##
**Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary.**
*(English)*
Zbl 1408.35227

The authors study the long-time behavior of the unique (classical) solution \((u,h)\) to the Cauchy free boundary problem of the diffusive logistic equation \(u_t=du_{xx}+A(x-ct)u-bu^2\) in \( D= \{ (x,t)\in \mathbb{R}^2_+: x < h(t) \} \). Here, \(u\) represents the population density of the concerned species and \(h\) represents the moving boundary to be determined, while \(c>0\) denotes the environment shifting speed and \(A\) is a Lipschitz continuous function, such that is strictly increasing over \(]-l_0,0[\) (\(l_0>0\)) and constant otherwise, in order to describe the shifting environment. At the fixed boundary \(x=0\), a no-flux boundary condition is assumed, while, at the free boundary, \(h'(t)=-\mu u_x(t,h(t))\) is stated. The constants \(d,b,\mu\) are assumed to be positive. The authors prove that, when \(c\geq c_0\), the species always dies out in the long-run, and, when \(0< c< c_0\), the long-time behavior of the species is determined by a trichotomy in the form (a) vanishing, (b) borderline spreading, or (c) spreading. To see which case, in the trichotomy, happens with a given initial population \(u_0\), the authors take \(u_0=\sigma \phi\), where \(\sigma\) is a positive parameter and \(\phi\in C^2([0,h_0])\) is such that \(\phi>0\) in \([0,h_0[\), \(\phi'(0)=\phi(h_0)=0\) and \(\phi'(h_0)<0\). Then, they show that there exists a critical \(\sigma_0\in ]0,+\infty]\) such that vanishing happens for \(\sigma< \sigma_0\), borderline spreading happens for \(\sigma=\sigma_0\), and spreading happens for \(\sigma>\sigma_0\). The proofs rely on comparison principles by taking the contradiction argument into account.

Reviewer: Luisa Consiglieri (Lisboa)

### MSC:

35R35 | Free boundary problems for PDEs |

35K20 | Initial-boundary value problems for second-order parabolic equations |

92B05 | General biology and biomathematics |

### Keywords:

diffusive logistic equation; long-time behavior; free boundary; spreading; invasive population; shifting environment; zero number result; super-subsolution
PDF
BibTeX
XML
Cite

\textit{Y. Du} et al., J. Dyn. Differ. Equations 30, No. 4, 1389--1426 (2018; Zbl 1408.35227)

### References:

[1] | Angenent, SB, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390, 79-96, (1988) · Zbl 0644.35050 |

[2] | Aronson, D. G.; Weinberger, H. F., Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, 5-49, (1975), Berlin, Heidelberg · Zbl 0325.35050 |

[3] | Berestycki, H.; Diekmann, O.; Nagelkerke, CJ; Zegeling, PA, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71, 399-429, (2009) · Zbl 1169.92043 |

[4] | Berestycki, H.; Rossi, L., Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space, Discrete Contin. Dyn. Syst., 21, 41-67, (2008) · Zbl 1173.35542 |

[5] | Berestycki, H.; Rossi, L., Reaction-diffusion equations for population dynamics with forced speed. II. Cylindrical-type domains, Discrete Contin. Dyn. Syst., 25, 19-61, (2009) · Zbl 1183.35166 |

[6] | Bouhours, J.; Nadin, G., A variational approach to reaction-diffusion equations with forced speed in dimension 1, Discrete Contin. Dyn. Syst., 35, 1843-1872, (2015) · Zbl 1325.35096 |

[7] | Bramson, Maury, Convergence of solutions of the Kolmogorov equation to travelling waves, Memoirs of the American Mathematical Society, 44, 0-0, (1983) · Zbl 0517.60083 |

[8] | Bunting, G.; Du, Y.; Krakowski, K., Spreading speed revisited: analysis of a free boundary model, Netw. Heterog. Media, 7, 583-603, (2012) · Zbl 1302.35194 |

[9] | Du, Y.: Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Vol. 1 Maximum Principle and Applications. World Scientific Publishing, Singapore (2006) · Zbl 1202.35043 |

[10] | Du, Y., Spreading profile and nonlinear Stefan problems, Bull. Inst. Math. Acad. Sin., 8, 413-430, (2013) · Zbl 1290.35343 |

[11] | Du, Y.; Guo, Z.; Peng, R., A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265, 2089-2142, (2013) · Zbl 1282.35419 |

[12] | Du, Y.; Liang, X., Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. Henri Poincare Anal. Non Lineaire, 32, 279-305, (2015) · Zbl 1321.35263 |

[13] | Du, Y., Lin, Z.: Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42, 377-405 (2010). Erratum: SIAM J. Math. Anal. 45, 1995-1996 (2013) · Zbl 1219.35373 |

[14] | Du, Y.; Lou, B., Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17, 2673-2724, (2015) · Zbl 1331.35399 |

[15] | Du, Y.; Lou, B.; Zhou, M., Nonlinear diffusion problems with free boundaries: convergence, transition speed and zero number arguments, SIAM J. Math. Anal., 47, 3555-3584, (2015) · Zbl 1321.35086 |

[16] | Du, Y.; Ma, L., Logistic type equations on \(\mathbb{R}^N\) by a squeezing method involving boundary blow-up solutions, J. Lond. Math. Soc., 64, 107-124, (2001) · Zbl 1018.35045 |

[17] | Du, Y.; Matsuzawa, H.; Zhou, M., Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46, 375-396, (2014) · Zbl 1296.35219 |

[18] | Du, Y.; Matsuzawa, H.; Zhou, M., Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., 103, 741-787, (2015) · Zbl 1310.35060 |

[19] | Fernandez, FJ, Unique continuation for parabolic operators. II, Commun. Partial Differ. Equ., 28, 1597-1604, (2003) · Zbl 1029.35050 |

[20] | Fisher, RA, The wave of advance of advantageous genes, Ann. Eugen., 7, 335-369, (1937) · JFM 63.1111.04 |

[21] | GĂ¤rtner, J., Location of wave fronts for the multidimensional KPP equation and Brownian first exit densities, Math. Nachr., 105, 317-351, (1982) · Zbl 0501.60083 |

[22] | Hamel, F.; Nolen, J.; Roquejoffre, J-M; Ryzhik, L., A short proof of the logarithmic Bramson correction in Fisher-KPP equations, Netw. Heterog. Media, 8, 275-289, (2013) · Zbl 1275.35067 |

[23] | Gu, H.; Lou, B.; Zhou, M., Long time behavior for solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269, 1714-1768, (2015) · Zbl 1335.35102 |

[24] | Kolmogorov, AN; Petrovski, IG; Piskunov, NS, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ. Math. Mech., 1, 1-25, (1937) |

[25] | Lau, K-S, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov, J. Differ. Equ., 59, 44-70, (1985) · Zbl 0584.35091 |

[26] | Lei, C.; Du, Y., Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model, Discrete Cont. Dyn. Syst. B, 22, 895-911, (2017) · Zbl 1360.35299 |

[27] | C. Lei, H. Nie, W. Dong, Y. Du: Spreading of two competing species governed by a free boundary model in a shifting environment (2017, preprint) · Zbl 1390.35377 |

[28] | Lei, C.; Lin, Z.; Zhang, Q., The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differ. Equ., 257, 145-166, (2014) · Zbl 1286.35274 |

[29] | Li, B.; Bewick, S.; Shang, J.; Fagan, W., Persistence and spread of a species with a shifting habitat edge, SIAM J. Appl. Math., 74, 1397-1417, (2014) · Zbl 1345.92120 |

[30] | Peng, R.; Zhao, X-Q, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33, 2007-2031, (2013) · Zbl 1273.35327 |

[31] | Sattinger, D., Monotone methods in nonlinear elliptic and parabolic boundary value problem, Indiana Univ. Math. J., 21, 979-1000, (1972) · Zbl 0223.35038 |

[32] | Uchiyama, K., The behavior of solutions of some non-linear diffusion equations for large time, J. Math. Kyoto Univ., 18, 453-508, (1978) · Zbl 0408.35053 |

[33] | Uchiyama, K., Asymptotic behaviour of solutions of reaction-diffusion equations with varying drift coefficients, Arch. Ration. Mech. Anal., 90, 291-311, (1983) · Zbl 0618.35058 |

[34] | Wang, MX, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differ. Equ., 258, 1252-1266, (2015) · Zbl 1319.35094 |

[35] | Wei, L., Zhang, G., Zhou, M.: Long time behavior for solutions of the diffusive logistic equation with advection and free boundary. Calc. Var. PDEs 55(4), 95 (2016) · Zbl 1377.35290 |

[36] | Zhou, P.; Xiao, DM, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differ. Equ., 256, 1927-1954, (2014) · Zbl 1316.35156 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.