Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 279-305.

We consider a radially symmetric free boundary problem with logistic nonlinear term. The spatial environment is assumed to be asymptotically periodic at infinity in the radial direction. For such a free boundary problem, it is known from [7] that a spreading-vanishing dichotomy holds. However, when spreading occurs, only upper and lower bounds are obtained in [7] for the asymptotic spreading speed. In this paper, we investigate one-dimensional pulsating semi-waves in spatially periodic media. We prove existence, uniqueness of such pulsating semi-waves, and show that the asymptotic spreading speed of the free boundary problem coincides with the speed of the corresponding pulsating semi-wave.

DOI : 10.1016/j.anihpc.2013.11.004
Classification : 35K20, 35R35, 35J60, 92B05
Mots-clés : Diffusive logistic equation, Free boundary, Periodic environment, Pulsating semi-wave, Spreading speed 
@article{AIHPC_2015__32_2_279_0,
     author = {Du, Yihong and Liang, Xing},
     title = {Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {279--305},
     publisher = {Elsevier},
     volume = {32},
     number = {2},
     year = {2015},
     doi = {10.1016/j.anihpc.2013.11.004},
     mrnumber = {3325238},
     zbl = {1321.35263},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.004/}
}
TY  - JOUR
AU  - Du, Yihong
AU  - Liang, Xing
TI  - Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 279
EP  - 305
VL  - 32
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.004/
DO  - 10.1016/j.anihpc.2013.11.004
LA  - en
ID  - AIHPC_2015__32_2_279_0
ER  - 
%0 Journal Article
%A Du, Yihong
%A Liang, Xing
%T Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 279-305
%V 32
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.004/
%R 10.1016/j.anihpc.2013.11.004
%G en
%F AIHPC_2015__32_2_279_0
Du, Yihong; Liang, Xing. Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 279-305. doi : 10.1016/j.anihpc.2013.11.004. http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.004/

[1] H. Berestycki, F. Hamel, Front propagation in periodic excitable media, Commun. Pure Appl. Math. 55 (2002), 949 -1032 | MR | Zbl

[2] H. Berestycki, F. Hamel, G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal. 255 no. 9 (2008), 2146 -2189 | MR | Zbl

[3] H. Berestycki, F. Hamel, N. Nadirashvili, The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc. 7 (2005), 173 -213 | EuDML | MR | Zbl

[4] H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol. 51 (2005), 75 -113 | MR | Zbl

[5] H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts, J. Math. Pures Appl. 84 (2005), 1101 -1146 | MR | Zbl

[6] G. Bunting, Y. Du, K. Krakowski, Spreading speed revisited: analysis of a free boundary model, Netw. Heterog. Media 7 (2012), 583 -603 | MR | Zbl

[7] Y. Du, Z.M. Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary II, J. Diff. Eqns. 250 (2011), 4336 -4366 | MR | Zbl

[8] Y. Du, Z.M. Guo, The Stefan problem for the Fisher–KPP equation, J. Diff. Eqns. 253 (2012), 996 -1035 | MR | Zbl

[9] Y. Du, Z.M. Guo, R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal. 265 (2013), 2089 -2142 | MR | Zbl

[10] Y. Du, Z.G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal. 42 (2010), 377 -405 , SIAM J. Math. Anal. 45 (2013), 1995 -1996 | MR | Zbl

[11] Y. Du, L.S. Liu, Remarks on the uniqueness problem for the logistic equation on the entire space, Bull. Aust. Math. Soc. 73 (2006), 129 -137 | MR

[12] A. Ducrot, T. Giletti, H. Matano, Existence and convergence to a propagating terrace in one-dimensional reaction–diffusion equations, Trans. Am. Math. Soc. (2013) | MR | Zbl

[13] J. Gärtner, M.I. Freidlin, On the propagation of concentration waves in periodic and random media, Sov. Math. Dokl. 20 (1979), 1282 -1286 | MR | Zbl

[14] F. Hamel, L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Eur. Math. Soc. 13 (2011), 345 -390 | EuDML | MR | Zbl

[15] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural'Ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI (1968) | MR

[16] X. Liang, X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal. 259 no. 4 (2010), 857 -903 | MR | Zbl

[17] G.M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore (1996) | MR

[18] H. Matano, Traveling waves in spatially random media, Mathematical Economics, RIMS Kokyuroku vol. 1337 (2003), 1 -9

[19] G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl. 188 (2009), 269 -295 | MR | Zbl

[20] G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction–diffusion equation, J. Diff. Eqns. 249 (2010), 1288 -1304 | MR | Zbl

[21] Wenxian Shen, Traveling waves in time dependent bistable equations, Diff. Integral Eqns. 19 (2006), 241 -278 | MR | Zbl

[22] H.F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol. 45 no. 6 (2002), 511 -548 | MR | Zbl

[23] Maolin Zhou, The asymptotic behavior of the Fisher–KPP equation with free boundary, preprint.

Cité par Sources :