March  2013, 8(1): 79-114. doi: 10.3934/nhm.2013.8.79

Traveling fronts guided by the environment for reaction-diffusion equations

1. 

CAMS, UMR 8557, EHESS, 190-198 avenue de France, 75244 Paris Cedex 13, France

2. 

LATP, UMR 7353, Aix-Marseille Université, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France

Received  May 2012 Revised  March 2013 Published  April 2013

This paper deals with the existence of traveling fronts for the reaction-diffusion equation: $$ \frac{\partial u}{\partial t} - \Delta u =h(u,y) \qquad t\in \mathbb{R}, \; x=(x_1,y)\in \mathbb{R}^N. $$ We first consider the case $h(u,y)=f(u)-\alpha g(y)u$ where $f$ is of KPP or bistable type and $\lim_{|y|\rightarrow +\infty}g(y)=+\infty$. This equation comes from a model in population dynamics in which there is spatial spreading as well as phenotypic mutation of a quantitative phenotypic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of threshold value $\alpha_0$ and of a nonzero asymptotic profile (a stationary limiting solution) $V(y)$ if and only if $\alpha<\alpha_0$. When this condition is met, we prove the existence of a traveling front. This allows us to completely identify the behavior of the solution of the parabolic problem in the KPP case.
    We also study here the case where $h(y,u)=f(u)$ for $|y|\leq L_1$ and $h(y,u) \approx - \alpha u$ for $|y|>L_2\geq L_1$. This equation provides a general framework for a model of cortical spreading depressions in the brain. We prove the existence of traveling front if $L_1$ is large enough and the non-existence if $L_2$ is too small.
Citation: Henri Berestycki, Guillemette Chapuisat. Traveling fronts guided by the environment for reaction-diffusion equations. Networks & Heterogeneous Media, 2013, 8 (1) : 79-114. doi: 10.3934/nhm.2013.8.79
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show all references

References:
[1]

arXiv:1211.3228, (2013). Google Scholar

[2]

Stroke, 31 (2000), 2901-2906. doi: 10.1161/01.STR.31.12.2901.  Google Scholar

[3]

J. Analyse Math., 38 (1980), 144-187.  Google Scholar

[4]

Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37. doi: 10.1007/BF01244896.  Google Scholar

[5]

J. Funct. Anal., 40 (1981), 1-29. doi: 10.1016/0022-1236(81)90069-0.  Google Scholar

[6]

In preparation., (2013). Google Scholar

[7]

In preparation, (2013). Google Scholar

[8]

Comm. Pures and Appl. Math., 62 (2009), 729-788. doi: 10.1002/cpa.20275.  Google Scholar

[9]

C. R. Math. Acad. Sci. Paris, 343 (2006), 711-716. doi: 10.1016/j.crma.2006.09.036.  Google Scholar

[10]

in "Perspectives in Nonlinear Partial Differential Equations", 446 of Contemp. Math., 101-123. Amer. Math. Soc., Providence, RI, (2007). doi: 10.1090/conm/446.  Google Scholar

[11]

Comm. Pure Appl. Math., to appear, (2012). doi: 10.1002/cpa.21389.  Google Scholar

[12]

in "Bifurcation and Nonlinear Eigenvalue Problems (Proc., Session, Univ. Paris XIII, Villetaneuse, 1978)", 782 of Lecture Notes in Math., 16-41. Springer, Berlin, (1980). doi: 10.1007/BFb0090426.  Google Scholar

[13]

Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[14]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497-572.  Google Scholar

[15]

J. Eur. Math. Soc. (JEMS), 8 (2006), 195-215. doi: 10.4171/JEMS/47.  Google Scholar

[16]

Discrete Contin. Dyn. Syst., 21 (2008), 41-67. doi: 10.3934/dcds.2008.21.41.  Google Scholar

[17]

J. Differential Equations, 236 (2007), 237-279. doi: 10.1016/j.jde.2007.01.021.  Google Scholar

[18]

Comm. Partial Differential Equations, 30 (2005), 1805-1816. doi: 10.1080/03605300500300006.  Google Scholar

[19]

Progress in Biophysics and Molecular Biology, 97 (2008), 4-27. doi: 10.1016/j.pbiomolbio.2007.10.004.  Google Scholar

[20]

Math. Models Methods Appl. Sci., 21 (2011), 2155-2177. doi: 10.1142/S0218202511005696.  Google Scholar

[21]

Trends Neurosci., 22 (1999), 535-540. Google Scholar

[22]

Prépublication du CMLA No. 2003-04, (2003). Google Scholar

[23]

19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[24]

Annual Review of Ecology Evolution and Systematics, 40 (2009), 481-501. doi: 10.1146/annurev.ecolsys.39.110707.173414.  Google Scholar

[25]

Am. Nat., 172 (2008), 233-247. doi: 10.1086/589459.  Google Scholar

[26]

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[30]

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[31]

Am. Nat., 150 (1997), 1-23. doi: 10.1086/286054.  Google Scholar

[32]

SIAM J. Math. Anal., 26 (1995), 1-20. doi: 10.1137/S0036141093246105.  Google Scholar

[33]

Publ. Res. Inst. Math. Sci., 15 (1979), 401-454. doi: 10.2977/prims/1195188180.  Google Scholar

[34]

Preprint, 2012. Google Scholar

[35]

Brain Res., 740 (1996), 268-274. doi: 10.1016/S0006-8993(96)00874-8.  Google Scholar

[36]

J. Neurobiol., 28 (1995), 433-444. doi: 10.1002/neu.480280404.  Google Scholar

[37]

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J. Differential Equations, 61 (1986), 380-397. doi: 10.1016/0022-0396(86)90112-9.  Google Scholar

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in "CEMRACS (2009): Mathematical Modelling in Medicine", 30 of ESAIM Proc., 44-52. EDP Sci., Les Ulis, (2010). doi: 10.1051/proc/2010005.  Google Scholar

[40]

Paul H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations,, Indiana Univ. Math. J., 23 (): 173.  doi: 10.1512/iumj.1973.23.23014.  Google Scholar

[41]

Proceedings of the National Academy of Sciences, 109 (2012), 8828-8833. doi: 10.1073/pnas.1201695109.  Google Scholar

[42]

Collection Méthodes. Hermann, Paris, 1979.  Google Scholar

[43]

J. Comput. Neurosci., 10 (2001), 99-120. Google Scholar

[44]

J. Differential Equations, 169 (2001), 493-548. Special issue in celebration of Jack K. Hale's 70th birthday, Part 4 (Atlanta, GA/Lisbon, 1998). doi: 10.1006/jdeq.2000.3906.  Google Scholar

[45]

Oxford University Press, New York, 2004. Google Scholar

[46]

Stroke, 33 (2002), 2738-2743. doi: 10.1161/01.STR.0000043073.69602.09.  Google Scholar

[47]

Int. J. Neurosci., 10 (1980), 145-164. doi: 10.3109/00207458009160493.  Google Scholar

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Comm. Partial Differential Equations, 18 (1993), 505-531. doi: 10.1080/03605309308820939.  Google Scholar

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Proceedings of the National Academy of Sciences, 101 (2004), 10249-10253. Google Scholar

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Preprint, 2012. Google Scholar

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