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`https://hdl.handle.net/1959.11/28558`

Title: | Logarithmic corrections in Fisher-KPP type porous medium equations |
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Contributor(s): | Du, Yihong (author) ; Quiros, Fernando (author); Zhou, Maolin (author) |

Publication Date: | 2020-04 |

Early Online Version: | 2019 |

DOI: | 10.1016/j.matpur.2019.12.008 |

Handle Link: | https://hdl.handle.net/1959.11/28558 |

Abstract: | We consider the large time behavior of solutions to the porous medium equation with a Fisher–KPP type reaction term and nonnegative, compactly supported initial function in L∞(RN) \ {0}:
(*) ut = Δum + u − u2 in Q := RN × R+, u(·,0) = u0 in RN, (*) with m > 1. It is well known that the spatial support of the solution u(·, t) to this problem remains bounded for all time t > 0 (whose boundary is called the free boundary), which is a main different feature of (*) to the corresponding semilinear case m = 1. Similar to the corresponding semilinear case m = 1, it is known that there is a minimal speed c* > 0 such that for any c ≥ c*, the equation admits a wavefront solution Φc(r): For any ν ∈ SN−1, v(x, t) := Φc(x · ν − ct) solves vt = Δvm+v−v2. When m = 1, it is well known that the long-time behavior of the solution with compact initial support can be well approximated by Φc∗ (|x| − c∗t + N+2 c* log t + O(1)), and the term N+2 c* log t is known as the logarithmic correction term. When m > 1, an analogous approximation has been an open question for N ≥ 2. In this paper, we answer this question by showing that there exists a constant c# > 0 independent of the dimension N and the initial function u0, such that for all large time, any solution of (*) is well approximated by Φc∗ (|x| − c∗t + (N−1)c# log t +O(1)). This is achieved by a careful analysis of the radial case, where the initial function u0 is radially symmetric, which enables us to give a formula for c# (involving integrals of Φc* (r)), and to replace the O(1) term by C + o(1) with C a constant depending on u0. The approximation for the general non-radial case is obtained by using the radial results and simple comparison arguments. We note that in sharp contrast to the m = 1 case, when N = 1, there is no logarithmic correction term for (*). |

Publication Type: | Journal Article |

Grant Details: | ARC/DP190103757 |

Source of Publication: | Journal de Mathematiques Pures et Appliquees, v.136, p. 415-455 |

Publisher: | Elsevier Masson |

Place of Publication: | France |

ISSN: | 1776-3371 0021-7824 |

Fields of Research (FoR) 2008: | 010110 Partial Differential Equations |

Fields of Research (FoR) 2020: | 490410 Partial differential equations |

Socio-Economic Objective (SEO) 2008: | 970101 Expanding Knowledge in the Mathematical Sciences |

Socio-Economic Objective (SEO) 2020: | 280118 Expanding knowledge in the mathematical sciences |

Peer Reviewed: | Yes |

HERDC Category Description: | C1 Refereed Article in a Scholarly Journal |

Appears in Collections: | Journal Article School of Science and Technology |

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