsating fronts. Next, we study nonlinearities asymptotically periodic in time with perturbation. We show that the large time behavior and the spreading properties can still be determined in this case.

Keywords: Reaction-diffusion equation, KPP nonlinearity, principal eigenvalue, pulsating front, spreading speed, medical models.
Mathematics Subject Classification: Primary: 35K57, 92C50; Secondary: 35C07.
Citation: Benjamin Contri. Fisher-KPP equations and applications to a model in medical sciences. Networks and Heterogeneous Media, 2018, 13 (1) : 119-153. doi: 10.3934/nhm.2018006
References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[2]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.  doi: 10.1002/cpa.3022.

[3]

H. Berestycki, F. Hamel and N. Nadirashvili, The principal eigenvalue of elliptic operators with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480.  doi: 10.1007/s00220-004-1201-9.

[4]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. I - Periodic framework, J. Europ. Math. Soc., 7 (2005), 173-213. 

[5]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: I - Species persistence, J. Math. Bio., 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.

[6]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: II - Biological invasions and pulsating traveling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.

[7]

B. Contri, Pulsating fronts for bistable on average reaction-diffusion equations in a time periodic environment, Journal of Mathematical Analysis and Applications, 437 (2016), 90-132.  doi: 10.1016/j.jmaa.2015.12.030.

[8]

W. Ding, F. Hamel and X. Zhao, Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat, Indiana Univ. Math. J., 66 (2017), 1189-1265.  doi: 10.1512/iumj.2017.66.6070.

[9]

J. Fang, X. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Func. Anal., 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.

[10]

J. Fang and X.-Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Europ. Math. Soc., 17 (2015), 2243-2288.  doi: 10.4171/JEMS/556.

[11]

P. C. Fife and J. B. McLeod, The approach of solutions of non-linear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. 

[12]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.

[13]

M. Freidlin and J. Gartner, On the propagation of concentration waves in periodic and random media, Sov. Math. Dokl., 20 (1979), 1282-1286. 

[14]

G. Frejacques, Travelling Waves In Infinite Cylinders With Time-periodic Coefficients, Ph. D thesis, Université d'Aix-Marseille, 2005.

[15]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Longman Scientific and Technical, 1991.

[16]

W. Hudson and B. Zinner, Existence of travelling waves for reaction-diffusion equations of fisher type in periodic media, Boundary Value Problems for Functional-Differential Equations, J. Henderson (ed. ), World Scientific, 1995,187–199.

[17]

A. N. Kolmogorov, I. 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, Bulletin Université D'Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), 1 (1937), 1-26. 

[18]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.

[19]

X. Liang and X.-Q Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.

[20]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295.  doi: 10.1007/s10231-008-0075-4.

[21]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.

[22]

G. Nadin, Existence and uniqueness of the solutions of a space-time periodic reaction-diffusion equation, J. Diff. Equations, 249 (2010), 1288-1304.  doi: 10.1016/j.jde.2010.05.007.

[23]

J. Nolen, M. Rudd and J. Xin, Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24.  doi: 10.4310/DPDE.2005.v2.n1.a1.

[24]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. Ⅱ. Existence, J. Diff. Equations, 159 (1999), 55-101.  doi: 10.1006/jdeq.1999.3652.

[25]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. Ⅰ. Stability and uniqueness, J. Diff. Equations, 159 (1999), 1-54.  doi: 10.1006/jdeq.1999.3651.

[26]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments, Theor. Pop. Bio., 30 (1986), 143-160.  doi: 10.1016/0040-5809(86)90029-8.

[27]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Bio, 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.

[28]

J. X. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Ration. Mech. Anal., 121 (1992), 205-233.  doi: 10.1007/BF00410613.

[29]

J. X. Xin, Analysis and modeling of front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[2]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.  doi: 10.1002/cpa.3022.

[3]

H. Berestycki, F. Hamel and N. Nadirashvili, The principal eigenvalue of elliptic operators with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480.  doi: 10.1007/s00220-004-1201-9.

[4]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. I - Periodic framework, J. Europ. Math. Soc., 7 (2005), 173-213. 

[5]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: I - Species persistence, J. Math. Bio., 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.

[6]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: II - Biological invasions and pulsating traveling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.

[7]

B. Contri, Pulsating fronts for bistable on average reaction-diffusion equations in a time periodic environment, Journal of Mathematical Analysis and Applications, 437 (2016), 90-132.  doi: 10.1016/j.jmaa.2015.12.030.

[8]

W. Ding, F. Hamel and X. Zhao, Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat, Indiana Univ. Math. J., 66 (2017), 1189-1265.  doi: 10.1512/iumj.2017.66.6070.

[9]

J. Fang, X. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Func. Anal., 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.

[10]

J. Fang and X.-Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Europ. Math. Soc., 17 (2015), 2243-2288.  doi: 10.4171/JEMS/556.

[11]

P. C. Fife and J. B. McLeod, The approach of solutions of non-linear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. 

[12]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.

[13]

M. Freidlin and J. Gartner, On the propagation of concentration waves in periodic and random media, Sov. Math. Dokl., 20 (1979), 1282-1286. 

[14]

G. Frejacques, Travelling Waves In Infinite Cylinders With Time-periodic Coefficients, Ph. D thesis, Université d'Aix-Marseille, 2005.

[15]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Longman Scientific and Technical, 1991.

[16]

W. Hudson and B. Zinner, Existence of travelling waves for reaction-diffusion equations of fisher type in periodic media, Boundary Value Problems for Functional-Differential Equations, J. Henderson (ed. ), World Scientific, 1995,187–199.

[17]

A. N. Kolmogorov, I. 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, Bulletin Université D'Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), 1 (1937), 1-26. 

[18]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.

[19]

X. Liang and X.-Q Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.

[20]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295.  doi: 10.1007/s10231-008-0075-4.

[21]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.

[22]

G. Nadin, Existence and uniqueness of the solutions of a space-time periodic reaction-diffusion equation, J. Diff. Equations, 249 (2010), 1288-1304.  doi: 10.1016/j.jde.2010.05.007.

[23]

J. Nolen, M. Rudd and J. Xin, Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24.  doi: 10.4310/DPDE.2005.v2.n1.a1.

[24]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. Ⅱ. Existence, J. Diff. Equations, 159 (1999), 55-101.  doi: 10.1006/jdeq.1999.3652.

[25]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. Ⅰ. Stability and uniqueness, J. Diff. Equations, 159 (1999), 1-54.  doi: 10.1006/jdeq.1999.3651.

[26]

N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments, Theor. Pop. Bio., 30 (1986), 143-160.  doi: 10.1016/0040-5809(86)90029-8.

[27]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Bio, 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.

[28]

J. X. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Ration. Mech. Anal., 121 (1992), 205-233.  doi: 10.1007/BF00410613.

[29]

J. X. Xin, Analysis and modeling of front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.

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