Some monotone properties for solutions to a reaction-diffusion model

Rui Li; Yuan Lou

doi:10.3934/dcdsb.2019126

Article Contents

2019, Volume 24, Issue 8: 4445-4455. Doi: 10.3934/dcdsb.2019126

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Some monotone properties for solutions to a reaction-diffusion model

Rui Li^1, , and
Yuan Lou^2,

1.
Institute for Mathematical Sciences, Renmin University of China, Beijing, 100876, China
2.
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA

^* Corresponding author: Rui Li
^* Corresponding author: Rui Li

Received: August 2018

Revised: November 2018

Early access: June 2019

Published: August 2019

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Abstract

Motivated by the recent investigation of a predator-prey model in heterogeneous environments [20], we show that the maximum of the unique positive solution of the scalar equation

$ \begin{equation} \begin{cases} \mu\Delta\theta+(m(x)-\theta)\theta = 0 \quad &\text{in} \quad \Omega,\\ \frac{\partial \theta}{\partial n} = 0 \quad &\text{on} \quad \partial\Omega \end{cases} \end{equation} $

is a strictly monotone decreasing function of the diffusion rate $ \mu $ for several classes of function $ m $, which substantially improves a result in [20]. However, the minimum of the positive solution of (1) is not always monotone increasing in the diffusion rate [15].

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Mathematics Subject Classification: 34D23, 92D25.

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References

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