Dynamical behavior of solutions of a free boundary problem

Di Zhang; Ningkui Sun; Xuemei Han; Di Zhang; Ningkui Sun; Xuemei Han

doi:10.3934/mmc.2024001

Mathematical Modelling and Control

2024, Volume 4, Issue 1: 1-8. doi: 10.3934/mmc.2024001

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Dynamical behavior of solutions of a free boundary problem

1.
School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
2.
Jinan Technician College, Jinan 250031, China

Received: 20 February 2023 Revised: 04 May 2023 Accepted: 07 June 2023 Published: 13 March 2024

This paper is concerned with the spreading properties for a reaction-diffusion equation with free boundary condition. We obtained a complete description of the long-time dynamical behavior of this problem. By introducing a parameter $ \sigma $ in the initial data, we revealed a threshold value $ \sigma^* $ such that spreading happens when $ \sigma > \sigma^* $ and vanishing happens when $ \sigma\leq \sigma^* $. There exists a unique $ L^* > 0 $ independent of the initial data such that $ \sigma^* = 0 $ if and only if the length of initial occupying interval is no smaller than $ 2L^* $. These theoretical results may have important implications for prediction and prevention of biological invasions.
- reaction-diffusion equation,
- free boundary problem,
- long time behavior,
- spreading phenomena
Citation: Di Zhang, Ningkui Sun, Xuemei Han. Dynamical behavior of solutions of a free boundary problem[J]. Mathematical Modelling and Control, 2024, 4(1): 1-8. doi: 10.3934/mmc.2024001

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Abstract

This paper is concerned with the spreading properties for a reaction-diffusion equation with free boundary condition. We obtained a complete description of the long-time dynamical behavior of this problem. By introducing a parameter $ \sigma $ in the initial data, we revealed a threshold value $ \sigma^* $ such that spreading happens when $ \sigma > \sigma^* $ and vanishing happens when $ \sigma\leq \sigma^* $. There exists a unique $ L^* > 0 $ independent of the initial data such that $ \sigma^* = 0 $ if and only if the length of initial occupying interval is no smaller than $ 2L^* $. These theoretical results may have important implications for prediction and prevention of biological invasions.

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