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Asymptotic Behavior of Solutions for Multi-stable Equation with Dirichlet Boundary Condition

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Abstract

We study a free boundary problem with a type of multi-stable nonlinearity and a Dirichlet boundary condition, the model describes the spreading of chemical substance or species, when the density of species/chemical substance exceeds some critical number, the species/chemical substance will spread outside, but the environment at the right moving boundary \(x=h(t)\) is not very kind for spreading because of some bad factors, this produces a decay rate. We mainly study the long time behavior of the solution, when the decay rate is small, we have four spreading cases: the solution is either big spreading, or small spreading, or in the big equilibrium state, or small equilibrium state. Besides this, we also have two different trichotomy results: big spreading, big equilibrium state (resp. small spreading) and small equilibrium state (resp. medium equilibrium state), we also have a big-small spreading result and another dichotomy result.

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Authors and Affiliations

  1. School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 200090, China

    Jingjing Cai, Pengchen Xu & Li Xu

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  1. Jingjing Cai
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  2. Pengchen Xu
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  3. Li Xu
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JC wrote the main manuscript text; the second author prepared figures. The third author reviewed the manuscript.

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Correspondence to Jingjing Cai.

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This research was sponsored by NSFC (No. 12071299).

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Cai, J., Xu, P. & Xu, L. Asymptotic Behavior of Solutions for Multi-stable Equation with Dirichlet Boundary Condition. Mediterr. J. Math. 20, 299 (2023). https://doi.org/10.1007/s00009-023-02495-y

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  • DOI: https://doi.org/10.1007/s00009-023-02495-y

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