Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-11T12:59:42.876Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behaviour of solutions of free boundary problems for Fisher-KPP equation

Published online by Cambridge University Press:  30 August 2016

JINGJING CAI  and
HONG GU
JINGJING CAI
Affiliation:
School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, China email: cjjing1983@163.com
HONG GU
Affiliation:
School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing, China email: honggu87@126.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study a free boundary problem for the Fisher-KPP equation: ut = uxx + f(u) (g(t) < x < h(t)) with free boundary conditions h′(t) = −ux(t, h(t)) − α and g′(t) = −ux(t, g(t)) + β for 0 < β < α. This problem can model the spreading of a biological or chemical species, where free boundaries represent the spreading fronts of the species. We investigate the asymptotic behaviour of bounded solutions. There are two parameters α0 and α* with 0 < α0 < α* which play key roles in the dynamics. More precisely, (i) in case 0 < β < α0 and 0 < α < α*, we obtain a trichotomy result: (i-1) spreading, i.e., h(t) − g(t) → +∞ and u(t, ⋅ + ct) → 1 with c ∈ (cL, cR), where cL and cR are the asymptotic spreading speed of g(t) and h(t), respectively, (cR > 0 > cL when 0 < β < α < α0; cR = 0 >cL when 0 < β < α = α0; 0 > cR > cL when α0 < α < α* and 0 < β < α0); (i-2) vanishing, i.e., limt→Th(t) = limt→Tg(t) and limt→T u(t, x) = 0, where T is some positive constant; (i-3) transition, i.e., g(t) → −∞, h(t) → −∞, 0 < limt→∞[h(t) − g(t)] < +∞ and u(t, x) → V*(x − c*t) with c* < 0, where V*(x − c*t) is a travelling wave with compact support and which satisfies the free boundary conditions. (ii) in case β ≥ α0 or α ≥ α*, vanishing happens for any solution.

Keywords

Fisher-KPP equationfree boundary problemzero number principlecompactly supported travelling wave
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 3 , June 2017 , pp. 435 - 469
Copyright
Copyright © Cambridge University Press 2016 

References

[1] Aronson, D. G. & Weinberger, H. F. (1975) Nonlinear diffusion in population genetics, conbustion, and nerve pulse propagation. In: Goldstein, Jerome A. (Ed.) Partial Differential Equations and Related Topics, Lecture Notes in Math, Vol. 446, Springer, Berlin, pp. 5–49.CrossRefGoogle Scholar
[2] Aronson, D. G. & Weinberger, H. F. (1978) Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76.Google Scholar
[3] Cai, J. (2014) Asymptotic behavior of solutions of Fisher-KPP equation with free boundary conditions. Nonl. Anal. 16, 170–177.CrossRefGoogle Scholar
[4] Cai, J., Lou, B. & Zhou, M. (2014) Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions. J. Dyn. Diff. Equat. 26, 1007–1028.Google Scholar
[5] Cui, S. & Friedman, A. (1999) Analysis of a mathematical model of protocell. J. Math. Anal. Appl. 236, 171–206.CrossRefGoogle Scholar
[6] Du, Y. & Guo, Z. (2012) The Stefan problem for the Fisher-KPP equation. J. Diff. Eqns. 253, 996–1035.Google Scholar
[7] Du, Y. & Lin, Z. G. (2010) Spreading-vanishing dichtomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42, 377–405.Google Scholar
[8] Du, Y. & Lou, B. (2015) Spreading and vanishing in nonlinear diffusion problems with free boundaries. J. Eur. Math. Soc. 17, 2673–2724.Google Scholar
[9] Du, Y., Lou, B. & Zhou, Z. (2015) Nonlinear diffusive problems with free boundaries: Convergence, transition speed and zero number arguments. SIAM J. Math. Anal. 47, 3555–3584.Google Scholar
[10] Du, Y., Matano, H. & Wang, K. (2014) Regularity and asymptotic behavior of nonlinear Stefan problems. Arch. Rational Mech. Anal. 212, 957–1010.Google Scholar
[11] Du, Y., Matsuzawa, H. & Zhou, M. (2014) Sharp estimate of the spreading speed determined by nonlinear free boundary problems. SIAM J. Math. Anal. 46, 375–396.Google Scholar
[12] Du, Y., Matsuzawa, H. & Zhou, M. (2015) Spreading speed and profile for nonlinear Stefan problems in high space dimensions. J. Math. Pures Appl. 103, 741–787.Google Scholar
[13] Friedman, A. & Hu, B. (1999) A Stefan problem for a protocell model. SIAM J. Math. Anal. 30, 912–926.CrossRefGoogle Scholar
[14] Gu, H., Lou, B. & Zhou, M. (2015) Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries. J. Funct. Anal. 269, 1714–1768.Google Scholar
[15] Hilhorst, D., Iida, M., Mimura, M. & Ninomiy, H. (2001) A competition-diffusion system approximation to the classical two-phase Stefan problems. Japan J. Ind. Appl. Math. 18, 161–180.Google Scholar
[16] Hilhorst, D., Iida, M., Mimura, M. & Ninomiy, H. (2008) Relative compactness in Lp of solutions of some 2m components competition-diffusion systems. Discrete Contin. Dyn. Syst. 21, 233–244.Google Scholar
[17] Lou, B. & Yang, J. Spatial segregation limit of competition systems and free boundary problems. preprint.Google Scholar
[18] Kaneko, Y. & Matsuzawa, H. (2015) Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations. J. Math. Anal. Appl. 428, 43–76.Google Scholar
[19] Schwegler, H. & Tarumi, K. (1986) The protocell: A mathematical model of self-maintenance. Biosystems 19, 307–315.Google Scholar
[20] Schwegler, H., Tarumi, K. & Gerstmann, B. (1985) Physico-chemical model of a protocell. J. Math. Biol. 22, 335–348.Google Scholar
[21] Zhang, H., Qu, C. & Hu, B. (2009) Bifurcation for a free boundary problem modeling a protocell. Nonl. Anal. 70, 2779–2795.Google Scholar