Spreading speed and profiles of solutions to a free boundary problem with Dirichlet boundary conditions

Yuki Kaneko, Yoshio Yamada

    Research output: Contribution to journalArticle

    Abstract

    We discuss a free boundary problem for a reaction–diffusion equation with Dirichlet boundary conditions on both fixed and free boundaries of a one-dimensional interval. The problem was proposed by Du and Lin (2010) to model the spreading of an invasive or new species by putting Neumann boundary condition on the fixed boundary. Asymptotic properties of spreading solutions for such problems have been investigated in detail by Du and Lou (2015) and Du, Matsuzawa and Zhou (2014). The authors (2011) studied a free boundary problem with Dirichlet boundary condition. In this paper we will derive sharp asymptotic properties of spreading solutions to the free boundary problem in the Dirichlet case under general conditions on f. It will be shown that the spreading speed is asymptotically constant and determined by a semi-wave problem and that the solution converges to a semi-wave near the spreading front as t→∞ provided that the semi-wave problem has a unique solution.

    Original languageEnglish
    Pages (from-to)1159-1175
    Number of pages17
    JournalJournal of Mathematical Analysis and Applications
    Volume465
    Issue number2
    DOIs
    Publication statusPublished - 2018 Sep 15

    Fingerprint

    Spreading Speed
    Free Boundary Problem
    Dirichlet Boundary Conditions
    Boundary conditions
    Asymptotic Properties
    Neumann Boundary Conditions
    Reaction-diffusion Equations
    Free Boundary
    Unique Solution
    Dirichlet
    Converge
    Interval
    Profile

    Keywords

    • Dirichlet boundary condition
    • Free boundary problem
    • Reaction–diffusion equation
    • Spreading speed

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

    Spreading speed and profiles of solutions to a free boundary problem with Dirichlet boundary conditions. / Kaneko, Yuki; Yamada, Yoshio.

    In: Journal of Mathematical Analysis and Applications, Vol. 465, No. 2, 15.09.2018, p. 1159-1175.

    Research output: Contribution to journalArticle

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